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Encyclopedia > Uniform polychoron
Schlegel diagram for the truncated 120-cell with tetrahedral cells visible.

In geometry, a uniform polychoron (plural: uniform polychora) is a polychoron or 4-polytope which is vertex-transitive and whose cells are uniform polyhedra. Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 573 KB, MIME type: image/png)truncated 120-cell [Schlegel diagram]], solid tetrahedron cells, hidden truncated dodecahedron cells. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 573 KB, MIME type: image/png)truncated 120-cell [Schlegel diagram]], solid tetrahedron cells, hidden truncated dodecahedron cells. ... Examples colored by the number of sides on each face. ... In geometry, the truncated 120-cell is a uniform polychoron. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... A uniform polytope is a vertex-uniform polytope made from uniform polytope facets and lower elements. ... In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... In mathematics, a vertex-transitive graph is a graph G such that, given any two vertices v1 and v2 of G, there is some automorphism f : G → G such that f ( v1 ) = v2. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...


This article contains the complete list of 64 non-prismatic convex uniform polychora, and describes two infinite sets of convex prismatic forms.

Contents

History of discovery

  • Regular polytopes: (convex faces)
    • 1852: Ludwig Schläfli proved in his manuscript Theorie der vielfachen Kontinuität" that there are exactly 6 regular polytopes in 4 dimensions and only 3 in 5 or more dimensions.
  • Regular star-polychora (star polyhedron cells and/or vertex figures)
    • 1852: Ludwig Schläfli also found 4 of the 10 regular star polychora, discounting 6 with cells or vertex figures {5/2,5} and {5,5/2}.
    • 1883: Edmund Hess completed the list of 10 of the nonconvex regular polychora, in his book (in German) Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder [1].
  • Semiregular polytopes: (convex polytopes)
    • 1900: Thorold Gosset enumerated the list of nonprismatic semiregular convex polytopes with regular cells (Platonic solids) in his publication On the Regular and Semi-Regular Figures in Space of n Dimensions.
    • 1912: E. L. Elte expanded on Gosset's work with the publication The Semiregular Polytopes of the Hyperspaces, including a special subset of polytopes with semiregular facets (Those constructible by a single ringed node of a Coxeter-Dynkin diagram.)
  • Convex uniform polytopes:
    • 1910: Alicia Boole Stott, in her publication Geometrical deduction of semiregular from regular polytopes and space fillings, expanded the definition by also allowing Archimedean solid and prism cells.
    • 1940: The search was expanded systematically by H.S.M. Coxeter in his publication Regular and Semi-Regular Polytopes.
    • Convex uniform polychora:
      • 1965: The complete list of convex forms was finally done by John Horton Conway and Michael Guy, in their publication Four-Dimensional Archimedean Polytopes, established by computer analysis, adding only one non-Wythoffian convex polychoron, the grand antiprism.
      • 1997: A complete enumeration of the names and elements of the convex uniform polychora is given online by George Olshevsky. [2]
      • 2004: A proof that the Conway-Guy set is complete was published by Marco Möller in his dissertation, Vierdimensionale Archimedische Polytope (in German).
  • Nonregular uniform star polychora: (similar to the nonconvex uniform polyhedra)
    • Ongoing: Thousands of nonconvex uniform polychora are known, but mostly unpublished. The list is presumed not to be complete, and there is no estimate of how long the complete list will be. Participating researchers include Jonathan Bowers, George Olshevsky and Norman Johnson.

A dodecahedron, one of the five Platonic solids. ... This article needs to be cleaned up to conform to a higher standard of quality. ... 2-dimensional renderings (ie. ... One of ten Schläfli-Hess polychora by orthographic projection. ... In geometry, the term star polyhedron does not seem to have been propely defined, even though it is in common use. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... This article needs to be cleaned up to conform to a higher standard of quality. ... Edmund Hess (1843 – 1903) was a German mathematician who discovered several regular polytopes. ... A semiregular polyhedron is a polyhedron with regular faces and a symmetry group which is transitive on its vertices. ... Thorold Gosset (1869–1962) was an English lawyer and an amateur mathematician. ... In geometry, a Platonic solid is a convex regular polyhedron. ... Coxeter groups in the plane with equivalent diagrams. ... Alicia Boole Stott (June 8, 1860 - December 17, 1940) was the third daughter of George Boole. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... H.S.M. Coxeter. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Michael Guy (born April 6, 1961 in Portland, Oregon) is an American musician who is best known as the lead guitarist of the House of Lords. ... Vertex figure: Dissected regular icosahedron, 14 faces: 12 () and 2 () In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform polychoron (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. ... George Olshevsky is a freelance editor, writer, publisher, paleontologist, and mathematician living in San Diego, California. ... The following list contains all 75 nonprismatic uniform polyhedra, 11 uniform tessellations in the plane, and a samplings of the infinite set of prisms and antiprisms. ... George Olshevsky is a freelance editor, writer, publisher, paleontologist, and mathematician living in San Diego, California. ... Norman W. Johnson is a mathematician, previously at Wheaton College, Norton, Massachusetts. ...

Regular polychora

The uniform polychora include two special subsets, which satisfy additional requirements:

  • The 16 regular polychora, with the property that all cells, faces, edges, and vertices are congruent:
    • 6 convex regular 4-polytopes;
    • 10 Schläfli-Hess polychora.

This page lists the regular polytopes in Euclidean space. ... In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ... One of ten Schläfli-Hess polychora by orthographic projection. ...

Convex uniform polychora

There are 64 convex uniform polychora, including the 6 regular convex polychora, and excluding the infinite sets of the duoprisms and the antiprismatic hyperprisms. A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. ...

  • 5 are polyhedral prisms based on the Platonic solids (1 overlap with regular since a cubic hyperprism is a tesseract)
  • 13 are polyhedral prisms based on the Archimedean solids
  • 9 are in the self-dual regular [3,3,3] group (5-cell) family.
  • 9 are in the self-dual regular [3,4,3] group (24-cell) family. (Excluding snub 24-cell)
  • 15 are in the regular [3,3,4] group (tesseract/16-cell) family (3 overlap with 24-cell family)
  • 15 are in the regular [3,3,5] group (120-cell/600-cell) family.
  • 1 special snub form in the [3,4,3] group (24-cell) family.
  • 1 special non-Wythoffian polychoron, the grand antiprism.
  • TOTAL: 68 - 4 = 64

In addition to the 64 above, there are 2 infinite prismatic sets that generate all of the remaining convex forms: In geometry, a Platonic solid is a convex regular polyhedron. ... In geometry, the tesseract, also called 8-cell or octachoron, is the four-dimensional analog of the (three-dimensional) cube, where motion along the fourth dimension is often a representation for bounded transformations of the cube through time. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... The pentachoron, or 5-cell, also called a pentatope or 4-simplex, is the simplest convex regular polychoron (a type of four-dimensional geometric figure). ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... In geometry, the tesseract, also called 8-cell or octachoron, is the four-dimensional analog of the (three-dimensional) cube, where motion along the fourth dimension is often a representation for bounded transformations of the cube through time. ... In geometry, a 16-cell, or orthoplex, is a regular convex polychoron, or polytope existing in four dimensions. ... In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is sometimes thought of as the 4-dimensional analog of the dodecahedron. ... Vertex figure: icosahedron In geometry, the 600-cell (or hexacosichoron) is the convex regular 4-polytope with Schläfli symbol {3,3,5}. It is sometimes thought of as the 4-dimensional analog of the icosahedron. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ...

In 4-dimensional geometry, a uniform antiprismatic prism is a uniform polychoron with two uniform antiprism cells in two parallel 3-space hyperplanes, connected by uniform prisms cells between pairs of faces. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. ...

The A4 [3,3,3] family - (5-cell)

The pictures are draw as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at pos.0 shown solid. Examples colored by the number of sides on each face. ...

Name Picture Coxeter-Dynkin
and Schläfli
symbols		 Cell counts by location Element counts
Pos. 3
(5)		 Pos. 2
(10)		 Pos. 1
(10)		 Pos. 0
(5)		 Cells Faces Edges Vertices
5-cell
{3,3,3}
(3.3.3)		 5 10 10 5
truncated 5-cell
t0,1{3,3,3}
(3.6.6)
(3.3.3)		 10 30 40 20
rectified 5-cell
t1{3,3,3}
(3.3.3.3)
(3.3.3)		 10 30 30 10
cantellated 5-cell
t0,2{3,3,3}
(3.4.3.4)
(3.4.4)
(3.3.3.3)		 20 80 90 30
cantitruncated 5-cell
t0,1,2{3,3,3}
(4.6.6)
(3.4.4)
(3.6.6)		 20 80 120 60
runcitruncated 5-cell
t0,1,3{3,3,3}
(3.6.6)
(4.4.6)
(3.4.4)
(3.4.3.4)		 30 120 150 60
*bitruncated 5-cell
t1,2{3,3,3}
(3.6.6)
(3.6.6)		 10 40 60 30
*runcinated 5-cell
t0,3{3,3,3}
(3.3.3)
(3.4.4)
(3.4.4)
(3.3.3)		 30 70 60 20
*omnitruncated 5-cell
t0,1,2,3{3,3,3}
(4.6.6)
(4.4.6)
(4.4.6)
(4.6.6)		 30 150 240 120

The 5-cell has diploid pentachoric symmetry, of order 120, isomorphic to the permutations of five elements, because all pairs of vertices are related in the same way. Examples colored by the number of sides on each face. ... Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... The pentachoron, or 5-cell, also called a pentatope or 4-simplex, is the simplest convex regular polychoron (a type of four-dimensional geometric figure). ... Image File history File links Size of this preview: 624 × 600 pixelsFull resolution (817 × 785 pixel, file size: 213 KB, MIME type: image/png)5-cell wireframe File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the truncated 5-cell or truncated pentatope is a uniform polychoron (4-dimensional polytope) bounded by 10 cells: 5 tetrahedra, and 5 truncated tetrahedra. ... Image File history File links Size of this preview: 637 × 599 pixelsFull resolution (930 × 875 pixel, file size: 295 KB, MIME type: image/png)Truncated pentachoron Schlegel diagram, tetrahedron cells shown File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the Rectified 5-cell is a polychoron composed of 5 regular tetrahedra and 5 regular octahedra cells. ... Image File history File links Size of this preview: 601 × 600 pixelsFull resolution (1001 × 999 pixel, file size: 369 KB, MIME type: image/png)Rectified 5-cell, Schlegel diagram, Tetrahedron cells shown File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the cantellated 5-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 459 KB, MIME type: image/png)Cantellated 5-cell, Octahedral cells shown File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the cantitruncated 5-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 419 KB, MIME type: image/png)Cantitruncated 5-cell, Truncated tetrahedral cells shown File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the runcitruncated 5-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 404 KB, MIME type: image/png)Runcitruncated 5-cell, Cuboctahedral cells shown File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... In geometry, the bitruncated 5-cell is a 4-dimensional polytope, or polychoron, composed of 10 cells in the shape of truncated tetrahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 231 KB, MIME type: image/png)Bitruncated 5-cell, Alternate cells shown File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Vertex figure: (Elongate equilateral-triangular antiprism) 2 () and 6 () In geometry, the runcinated pentatope is a 4-dimensional convex uniform polytope (or uniform polychoron) constructed by expanding the cells of a pentatope radially and filling in the gaps with triangular prisms (which are the face prisms and edge figures) and... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 308 KB, MIME type: image/png)Runcinated_5-cell, tetrahedron cells shown, triangular prisms hidden. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the omnitruncated 5-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 383 KB, MIME type: image/png)Omnitruncated 5-cell, Half of truncated octahedron shown File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ...


The three forms marked with an asterisk have the higher extended pentachoric symmetry, of order 240, because the element corresponding to any element of the underlying 5-cell can be exchanged with one of those corresponding to an element of its dual.


The C4 [4,3,3] family - (tesseract/16-cell)

Tesseract family

The pictures are draw as Schlegel diagram perspective projections, centered on the cell at pos. 3, with a consistent orientation, and the 16 cells at pos. 0 shown solid, alternately colored. Examples colored by the number of sides on each face. ...

Name Picture Coxeter-Dynkin
and Schläfli
symbols		 Cell counts by location Element counts
Pos. 3
(8)		 Pos. 2
(24)		 Pos. 1
(32)		 Pos. 0
(16)		 Cells Faces Edges Vertices
8-cell
or tesseract
{4,3,3}
(4.4.4)		 8 24 32 16
truncated 8-cell
t0,1{4,3,3}
(3.8.8)
(3.3.3)		 24 88 128 64
rectified 8-cell
t1{4,3,3}
(3.4.3.4)
(3.3.3)		 24 88 96 32
cantellated 8-cell
t0,2{4,3,3}
(3.4.4.4)
(3.4.4)
(3.3.3.3)		 56 248 288 96
cantitruncated 8-cell
t0,1,2{4,3,3}
(4.6.8)
(3.4.4)
(3.6.6)		 56 248 384 192
runcitruncated 8-cell
t0,1,3{4,3,3}
(3.8.8)
(4.4.8)
(3.4.4)
(3.4.3.4)		 80 368 480 192
bitruncated 8-cell
(also bitruncated 16-cell)
t1,2{4,3,3}
(4.6.6)
(3.6.6)		 24 120 192 96
runcinated 8-cell
(also runcinated 16-cell)
t0,3{4,3,3}
(4.4.4)
(4.4.4)
(3.4.4)
(3.3.3)		 80 208 192 64
omnitruncated 8-cell
(also omnitruncated 16-cell)
t0,1,2,3{3,3,4}
(4.6.8)
(4.4.8)
(4.4.6)
(4.6.6)		 80 464 768 384

Examples colored by the number of sides on each face. ... Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, the tesseract, also called 8-cell or octachoron, is the four-dimensional analog of the (three-dimensional) cube, where motion along the fourth dimension is often a representation for bounded transformations of the cube through time. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 294 KB, MIME type: image/png)tesseract [Schlegel diagram]], wireframe. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... In geometry, a truncated tesseract is a uniform polychoron (4-dimensional uniform polytope) which is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 380 KB, MIME type: image/png)truncated_tesseract [Schlegel diagram]], solid tetrahedron cells, hidden truncated cube cells. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the rectified tesseract is a uniform polychoron (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 392 KB, MIME type: image/png)Rectified tesseract Schlegel diagram, solid tetrahedron cells, hidden truncated cube cells. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the cantellated tesseract is a uniform polychora or 4-dimensional polytope bounded by 56 cells: 8 small rhombicuboctahedra, 16 octahedra, and 32 triangular prisms. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 461 KB, MIME type: image/png)Cantellated tesseract Schlegel diagram, solid octahedron cells File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 227 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the cantitruncated tesseract is a uniform polychoron (or uniform 4-dimensional polytope) that is bounded by 56 cells: 8 great rhombicuboctahedra, 16 truncated tetrahedra, and 32 triangular prisms. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 366 KB, MIME type: image/png)Cantitruncated tesseract Schlegel diagram, solid truncated tetrahedron cells File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the runcitruncated tesseract is a uniform polychoron (uniform 4-dimensional polytope) bounded by 80 cells: 8 truncated cubes, 16 cuboctahedra, 24 octagonal prisms, and 32 triangular prisms. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 454 KB, MIME type: image/png)Runcitruncated tesseract Schlegel diagram, solid cuboctahedron cells File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Image File history File links Octagonal_prism. ... In geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... In geometry, the bitruncated tesseract is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 217 KB, MIME type: image/png)Bitruncated tesseract Schlegel diagram, solid Truncated tetrahedron cells File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the runcinated tesseract or runcinated 16-cell is a 4-dimensional convex uniform polytope (or polychoron) made of 16 tetrahedra, 32 cubes, and 32 triangular prisms. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 464 KB, MIME type: image/png)Runcinated tesseract Schlegel diagram, solid tetrahedron cells File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the omnitruncated tesseract is a uniform polychoron (or uniform 4-dimensional polytope) bounded by 80 cells: 8 great rhombicuboctahedra, 16 truncated octahedra, 24 octagonal prisms, and 32 hexagonal prisms. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 393 KB, MIME type: image/png)Omnitruncated tesseract Schlegel diagram, solid truncated octahedron cells File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Image File history File links Octagonal_prism. ... In geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ...

16-cell family

The pictures are drawn as Schlegel diagram perspective projections, centered on the cell at pos. 0, with a consistent orientation, and the 8 cells at pos. 3 shown solid, bicolored in two prismatic sets. Examples colored by the number of sides on each face. ...

Name Picture Coxeter-Dynkin
and Schläfli
symbols		 Cell counts by location Element counts
Pos. 3
(8)		 Pos. 2
(24)		 Pos. 1
(32)		 Pos. 0
(16)		 Cells Faces Edges Vertices
16-cell
{3,3,4}
(3.3.3)		 16 32 24 8
truncated 16-cell
t0,1{3,3,4}
(3.3.3.3)
(3.6.6)		 24 96 120 48
*rectified 16-cell
(Same as 24-cell)
t1{3,3,4}
(3.3.3.3)
(3.3.3.3)		 24 96 96 24
*cantellated 16-cell
(Same as rectified 24-cell)
t0,2{3,3,4}
(3.4.3.4)
(4.4.4)
(3.4.3.4)		 48 240 288 96
*cantitruncated 16-cell
(Same as truncated 24-cell)
t0,1,2{3,3,4}
(4.6.6)
(4.4.4)
(4.6.6)		 48 240 384 192
runcitruncated 16-cell
t0,1,3{3,3,4}
(3.4.4.4)
(4.4.4)
(4.4.6)
(3.6.6)		 80 368 480 192
bitruncated 16-cell
(also bitruncated 8-cell)
t1,2{3,3,4}
(4.6.6)
(3.6.6)		 24 120 192 96
runcinated 16-cell
(also runcinated 8-cell)
t0,3{3,3,4}
(4.4.4)
(4.4.4)
(3.4.4)
(3.3.3)		 80 208 192 64
omnitruncated 16-cell
(also omnitruncated 8-cell)
t0,1,2,3{3,3,4}
(4.6.8)
(4.4.8)
(4.4.6)
(4.6.6)		 80 464 768 384
Alternated cantitruncated 16-cell
(Same as the snub 24-cell)
h0,1,2{3,3,4}
(3.3.3.3.3)
(3.3.3)		 (96)
(3.3.3)
(3.3.3.3.3)		 144 480 432 96

This family has diploid hexadecachoric symmetry, of order 24*16=384: 4!=24 permutations of the four axes, 24=16 for reflection in each axis. Examples colored by the number of sides on each face. ... Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, a 16-cell, or orthoplex, is a regular convex polychoron, or polytope existing in four dimensions. ... Image File history File links Size of this preview: 586 × 600 pixelsFull resolution (768 × 786 pixel, file size: 259 KB, MIME type: image/png)16-cell Schlegel diagram, wireframe See: Stella (software) ‹ The template below has been proposed for deletion. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the truncated 16-cell or cantic tesseract is a uniform polychoron (4-dimensional uniform polytope) which is bounded by 24 cells: 8 regular octahedra, and 16 truncated tetrahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 328 KB, MIME type: image/png)truncated 16-cell Schlegel diagram, solid octahedron cells, hidden truncated tetrahedron cells. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope, or polychoron, with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular polychoron without a good 3-dimensional analog. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 300 KB, MIME type: image/png)Rectified 16-cell Schlegel diagram, solid pos. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the rectified 24-cell is a uniform 4-dimensional polytope (or uniform polychoron), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 462 KB, MIME type: image/png)Cantitruncated 16-cell Schlegel diagram, solid pos. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... In geometry, the truncated 24-cell is a uniform 4-dimensional polytope (or uniform polychoron), which is bounded by 48 cells: 24 cubes, and 24 truncated octahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 402 KB, MIME type: image/png)Centellated 16-cell Schlegel diagram, solid pos. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... In geometry, the runcitruncated 16-cell is a convex uniform polychoron, or 4-dimensional polytope, bounded by 80 cells: 8 small rhombicuboctahedra, 16 truncated tetrahedra, 24 cubes, and 32 hexagonal prisms. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (925 × 925 pixel, file size: 403 KB, MIME type: image/png)Runcitruncated 16-cell Schlegel diagram, solid rhombicuboctahedron cells File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 227 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the bitruncated tesseract is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 310 KB, MIME type: image/png)Bitruncated 16-cell Schlegel diagram, solid pos. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the runcinated tesseract or runcinated 16-cell is a 4-dimensional convex uniform polytope (or polychoron) made of 16 tetrahedra, 32 cubes, and 32 triangular prisms. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 499 KB, MIME type: image/png)Runcinated 16-cell Schlegel diagram, solid pos. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the omnitruncated tesseract is a uniform polychoron (or uniform 4-dimensional polytope) bounded by 80 cells: 8 great rhombicuboctahedra, 16 truncated octahedra, 24 octagonal prisms, and 32 hexagonal prisms. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1500 × 1500 pixel, file size: 550 KB, MIME type: image/png)Omnitruncated 16-cell Schlegel diagram, solid truncated cuboctahedron cells File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Image File history File links Octagonal_prism. ... In geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... In geometry, the snub 24-cell is a convex uniform polychoron composed of 120 regular tetrahedra and 24 icosahedra cells. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1500 × 1500 pixel, file size: 995 KB, MIME type: image/png)Snub 24-cell Schlegel diagram, colored as alternated cantruncated 16-cell. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_hole. ... Image File history File links CDW_3. ... Image File history File links CDW_hole. ... Image File history File links CDW_3. ... Image File history File links CDW_hole. ... Image File history File links Download high resolution version (1000x1000, 274 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: Colors edited from icosahedron to show Chiral tetrahedral symmetry. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high resolution version (1000x1000, 274 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: Colors edited from icosahedron to show Chiral tetrahedral symmetry. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...


(*) Just as rectifying the tetrahedron produces the octahedron, rectifying the 16-cell produces the 24-cell, the regular member of the following family. A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ...


The snub 24-cell is repeat to this family for completeness. It is an alternation of the cantitruncated 16-cell or truncated 24-cell. The truncated octahedral cells become icosahedra. The cube becomes a tetrahedron, and 96 new tetrahedra are created in the gaps from the removed vertices.


The F4 [3,4,3] family - (24-cell)

Name Picture Coxeter-Dynkin
and Schläfli
symbols		 Cell counts by location Element counts
Pos. 3
(24)		 Pos. 2
(96)		 Pos. 1
(96)		 Pos. 0
(24)		 Cells Faces Edges Vertices
24-cell
(Same as rectified 16-cell)
{3,4,3}
(3.3.3.3)		 24 96 96 24
truncated 24-cell
(Same as cantitruncated 16-cell)
t0,1{3,4,3}
(4.6.6)
(4.4.4)		 48 240 384 192
rectified 24-cell
(Same as cantellated 16-cell)
t1{3,4,3}
(3.4.3.4)
(4.4.4)		 48 240 288 96
cantellated 24-cell
t0,2{3,4,3}
(3.4.4.4)
(3.4.4)
(3.4.3.4)		 144 720 864 288
cantitruncated 24-cell
t0,1,2{3,4,3}
(4.6.8)
(3.4.4)
(3.8.8)		 144 720 1152 576
runcitruncated 24-cell
t0,1,3{3,4,3}
(4.6.6)
(4.4.6)
(3.4.4)
(3.4.4.4)		 240 1104 1440 576
*bitruncated 24-cell
t1,2{3,4,3}
(3.8.8)
(3.8.8)		 48 336 576 288
*runcinated 24-cell
t0,3{3,4,3}
(3.3.3.3)
(3.4.4)
(3.4.4)
(3.3.3.3)		 240 672 576 144
*omnitruncated 24-cell
t0,1,2,3{3,4,3}
(4.6.8)
(4.4.6)
(4.4.6)
(4.6.8)		 240 1392 2304 1152
Alternated truncated 24-cell
†(Same as snub 24-cell)
h0,1{3,4,3}
(3.3.3.3.3)
(3.3.3)
(oblique)
(3.3.3)		 144 480 432 96

This family has diploid icositetrachoric symmetry, of order 24*48=1152: the 48 symmetries of the octahedron for each of the 24 cells. Examples colored by the number of sides on each face. ... Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1050 × 1050 pixel, file size: 568 KB, MIME type: image/png) Stereographic projection of the 24-cell, a 4-dimensional polytope. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the truncated 24-cell is a uniform 4-dimensional polytope (or uniform polychoron), which is bounded by 48 cells: 24 cubes, and 24 truncated octahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 478 KB, MIME type: image/png)truncated 24-cell [Schlegel diagram]], solid cubes cells, hidden truncated octahedron cells. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... In geometry, the rectified 24-cell is a uniform 4-dimensional polytope (or uniform polychoron), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. ... Image File history File links Rectified_24cell. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... In geometry, the cantellated 24-cell is a uniform polychoron. ... Image File history File links No higher resolution available. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 227 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... In geometry, the cantitruncated 24-cell is a uniform polychoron. ... Image File history File links No higher resolution available. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... In geometry, the runcitruncated 24-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 343 KB, MIME type: image/png)Runcitruncated_16-cell projected image created from yet beta released Stella4D software by Rob Webb Only triangular prisms shown File historyClick on a date/time to view... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 227 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Projection of the bitruncated 24-cell into 2D In geometry, the bitruncated 24-cell is a 4-dimensional uniform polytope (or polychoron) derived from the 24-cell. ... Image File history File links Bitruncated_xylotetron_stereographic_close-up. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... In geometry, the runcinated 24-cell is a uniform polychoron. ... Image File history File links No higher resolution available. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the omnitruncated 24-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 645 KB, MIME type: image/png)Omnitruncated_24-cell projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated cuboctahedron, or great rhombicuboctahedron, is an Archimedean solid. ... In geometry, the snub 24-cell is a convex uniform polychoron composed of 120 regular tetrahedra and 24 icosahedra cells. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links CDW_3. ... Image File history File links CDW_hole. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 252 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...


*Like the 5-cell, the 24-cell is self-dual, and so the three asterisked forms have twice as many symmetries, bringing their total to 2304 (the extended icositetrachoric group).


†The snub 24-cell here, despite its common name, is not analogous to the snub cube; rather, is derived by an alternation of the truncated 24-cell. Its symmetry number is only 576 (the ionic diminished icositetrachoric group). The snub cube, or snub cuboctahedron, is an Archimedean solid. ... Two snub cubes from great rhombicuboctahedron See that red and green dots are placed at alternate vertices. ...


The G4 [5,3,3] family - (120-cell/600-cell)

Name Picture Coxeter-Dynkin
and Schläfli
symbols		 Cell counts by location Element counts
Pos. 3
(120)		 Pos. 2
(720)		 Pos. 1
(1200)		 Pos. 0
(600)		 Cells Faces Edges Vertices
120-cell
{5,3,3}
(5.5.5)		 120 720 1200 600
600-cell
{3,3,5}
(3.3.3)		 600 1200 720 120
truncated 120-cell
t0,1{5,3,3}
(3.10.10)
(3.3.3)		 720 3120 4800 2400
truncated 600-cell
t0,1{3,3,5}
(3.3.3.3.3)
(3.6.6)		 720 3600 4320 1440
rectified 120-cell
t1{5,3,3}
(3.5.3.5)
(3.3.3)		 720 3120 3600 1200
rectified 600-cell
t1{3,3,5}
(3.3.3.3.3)
(3.3.3.3)		 720 3600 3600 720
cantellated 120-cell
t0,2{5,3,3}
(3.4.5.4)
(3.4.4)
(3.3.3.3)		 1920 9120 10800 3600
cantellated 600-cell
t0,2{3,3,5}
(3.5.3.5)
(4.4.5)
(3.4.3.4)		 1440 8640 10800 3600
cantitruncated 120-cell
t0,1,2{5,3,3}
(4.6.10)
(3.4.4)
(3.6.6)		 1920 9120 14400 7200
cantitruncated 600-cell
t0,1,2{3,3,5}
(5.6.6)
(4.4.5)
(4.6.6)		 1440 8640 14400 7200
runcitruncated 120-cell
t0,1,3{5,3,3}
(3.10.10)
(4.4.10)
(3.4.4)
(3.4.3.4)		 2640 13440 18000 7200
runcitruncated 600-cell
t0,1,3{3,3,5}
(3.4.5.4)
(4.4.5)
(4.4.6)
(3.6.6)		 2640 13440 18000 7200
bitruncated 120-cell
(also bitruncated 600-cell)
t1,2{5,3,3}
(5.6.6)
(3.6.6)		 720 4320 7200 3600
runcinated 120-cell
(also runcinated 600-cell)
t0,3{5,3,3}
(5.5.5)
(4.4.5)
(3.4.4)
(3.3.3)		 2640 7440 7200 2400
omnitruncated 120-cell
(also omnitruncated 600-cell)
t0,1,2,3{5,3,3}
(4.6.10)
(4.4.10)
(4.4.6)
(4.6.6)		 2640 17040 28800 14400

This family has diploid hexacosichoric symmetry, of order 120*120=24*600=14400: 120 for each of the 120 dodecahedra, or 24 for each of the 600 tetrahedra. Examples colored by the number of sides on each face. ... Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, the 120-cell (or hecatonicosachoron) is the convex regular 4-polytope with Schläfli symbol {5,3,3}. It is sometimes thought of as the 4-dimensional analog of the dodecahedron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1150 × 1150 pixel, file size: 781 KB, MIME type: image/png) Stereographic projection of the 120-cell, a 4-dimensional polytope. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 211 KB, MIME type: image/png)image for Dodecahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... Vertex figure: icosahedron In geometry, the 600-cell (or hexacosichoron) is the convex regular 4-polytope with Schläfli symbol {3,3,5}. It is sometimes thought of as the 4-dimensional analog of the icosahedron. ... Image File history File links Download high-resolution version (1100x1100, 413 KB) Stereographic projection of the 600-cell, a 4-dimensional polytope. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the truncated 120-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 573 KB, MIME type: image/png)truncated 120-cell [Schlegel diagram]], solid tetrahedron cells, hidden truncated dodecahedron cells. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 209 KB, MIME type: image/png)Truncated dodecahedron, rendered by Java applet I wrote. ... The truncated dodecahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Central part of Schlegel diagram and some of 120 red icosahedra. ... Image File history File links Size of this preview: 611 × 599 pixelsFull resolution (922 × 904 pixel, file size: 448 KB, MIME type: image/png)truncated 600-cell Schlegel diagram, solid icosahedral cells, hidden truncated tetrahedron cells. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 252 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the rectified 120-cell is a convex uniform polychoron composed of 600 regular tetrahedra and 120 icosidodecahedra cells. ... Image File history File links Download high-resolution version (1100x1100, 972 KB) Stereographic projection of the rectified 120-cell, a 4-dimensional polytope. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 258 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the 720-cell is a Semiregular Polytope composed of 600 regular octahedra and 120 icosahedra cells. ... Image File history File links Download high-resolution version (1150x1150, 968 KB) Stereographic projection of the rectified 600-cell, a 4-dimensional polytope. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 252 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the cantellated 120-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (800 × 800 pixel, file size: 1,010 KB, MIME type: image/png) File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high resolution version (1000x1000, 289 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the cantellated 600-cell is a uniform polychoron. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 258 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... Image File history File links Pentagonal_prism. ... In geometry, the pentagonal prism is the third in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... In geometry, the cantitruncated 120-cell is a uniform polychoron. ... Image File history File links No higher resolution available. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high resolution version (640x640, 34 KB) Summary Great rhombicosidodecahedron, U67 External Links VRML model 1 VRML model 2 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated icosidodecahedron, or great rhombicosidodecahedron, is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the cantitruncated 600-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 1. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 238 KB)Image for Truncated icosahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated icosahedron is an Archimedean solid. ... Image File history File links Pentagonal_prism. ... In geometry, the pentagonal prism is the third in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... In geometry, the runciruncated 120-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 1. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 209 KB, MIME type: image/png)Truncated dodecahedron, rendered by Java applet I wrote. ... The truncated dodecahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 154 KB)Decagonal prism Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In geometry, the decagonal prism is the eighth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... In geometry, the runcitruncated 600-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 1. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high resolution version (1000x1000, 289 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ... Image File history File links Pentagonal_prism. ... In geometry, the pentagonal prism is the third in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the bitruncated 120-cell is a uniform polychoron. ... Image File history File links Bitruncated_cosmotetron_stereographic_close-up. ... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links Download high-resolution version (1000x1000, 238 KB)Image for Truncated icosahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated icosahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the runcinated 120-cell is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 1. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 211 KB, MIME type: image/png)image for Dodecahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... Image File history File links Pentagonal_prism. ... In geometry, the pentagonal prism is the third in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the omnitruncated 120-cell is a convex uniform polychoron composed of 2640 cells: 120 truncated icosidodecahedra, 600 truncated octahedra, 720 decagonal prisms, and 1200 pentagonal prisms. ... Image File history File links Size of this preview: 626 × 600 pixelsFull resolution (1200 × 1150 pixel, file size: 1. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links Download high resolution version (640x640, 34 KB) Summary Great rhombicosidodecahedron, U67 External Links VRML model 1 VRML model 2 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated icosidodecahedron, or great rhombicosidodecahedron, is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 154 KB)Decagonal prism Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In geometry, the decagonal prism is the eighth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ...


The B4 [31,1,1] group family

This family introduces no new uniform polyhedra, but it is worthy to repeat these alternative constructions.

Name Picture Coxeter-Dynkin Cell counts by location Element counts
Pos. 0
(8)		 Pos. 1
(24)		 Pos. 0'
(8)		 Pos. 3
(8)		 Pos. Alt
(96)		 Cells Faces Edges Vertices
16-cell

t0{31,1,1} Examples colored by the number of sides on each face. ... Coxeter groups in the plane with equivalent diagrams. ... In geometry, a 16-cell, or orthoplex, is a regular convex polychoron, or polytope existing in four dimensions. ... Image File history File links Size of this preview: 586 × 600 pixelsFull resolution (768 × 786 pixel, file size: 259 KB, MIME type: image/png)16-cell Schlegel diagram, wireframe See: Stella (software) ‹ The template below has been proposed for deletion. ... Image File history File links CD_ring. ... Image File history File links No higher resolution available. ... Image File history File links CD_downbranch-00. ... Image File history File links No higher resolution available. ... Image File history File links CD_dot. ...


(3.3.3)
(3.3.3)		 16 32 24 8
truncated 16-cell

t0,1{31,1,1} Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the truncated 16-cell or cantic tesseract is a uniform polychoron (4-dimensional uniform polytope) which is bounded by 24 cells: 8 regular octahedra, and 16 truncated tetrahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 328 KB, MIME type: image/png)truncated 16-cell Schlegel diagram, solid octahedron cells, hidden truncated tetrahedron cells. ... Image File history File links CD_ring. ... Image File history File links No higher resolution available. ... Image File history File links CD_downbranch-10. ... Image File history File links No higher resolution available. ... Image File history File links CD_dot. ...


(3.3.3.3)
(3.6.6)
(3.6.6)		 24 96 120 48
rectified 8-cell

t0,2{31,1,1} Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the rectified tesseract is a uniform polychoron (4-dimensional polytope) bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 392 KB, MIME type: image/png)Rectified tesseract Schlegel diagram, solid tetrahedron cells, hidden truncated cube cells. ... Image File history File links CD_ring. ... Image File history File links No higher resolution available. ... Image File history File links CD_downbranch-01. ... Image File history File links No higher resolution available. ... Image File history File links CD_dot. ...


(3.3.3)
(3.3.3)
(3.4.3.4)		 24 88 96 32
bitruncated 8-cell

t0,1,2{31,1,1} Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... In geometry, the bitruncated tesseract is a uniform polychoron. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 310 KB, MIME type: image/png)Bitruncated 16-cell Schlegel diagram, solid pos. ... Image File history File links CD_ring. ... Image File history File links No higher resolution available. ... Image File history File links CD_downbranch-11. ... Image File history File links No higher resolution available. ... Image File history File links CD_dot. ...


(3.6.6)
(3.6.6)
(4.6.6)		 24 120 192 96
24-cell

t1{31,1,1} Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... In geometry, the 24-cell (or icositetrachoron) is the convex regular 4-polytope with Schläfli symbol {3,4,3}. The 24-cell is the unique convex regular 4-polytope without a good 3-dimensional analog. ... Image File history File links Size of this preview: 633 × 599 pixelsFull resolution (862 × 816 pixel, file size: 489 KB, MIME type: image/png)24-cell Schlegel diagram, wireframe File historyClick on a date/time to view the file as it appeared at that time. ... Image File history File links CD_dot. ... Image File history File links No higher resolution available. ... Image File history File links CD_downbranch-10. ... Image File history File links No higher resolution available. ... Image File history File links CD_dot. ...


(3.3.3.3)
(3.3.3.3)
(3.3.3.3)		 24 96 96 24
rectified 24-cell

t0,2,3{31,1,1} Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the rectified 24-cell is a uniform 4-dimensional polytope (or uniform polychoron), which is bounded by 48 cells: 24 cubes, and 24 cuboctahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 462 KB, MIME type: image/png)Cantitruncated 16-cell Schlegel diagram, solid pos. ... Image File history File links CD_ring. ... Image File history File links No higher resolution available. ... Image File history File links CD_downbranch-01. ... Image File history File links No higher resolution available. ... Image File history File links CD_ring. ...


(3.4.3.4)
(4.4.4)
(3.4.3.4)
(3.4.3.4)		 48 240 288 96
truncated 24-cell

t0,1,2,3{31,1,1} Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... In geometry, the truncated 24-cell is a uniform 4-dimensional polytope (or uniform polychoron), which is bounded by 48 cells: 24 cubes, and 24 truncated octahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 478 KB, MIME type: image/png)truncated 24-cell [Schlegel diagram]], solid cubes cells, hidden truncated octahedron cells. ... Image File history File links CD_ring. ... Image File history File links No higher resolution available. ... Image File history File links CD_downbranch-11. ... Image File history File links No higher resolution available. ... Image File history File links CD_ring. ...


(4.6.6)
(4.4.4)
(4.6.6)
(4.6.6)		 48 240 384 192
snub 24-cell

s{31,1,1} Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... In geometry, the snub 24-cell is a convex uniform polychoron composed of 120 regular tetrahedra and 24 icosahedra cells. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ...


(3.3.3.3.3)
(3.3.3)
(3.3.3.3.3)
(3.3.3.3.3)
(3.3.3)		 144 480 432 96

Here again the Snub 24-cell represents an alternated truncation of the truncated 24-cell, creating 96 new tetrahedra at the position of the deleted vertices. In contrast to its appearance within former groups as partly snubbed polychoron, only within this symmetry group it has the full analogy to the Kepler snubs, i.e. the snub cube and the snub dodecahedron. Image File history File links Download high resolution version (1000x1000, 274 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: Colors edited from icosahedron to show Chiral tetrahedral symmetry. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high resolution version (1000x1000, 274 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: Colors edited from icosahedron to show Chiral tetrahedral symmetry. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Download high resolution version (1000x1000, 274 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: Colors edited from icosahedron to show Chiral tetrahedral symmetry. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, the snub 24-cell is a convex uniform polychoron composed of 120 regular tetrahedra and 24 icosahedra cells. ... The snub cube, or snub cuboctahedron, is an Archimedean solid. ... The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid. ...


The grand antiprism

There is one non-Wythoffian uniform convex polychoron, known as the grand antiprism, consisting of 20 pentagonal antiprisms forming two perpendicular rings joined by 300 tetrahedra. It is loosely analogous to the three-dimensional antiprisms, which consist of two parallel polygons joined by a band of triangles; unlike them, the grand antiprism is not a member of an infinite family of uniform polytopes. Vertex figure: Dissected regular icosahedron, 14 faces: 12 () and 2 () In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform polychoron (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. ... In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... Look up polygon in Wiktionary, the free dictionary. ... A triangle. ...


Its symmetry number is 400 (the ionic diminished Coxeter group).

Name Picture Coxeter-Dynkin
and Schläfli
symbols		 Cells by type Element counts
Cells Faces Edges Vertices
grand antiprism No symbol 300 (3.3.3) 20 (3.3.3.5) 320 20 {5}
700 {3}		 500 100

Examples colored by the number of sides on each face. ... Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... Vertex figure: Dissected regular icosahedron, 14 faces: 12 () and 2 () In geometry, the grand antiprism or pentagonal double antiprismoid is a uniform polychoron (4-dimensional uniform polytope) bounded by 320 cells: 20 pentagonal antiprisms, and 300 tetrahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1100 × 1100 pixel, file size: 404 KB, MIME type: image/png) Stereographic projection of the Grand antiprism (faces not shown). ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ... Image File history File links Pentagonal_antiprism. ... Look up pentagon in Wiktionary, the free dictionary. ... A triangle. ...

Prismatic uniform polychora

There are three infinite families of uniform polychora that are considered prismatic, in that they generalize the properties of the 3-dimensional prisms. A prismatic polytope is a Cartesian product of two polytopes of lower dimension. The third form is contained within the duprism set, except for the snubbed form - prisms of antiprisms. In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... In mathematics, the Cartesian product is a direct product of sets. ...

Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. ... Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ...

Polyhedral prisms

The more obvious family of prismatic polychora is the polyhedral prisms, i.e. products of a polyhedron with a line segment. The cells of such a polychoron are two identical uniform polyhedra lying in parallel hyperplanes (the base cells) and a layer of prisms joining them (the lateral cells). This family includes prisms for the 75 nonprismatic uniform polyhedra (of which 18 are convex; one of these, the cube-prism, is listed above as the tesseract), The geometric definition of a line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. ... A hyperplane is a concept in geometry. ... A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...


There are 18 convex polyhedral prisms created 5 Platonic solid and 13 Archimedean solid) as well as for the infinite families of three-dimensional prisms and antiprisms. The symmetry number of a polyhedral prism is twice that of the base polyhedron. In geometry, a Platonic solid is a convex regular polyhedron. ... In geometry an Archimedean solid or semi-regular solid is a semi-regular convex polyhedron composed of two or more types of regular polygon meeting in identical vertices. ... In geometry, an n-sided prism is a polyhedron made of an n-sided polygonal base, a translated copy, and n faces joining corresponding sides. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...


Tetrahedral prisms: A3xA1 - [3,3] x [ ]

Name Picture Coxeter-Dynkin
and Schläfli
symbols		 Cells by type Element counts
Cells Faces Edges Vertices
Tetrahedral prism
t0{3,3}x{}		 2
3.3.3		 4
3.4.4		 6 8 {3}
6 {4}		 16 8
Truncated tetrahedral prism
t0,1{3,3}x{}		 2
3.6.6		 4
3.4.4		 4
4.4.6		 10 8 (3)
8 {4}
8 {6}		 48 24
Rectified tetrahedral prism
(Same as Octahedral prism)
t1{3,3}x{}		 2
3.3.3.3		 4
3.4.4		 6 16 {3}
12 {4}		 30 12
Cantellated tetrahedral prism
(Same as cuboctahedral prism)
t0,2{3,3}x{}		 2
3.4.3.4		 8
3.4.4		 6
4.4.4		 16 16 {3}
36 {4}		 60 24
Cantitruncated tetrahedral prism
(Same as truncated octahedral prism)
t0,1,2{3,3}x{}		 2
4.6.6		 8
3.4.4		 6
4.4.4		 16 48 {4}
16 {6}		 96 48
Snub tetrahedral prism
(Same as icosahedral prism)
s{3,3}x{}		 2
3.3.3.3.3		 20
3.4.4		 22 40 {3}
30 {4}		 72 24

Examples colored by the number of sides on each face. ... Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... Transpaarent Schlegel diagram An orthographic projection showing the pair of parallel tetrahedra projected as a quadrilateral divided into yellow and blue triangular faces. ... Image File history File links Size of this preview: 598 × 599 pixelsFull resolution (803 × 805 pixel, file size: 299 KB, MIME type: image/png)Uniform polychoron image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 187 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 365 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 202 KB)Image for Truncated tetrahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated tetrahedron is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... In geometry, a octahedral hyperprism is a convex uniform polychoron (four dimensional polytope). ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 463 KB, MIME type: image/png)Uniform polychoron image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Transparent Schlegel diagram In geometry, a cuboctahedral hyperprism is a convex uniform polychoron (four dimensional polytope). ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 502 KB, MIME type: image/png)Uniform polychoron image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 505 KB, MIME type: image/png)Uniform polychoron image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 538 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_hole. ... Image File history File links CDW_3. ... Image File history File links CDW_hole. ... Image File history File links CDW_3. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 252 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ...

Octahedral prisms: C3xA1 - [4,3] x [ ]

Name Picture Coxeter-Dynkin
and Schläfli
symbols		 Cells by type Element counts
Cells Faces Edges Vertices
Cubic prism
(Same as tesseract)
(Same as 4-4 duoprism)
t0{4,3}x{}		 2
4.4.4		 6
4.4.4		 8 24 {4} 32 16
Octahedral prism
(Same as rectified tetrahedral prism)
(Same as square antiprismatic prism)
t2{4,3}x{}		 2
3.3.3.3		 8
3.4.4		 10 16 {3}
12 {4}		 30 12
Cuboctahedral prism
(Same as cantellated tetrahedral prism)
t1{4,3}x{}		 2
3.4.3.4		 8
3.4.4		 6
4.4.4		 16 16 {3}
36 {4}		 60 24
Truncated cubic prism
t0,1{4,3}x{}		 2
3.8.8		 8
3.4.4		 6
4.4.8		 16 16 {3}
36 {4}
12 {8}		 96 48
Truncated octahedral prism
(Same as cantitruncated tetrahedral prism)
t1,2{4,3}x{}		 2
4.6.6		 6
4.4.4		 8
4.4.6		 16 48 {4}
16 {6}		 96 48
Rhombicuboctahedral prism
t0,2{4,3}x{}		 2
3.4.4.4		 8
3.4.4		 18
4.4.4		 28 16 {3}
84 {4}		 120 96
Truncated cuboctahedral prism
t0,1,2{4,3}x{}		 2
4.6.8		 12
4.4.4		 8
4.4.6		 6
4.4.8		 28 96 {4}
16 {6}
12 {8}		 192 96
Snub cubic prism
s{4,3}x{}		 2
3.3.3.3.4		 32
3.4.4		 6
4.4.4		 40 64 {3}
72 {4}		 144 48

Examples colored by the number of sides on each face. ... Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... In geometry, the tesseract, also called 8-cell or octachoron, is the four-dimensional analog of the (three-dimensional) cube, where motion along the fourth dimension is often a representation for bounded transformations of the cube through time. ... Image File history File links Hypercube. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... In geometry, a octahedral hyperprism is a convex uniform polychoron (four dimensional polytope). ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 463 KB, MIME type: image/png)Uniform polychoron image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 201 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Transparent Schlegel diagram In geometry, a cuboctahedral hyperprism is a convex uniform polychoron (four dimensional polytope). ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 502 KB, MIME type: image/png)Uniform polychoron image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 231 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 438 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 200 KB)Truncated hexahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Octagonal_prism. ... In geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 505 KB, MIME type: image/png)Uniform polychoron image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared... Image File history File links CDW_dot. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 216 KB)image for Truncated octahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated octahedron is an Archimedean solid. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 407 KB, MIME type: image/png)Uniform polychoron image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 227 KB)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 443 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high resolution version (638x639, 23 KB) Summary Great rhombicuboctahedron, U17 Licensing I, the creator of this work, hereby release it into the public domain. ... A colored model Net (polyhedron) The truncated cuboctahedron is an Archimedean solid. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Octagonal_prism. ... In geometry, the octagonal prism is the sixth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 499 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_hole. ... Image File history File links CDW_4. ... Image File history File links CDW_hole. ... Image File history File links CDW_3. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 271 KB, MIME type: image/png) File historyClick on a date/time to view the file as it appeared at that time. ... The snub cube, or snub cuboctahedron, is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ...

Icosahedral prisms: G3xA1 - [5,3] x [ ]

Name Picture Coxeter-Dynkin
and Schläfli
symbols		 Cells by type Element counts
Cells Faces Edges Vertices
Dodecahedral prism
t0{5,3}x{}		 2
5.5.5		 12
4.4.5		 14 30 {4}
24 {5}		 80 40
Icosahedral prism
(same as snub tetrahedral prism)
t2{5,3}x{}		 2
3.3.3.3.3		 20
3.4.4		 22 40 {3}
30 {4}		 72 24
Icosidodecahedral prism
t1{5,3}x{}		 2
3.5.3.5		 20
3.4.4		 12
4.4.5		 34 40 {3}
60 {4}
24 {5}		 150 60
Truncated dodecahedral prism
t0,1{5,3}x{}		 2
3.10.10		 20
3.4.4		 12
4.4.5		 34 40 {3}
90 {4}
24 {10}		 240 120
Truncated icosahedral prism
t1,2{5,3}x{}		 2
5.6.6		 12
4.4.5		 20
4.4.6		 34 90 {4}
24 {5}
40 {6}		 240 120
Rhombicosidodecahedral prism
t0,2{5,3}x{}		 2
3.4.5.4		 20
3.4.4		 30
4.4.4		 12
4.4.5		 64 40 {3}
180 {4}
24 {5}		 300 120
Truncated icosidodecahedral prism
t0,1,2{5,3}x{}		 2
4.6.4.10		 30
4.4.4		 20
4.4.6		 12
4.4.10		 64 240 {4}
40 {6}
24 {5}		 480 240
Snub dodecahedral prism
s{5,3}x{}		 2
3.3.3.3.5		 80
3.4.4		 12
4.4.5		 94 240 {4}
40 {6}
24 {10}		 360 120

Examples colored by the number of sides on each face. ... Coxeter groups in the plane with equivalent diagrams. ... In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... Transparent Schlegel diagram An orthographic projection with a wireframe model and has half of the pentagonal faces colored to show the two dodecahedra. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 432 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 211 KB, MIME type: image/png)image for Dodecahedron See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to... A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex. ... Image File history File links Pentagonal_prism. ... In geometry, the pentagonal prism is the third in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 538 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 252 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... [Etymology: 16th century: from Greek eikosaedron, from eikosi twenty + -edron -hedron], icosahedral adjective An icosahedron noun (plural: -drons, -dra ) is any polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 628 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (1000 × 1000 pixel, file size: 258 KB, MIME type: image/png)See: Stella (software) File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Pentagonal_prism. ... In geometry, the pentagonal prism is the third in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 432 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 209 KB, MIME type: image/png)Truncated dodecahedron, rendered by Java applet I wrote. ... The truncated dodecahedron is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Download high-resolution version (1000x1000, 154 KB)Decagonal prism Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In geometry, the decagonal prism is the eighth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 473 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_dot. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high-resolution version (1000x1000, 238 KB)Image for Truncated icosahedron File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... The truncated icosahedron is an Archimedean solid. ... Image File history File links Pentagonal_prism. ... In geometry, the pentagonal prism is the third in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 544 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high resolution version (1000x1000, 289 KB) Licensing Robert Webb produced this image for Wikipedia upon my request, and offered the credit statement below: File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... A colored model Net (polyhedron) The rhombicosidodecahedron, or small rhombicosidodecahedron, is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Pentagonal_prism. ... In geometry, the pentagonal prism is the third in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 539 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Download high resolution version (640x640, 34 KB) Summary Great rhombicosidodecahedron, U67 External Links VRML model 1 VRML model 2 Licensing I, the creator of this work, hereby release it into the public domain. ... The truncated icosidodecahedron, or great rhombicosidodecahedron, is an Archimedean solid. ... Hexahedron (sometimes called cube), rendered by Java applet I wrote. ... A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ... Image File history File links Hexagonal_prism. ... In geometry, the hexagonal prism is a prism with hexagonal base. ... Image File history File links Download high-resolution version (1000x1000, 154 KB)Decagonal prism Licensing File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... In geometry, the decagonal prism is the eighth in an infinite set of prisms formed by square sides and two regular polygon caps. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 595 KB, MIME type: image/png)Uniform polychoron projected image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it... Image File history File links CDW_hole. ... Image File history File links CDW_5. ... Image File history File links CDW_hole. ... Image File history File links CDW_3. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 333 KB, MIME type: image/png)Named with ccw because commons name conflict without it, although cw form not uploaded. ... The snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid. ... Image File history File links Triangular_prism. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Pentagonal_prism. ... In geometry, the pentagonal prism is the third in an infinite set of prisms formed by square sides and two regular polygon caps. ...

Duoprisms: D2pxD2q - [p] x [q]

The simplest of the duoprisms, the 3,3-duoprism, in Schlegel diagram, one of 6 triangular prism cells shown.

The second is the infinite family of uniform duoprisms, products of two regular polygons. Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 232 KB, MIME type: image/png)Duoprism image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared at... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 232 KB, MIME type: image/png)Duoprism image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared at... Examples colored by the number of sides on each face. ... For the optical prism, see Triangular prism (optics). ... A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. ... A regular pentagon A regular polygon is a simple polygon (a polygon which does not intersect itself anywhere) which is equiangular (all angles are equal) and equilateral (all sides have the same length). ...


They have a Coxeter-Dynkin diagram as: Coxeter groups in the plane with equivalent diagrams. ... Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ...


This family overlaps with the first: when one of the two "factor" polygons is a square, the product is equivalent to a hyperprism whose base is a three-dimensional prism. The symmetry number of a duoprism whose factors are a p-gon and a q-gon (a "p,q-duoprism") is 4pq if p≠q; if the factors are both p-gons, the symmetry number is 8p2. The tesseract can also be considered a 4,4-duoprism.


The elements of a p,q-duoprism (p ≥ 3, q ≥ 3) are:

  • Cells: p q-gonal prisms, q p-gonal prisms
  • Faces: pq squares, p q-gons, q p-gons
  • Edges: 2pq
  • Vertices: pq

There is no uniform analogue in four dimensions to the infinite family of three-dimensional antiprisms. An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ...


Infinite set of p-q duoprism - - p q-gonal prisms, q p-gonal prisms: Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ...

  • 3-3 duoprism - - 6 triangular prisms
  • 3-4 duoprism - - 3 cubes, 4 triangular prisms
  • 3-5 duoprism - - 3 pentagonal prisms, 5 triangular prisms
  • 3-6 duoprism - - 3 hexagonal prisms, 6 triangular prisms
  • 4-4 duoprism - - 8 cubes (same as tesseract)
  • 4-5 duoprism - - 4 pentagonal prisms, 5 cubes
  • 4-6 duoprism - - 4 hexagonal prisms, 6 cubes
  • 5-5 duoprism - - 10 pentagonal prisms
  • 5-6 duoprism - - 5 hexagonal prisms, 6 pentagonal prisms
  • 6-6 duoprism - - 12 hexagonal prisms
  • ...

Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_dot. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_dot. ...

Polygonal prismatic prisms: D2xA1xA1 - [p] x [ ] x [ ]

The infinte set of uniform prismatic prism overlap with the 4-p duoprisms: (p≥3) - - p cubes and 4 p-gonal prisms - (All are the same as 4-p duoprism) Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ...

  • Triangular prismatic prism - - 3 cubes and 4 triangular prisms - (same as 3-4 duoprism)
  • Square prismatic prism - - 4 cubes and 4 cubes - (same as 4-4 duoprism and same as tesseract)
  • Pentagonal prismatic prism - - 5 cubes and 4 pentagonal prisms - (same as 4-5 duoprism)
  • Hexagonal prismatic prism - - 6 cubes and 4 hexagonal prisms - (same as 4-6 duoprism)
  • Heptagonal prismatic prism - - 7 cubes and 4 heptagonal prisms - (same as 4-7 duoprism)
  • Octagonal prismatic prism - - 8 cubes and 4 octagonal prisms - (same as 4-8 duoprism)
  • ...

The infinite sets of uniform antiprismatic prisms are constructed from two parallel uniform antiprisms): (p≥3) - - 2 p-gonal antiprisms, connected by 2 p-gonal prisms and 2p triangular prisms. Image File history File links CDW_ring. ... Image File history File links CDW_3. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_ring. ... Image File history File links CDW_4. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_ring. ... Image File history File links CDW_5. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_ring. ... Image File history File links CDW_6. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_ring. ... Image File history File links No higher resolution available. ... Image File history File links CDW_dot. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... In 4-dimensional geometry, a uniform antiprismatic prism is a uniform polychoron with two uniform antiprism cells in two parallel 3-space hyperplanes, connected by uniform prisms cells between pairs of faces. ... An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ...

  1. Square antiprismatic prism - - 2 square antiprisms connected by 2 cubes and 8 triangular prisms
  2. Pentagonal antiprismatic prism - - 2 pentagonal antiprisms connected by 2 pentagonal prisms and 10 triangular prisms
  3. Hexagonal antiprismatic prism - - 2 hexagonal antiprisms connected by 2 hexagonal prisms and 12 triangular prisms
  4. Heptagonal antiprismatic prism - - 2 heptagonal antiprisms connected by 2 heptagonal prisms and 14 triangular prisms
  5. Octagonal antiprismatic prism - - 2 octagonal antiprisms connected by 2 octagonal prisms and 16 triangular prisms
  6. ...

A p-gonal antiprismatic prism has 4p triangle, 4p square and 4 p-gon faces. It has 10p edges, and 4p vertices. Image File history File links CDW_hole. ... Image File history File links CDW_3. ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... For the optical prism, see Triangular prism (optics). ... Image File history File links CDW_hole. ... Image File history File links CDW_4. ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... In geometry, the Square antiprism is the second in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 440 KB, MIME type: image/png)Uniform polychoron image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared... Image File history File links CDW_hole. ... Image File history File links CDW_5. ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... In geometry, the pentagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ... For the optical prism, see Triangular prism (optics). ... Image File history File links Size of this preview: 600 × 600 pixelsFull resolution (1000 × 1000 pixel, file size: 436 KB, MIME type: image/png)Uniform polychoron image created from beta release of Stella4D software by Rob Webb File historyClick on a date/time to view the file as it appeared... Image File history File links CDW_hole. ... Image File history File links CDW_6. ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... In geometry, the hexagonal antiprism is the third in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ... For the optical prism, see Triangular prism (optics). ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... In geometry, the heptagonal antiprism is the fourth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ... For the optical prism, see Triangular prism (optics). ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links No higher resolution available. ... Image File history File links CDW_hole. ... Image File history File links CDW_2. ... Image File history File links CDW_ring. ... In geometry, the octagonal antiprism is the fifth in an infinite set of antiprisms formed by an even-numbered sequence of triangle sides closed by two polygon caps. ... For the optical prism, see Triangular prism (optics). ...


Geometric derivations for 46 nonprismatic Wythoffian uniform polychora

Summary chart of truncation operations
Example locations of kaleidoscopic generator point on fundamental domain.

The 46 Wythoffian polychora include the six convex regular polychora. The other forty can be derived from the regular polychora by geometric operations which preserve most or all of their symmetries, and therefore may be classified by the symmetry groups that they have in common. Image File history File links Size of this preview: 560 × 599 pixelsFull resolution (644 × 689 pixel, file size: 58 KB, MIME type: image/png) La bildo estas kopiita de wikipedia:en. ... Image File history File links Size of this preview: 560 × 599 pixelsFull resolution (644 × 689 pixel, file size: 58 KB, MIME type: image/png) La bildo estas kopiita de wikipedia:en. ... Image File history File links Size of this preview: 800 × 547 pixelsFull resolution (1900 × 1300 pixel, file size: 126 KB, MIME type: image/png)I made this LARGE image to better map the fundamental triangle generator points from Image:Polychoron_truncation_chart. ... Image File history File links Size of this preview: 800 × 547 pixelsFull resolution (1900 × 1300 pixel, file size: 126 KB, MIME type: image/png)I made this LARGE image to better map the fundamental triangle generator points from Image:Polychoron_truncation_chart. ... In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ... Sphere symmetry group o. ... The symmetry group of an object (e. ...


The geometric operations that derive the 40 uniform polychora from the regular polychora are truncating operations. A polychoron may be truncated at the vertices, edges or faces, leading to addition of cells corresponding to those elements, as shown in the columns of the tables below.


The Coxeter-Dynkin diagram shows the four mirrors of the Wythoffian kaleidoscope as nodes, and the edges between the nodes are labeled by an integer showing the angle between the mirrors. (180/n degrees) Circled nodes show which mirrors are active for each form. That is mirrors for which the generating point is located off the mirror. Coxeter groups in the plane with equivalent diagrams. ...

Operation Schläfli
symbol		 Coxeter-
Dynkin
diagram		 Description
Parent t0{p,q,r} Original regular form {p,q,r}
Rectification t1{p,q,r} Truncation operation applied until the original edges are degenerated into points.
Birectification t2{p,q,r} Face are fully truncated to points. Same as rectified dual.
Trirectification
(dual)		 t3{p,q,r} Cells are truncated to points. Regular dual {r,q,p}
Truncation t0,1{p,q,r} Each vertex cut off so that the middle of each original edge remains. Where the vertex was, there appears a new cell, the parent's vertex figure. Each original cell is likewise truncated.
Bitruncation t1,2{p,q,r} A truncation between a rectified form and the dual rectified form
Tritruncation t2,3{p,q,r} Truncated dual {r,q,p}
Cantellation t0,2{p,q,r} A truncation applied to edges and vertices and defines a progression between the regular and dual rectified form.
Bicantellation t1,3{p,q,r} Cantellated dual {r,q,p}
Runcination
(or expansion)		 t0,3{p,q,r} A truncation applied to the cells, faces, and edges and defines a progression between a regular form and the dual.
Cantitruncation t0,1,2{p,q,r} Both the cantellation and truncation operations applied together.
Bicantitruncation t1,2,3{p,q,r} Cantitruncated dual {r,q,p}
Runcitruncation t0,1,3{p,q,r} Both the runcination and truncation operations applied together.
Runcicantellation t0,1,3{p,q,r} Runcitruncated dual {r,q,p}
Omnitruncation
(or more specifically runcicantitruncated)		 t0,1,2,3{p,q,r} Has all three operators applied.

See also convex uniform honeycombs, some of which illustrate these operations as applied to the regular cubic honeycomb. In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ... Coxeter groups in the plane with equivalent diagrams. ... Image File history File links No higher resolution available. ... In Euclidean geometry, rectification is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the others. ... Image File history File links No higher resolution available. ... A truncated cube - faces double in sides, and vertices replaced by new faces. ... Image File history File links No higher resolution available. ... In geometry, a vertex figure is most easily thought of as the cut surface exposed when a corner of a polytope is cut off in a certain way. ... A bitruncated cubic honeycomb - Cubic cells become orange truncated octahedra, and vertices are replaced by blue truncated octahedra. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... A cantellated cube - Red faces are reduced. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... A runcinated cubic honeycomb (partial) - The original cells (purple cubes) are reduced in size. ... An example of expanding pentagon into a decagon by moving edges away from the center and inserting new edges in the gaps. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... See: For polyhedra: Uniform_polyhedron#Definition_of_operations For polychora: Uniform_polychoron#Geometric_derivations_for_46_Wythoffian_polychora ... Image File history File links No higher resolution available. ... In geometry, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. ... The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ...


If two polytopes are duals of each other (such as the tesseract and 16-cell, or the 120-cell and 600-cell), then bitruncating, runcinating or omnitruncating either produces the same figure as the same operation to the other. Thus where only the participle appears in the table it should be understood to apply to either parent. In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the others. ...


See also

In geometry, a four-dimensional polytope is sometimes called a polychoron (plural: polychora) (from Greek poly meaning many and choros meaning room or space), 4-polytope, or polyhedroid. ... Semiregular polychora are vertex-uniform 4-dimensional polytopes constructed from regular polyhedral cells. ... A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. ...

References

  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • A. Boole Stott: Geometrical deduction of semiregular from regular polytopes and space fillings, Verhandelingen of the Koninklijke academy van Wetenschappen width unit Amsterdam, Eerste Sectie 11,1, Amsterdam, 1910
  • Elte, E. L. (1912), The Semiregular Polytopes of the Hyperspaces, Groningen: University of Groningen, ISBN 141817968X
  • H.S.M. Coxeter:
    • H.S.M. Coxeter, M.S. Longuet-Higgins und J.C.P. Miller: Uniform Polyhedra, Philosophical Transactions of the Royal Society of London, Londne, 1954
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6 [3]
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • M. Möller: Definitions and computations to the Platonic and Archimedean polyhedrons, thesis (diploma), University of Hamburg, 2001
  • B. Grünbaum Convex polytopes, New York ; London : Springer, c2003. ISBN 0-387-00424-6.
    Second edition prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler.

Thorold Gosset (1869–1962) was an English lawyer and an amateur mathematician. ... Alicia Boole Stott (June 8, 1860 - December 17, 1940) was the third daughter of George Boole. ... H.S.M. Coxeter. ... John Horton Conway (born December 26, 1937, Liverpool, England) is a prolific mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory. ... Michael Guy (born April 6, 1961 in Portland, Oregon) is an American musician who is best known as the lead guitarist of the House of Lords. ... Norman W. Johnson is a mathematician, previously at Wheaton College, Norton, Massachusetts. ... Branko Grünbaum is a mathematician who works mainly in geometry and is considered a founder of discrete geometry. ...

External links


  Results from FactBites:
 
Polychoron - Wikipedia, the free encyclopedia (588 words)
A polychoron is a closed four-dimensional figure with vertices, edges, faces, and cells.
A polychoron is uniform if it has a symmetry group under which all vertices are equivalent, and its cells are uniform polyhedra.
A uniform 3-space tessellation is one whose vertices are related by a space group and whose cells are uniform polyhedra.
Wikipedia: Polychoron (313 words)
A polychoron (plural: polychora) (from Greek poly meaning "many" and choros meaning "room" or "space") is a four-dimensional polytope, also known as a 4-polytope, or polyhedroid.
The two-dimensional analogue of a polychoron is a polygon, and the three-dimensional analogue is a polyhedron.
A uniform cell is a cell that is vertex-transitive, with each face made up of regular polygons.
  More results at FactBites »


 

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