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. The two-dimensional analogue of a polychoron is a polygon, and the three-dimensional analogue is a polyhedron. Geometry (Greek γεωμετρία; geo = earth, metria = measure) arose as the field of knowledge dealing with spatial relationships. ... In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... Look up Polygon in Wiktionary, the free dictionary For other use please see Polygon (disambiguation) A polygon (literally many angle, see Wiktionary for the etymology) is a closed planar path composed of a finite number of sequential line segments. ... A polyhedron is a geometric shape which in mathematics is defined by three related meanings. ...

Note that the use of the term polychoron is not entirely standard. Its use has been advocated by Norman Johnson and George Olshevsky. See the Uniform Polychora Project. Norman Lloyd Johnson was born in Ilford Essex, England, in 1917. ... The Uniform Polychora Project in geometry is a collaborative effort to recognize and standardize terms used to describe objects in higher-dimensional spaces. ...

A polychoron has vertices, edges, faces, and cells. A vertex is where two or more edges meet. An edge is where two or more faces meet, and a face is where two or more cells meet. A cell is the three-dimensional analogue of a face, and is therefore a polyhedron. In geometry, a vertex (Latin: whirl, whirlpool; plural vertices) is a corner of a polygon (where two sides meet) or of a polyhedron (where three or more faces and an equal number of edges meet). ... Edge may have <math>one</math> of the following special meanings, in addition to its dictionary definition: wiktionary:edge. ... A face is a polygonal component of a higher dimensional polytope. ... A cell is a three-dimensional object that is part of a higher-dimensional object, such as a polychoron. ...

A polychoron is a closed four-dimensional figure bounded by cells with the requirements that:

1. Each face must join exactly two cells.
2. Adjacent cells are not in the same three-dimensional space.
3. The figure is not a compound of other figures which meet the requirements.

Regular polychora

A polychoron is said to be regular if all its cells, faces, edges, and vertices, are congruent. There are exactly six convex regular polychora. These are the analogs of the five Platonic solids. In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ... In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ...

1. pentachoron (with 5 tetrahedral cells) (also called a "4-simplex")
2. tesseract (with 8 cubic cells) (also called a "hypercube")
3. hexadecachoron (with 16 tetrahedral cells)
4. icositetrachoron (with 24 octahedral cells)
5. hecatonicosachoron (with 120 dodecahedral cells)
6. hexacosichoron (with 600 tetrahedral cells)

There are ten non-convex regular polychora: The pentachoron, also called a pentatope or 4-simplex, is the simplest convex regular polychoron (a type of four-dimensional geometric figure). ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In geometry, a simplex or n-simplex is an n-dimensional analogue of a triangle. ... A 3-dimensional projection of a tesseract. ... Three dimensions A cube (or hexahedron) is a Platonic solid composed of six square faces, with three meeting at each vertex. ... In geometry, the tesseract, or hypercube, is a regular, convex polychoron with eight cubical cells. ... In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... In mathematics, the 24-cell is the 4-dimensional convex regular polytope with 24 facets. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the 120-cell, or hecatonicosachoron, is the convex, regular polychoron (a 4-dimensional polytope) with 120 cells (or facets). ... A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... In mathematics, the 600-cell is the 4-dimensional convex regular polytope with 600 facets. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

1. faceted hexacosichoron (also called icosahedral hecatonicosachoron)
2. great hecatonicosachoron
3. grand hecatonicosachoron
4. small stellated hecatonicosachoron
5. great grand hecatonicosachoron
6. great stellated hecatonicosachoron
7. grand stellated hecatonicosachoron
8. great faceted hexacosichoron
9. grand hexacosichoron
10. great grand stellated hecatonicosachoron

An icosahedron [ˌaıkəsəhiːdrən] noun (plural: -drons, -dra [-drə]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ...

Uniform polychora

A polychoron is said to be uniform if it is vertex-uniform (i.e. there is a symmetry taking any vertex to any other) and its cells are uniform polyhedra. A uniform polyhedron is a polyhedron that is vertex-transitive, with each face made up of regular polygons. (These include the 5 Platonic solids, 13 Archimedean solids, 4 Kepler-Poinsot solids, and 53 other nonregular, nonconvex forms). The symmetry group of an object (e. ... In solid geometry and some ancient physical theories, a Platonic solid is a convex polyhedron with all its faces being congruent regular polygons, and the same number of faces meeting at each of its vertices. ... 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. ... A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ...

The Uniform Polychora Project has classified the 8,186 currently known uniform polychora into 29 groups. There may be more. The Uniform Polychora Project in geometry is a collaborative effort to recognize and standardize terms used to describe objects in higher-dimensional spaces. ...

There is a technique called the Coxeter-Dynkin system for performing Wythoff's construction for producing uniform polytopes. This method allows the polychora to be effectively enumerated.

Convex Uniform Polychora

There are forty-six Wythoffian convex non-prismatic uniform polychora. These include the six regular polychora, and have symmetry groups derived from them.

The convex uniform polychora with pentatope symmetry include: The pentachoron or pentatope is the simplest regular polychoron (a type of four-dimensional figure used in geometry), analog of the planar triangle and solid tetrahedron. ...

1. The pentatope itself;
2. The truncated pentatope (with 5 tetrahedra and 5 truncated tetrahedra);
3. The rectified pentatope (5 tetrahedra and 5 octahedra);
4. The bitruncated pentatope (10 truncated tetrahedra);
5. The runcinated pentatope (10 tetrahedra and 20 triangular prisms).

The polychora with 16-cell and tesseract symmetry include: The pentachoron or pentatope is the simplest regular polychoron (a type of four-dimensional figure used in geometry), analog of the planar triangle and solid tetrahedron. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... The truncated tetrahedron is an Archimedean solid. ... In geometry, the Rectified 5-cell is a polychoron composed of 5 regular tetrahedra and 5 regular octahedra cells. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, the runcinated pentatope is a 4-dimensional convex uniform polytope (or 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 tetrahedra (which are vertex figures). ... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... (Redirected from 16-cell) In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... A 3-dimensional projection of a tesseract. ...

1. The 16-cell itself;
2. The tesseract itself;
3. The truncated 16-cell (with 8 octahedra and 16 truncated tetrahedra);
4. The rectified 16-cell, which is identical to the 24-cell (24 octahedra);
5. The bitruncated 16-cell, which is identical to the bitruncated tesseract (8 truncated octahedra and 16 truncated tetrahedra);
6. The truncated tesseract (16 tetrahedra and 8 truncated cubes);
7. The rectified tesseract (16 tetrahedra and 8 cuboctahedra);
8. The runcinated tesseract, which is identical to the runcinated 16-cell (32 cubes, 32 triangular prisms, and 16 tetrahedra).

The polychora with 24-cell symmetry include: (Redirected from 16-cell) In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ... A 3-dimensional projection of a tesseract. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... The truncated tetrahedron 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. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ... 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. ... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... 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. ...

1. The 24-cell itself;
2. The truncated 24-cell (with 24 cubes and 24 truncated octahedra);
3. The rectified 24-cell (24 cubes and 24 cuboctahedra);
4. The bitruncated 24-cell (48 truncated cubes);
5. The runcinated 24-cell (48 octahedra and 192 triangular prisms).

The polychora with 120-cell and 600-cell symmetry include: 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. ... A cube (or regular hexahedron) is a three-dimensional Platonic solid composed of six square faces, with three meeting at each vertex. ... The truncated octahedron, also known as a Mecon, is an Archimedean solid. ... A cuboctahedron is a polyhedron with eight triangular faces and six 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. ... The truncated cube, or truncated hexahedron, is an Archimedean solid. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ... 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. ... In mathematics, the 600-cell is the 4-dimensional convex regular polytope with 600 facets. ...

1. The 120-cell itself;
2. The 600-cell itself;
3. The truncated 120-cell (with 120 truncated dodecahedra and 600 tetrahedra);
4. The rectified 120-cell (120 icosidodecahedra and 600 tetrahedra);
5. The bitruncated 120-cell, which is identical to the bitruncated 600-cell (120 truncated icosahedra and 600 truncated tetrahedra);
6. The truncated 600-cell (600 truncated tetrahedra and 120 icosahedra);
7. The rectified 600-cell (600 octahedra and 120 icosahedra);
8. The runcinated 600-cell, identical to the runcinated 120-cell (120 dodecahedra, 600 tetrahedra, 720 pentagonal prisms, and 1200 triangular prisms).

(Note: these lists are not exhaustive.) 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. ... In mathematics, the 600-cell is the 4-dimensional convex regular polytope with 600 facets. ... The truncated dodecahedron is an Archimedean solid. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ... An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ... The truncated icosahedron is an Archimedean solid. ... The truncated tetrahedron is an Archimedean solid. ... The truncated tetrahedron is an Archimedean solid. ... An icosahedron [ËŒaıkÉ™sÉ™hiːdrÉ™n] noun (plural: -drons, -dra [-drÉ™]) is a polyhedron having 20 faces, but usually a regular icosahedron is meant. ... In geometry, the 720-cell is a Semiregular Polytope composed of 600 regular octahedra and 120 icosahedra cells. ... An octahedron (plural: octahedra) is a polyhedron with eight faces. ... A dodecahedron is literally a polyhedron with 12 faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve pentagonal faces, with three meeting at each vertex. ... In geometry, a prism is a polyhedron made of two parallel copies of some polygonal base joined by faces that are rectangles or parallelograms. ...

There is a forty-seventh anomalous polychoron, the grand antiprism, which is non-Wythoffian, consisting of 20 pentagonal antiprisms and 300 tetrahedra. An antiprism is a polyhedron composed of two parallel copies of some particular polygon, connected by an alternating band of triangles. ... For academic journal, see Tetrahedron A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...

The remaining convex uniform polychora may be grouped into two infinite families: the duoprisms and the polyhedral prisms. The polyhedral prisms consist of pairs of any of the 3D uniform polyhedra (including antiprisms), joined to each other by suitable polygonal prisms. A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. ...

See also

• Convex regular polychoron
• List of regular polytopes
• Semiregular 4-polytopes Subset of uniform polychora with regular polyhedron cells.
• The 3-sphere (or glome) is another commonly discussed figure that resides in 4-dimensional space. This is not a polychoron, since it is not made up of polyhedral cells.
• The duocylinder is a figure in 4-dimensional space related to the duoprisms. It is also not a polychoron because its bounding volumes are not polyhedral.

In mathematics, a convex regular 4-polytope (or polychoron) is 4-dimensional polytope which is both a regular and convex. ... This page lists the regular polytopes in Euclidean space. ... Semiregular 4-polytopes are 4 dimensional geometric shapes constructed by platonic solid cells such that every edge contains the same sequence of cells, and for every two vertices there in an isometry mapping one into the other. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. ... The duocylinder is a geometric object embedded in 4-dimensional Euclidean space, defined as the Cartesian product of two circles of radius r: It is bounded by two mutually perpendicular 3-manifolds with torus-like surfaces, described by the equations: and The duocylinder is so-called because these two bounding... A duoprism is a 4-dimensional figure resulting from the Cartesian product of two polygons in the 2-dimensional Euclidean space. ...

External links

• Polychoron on Mathworld
• Four dimensional figures page
• Multidimensional glossary – compiled by George Olshevsky