 A truncated cube - faces double in sides, and vertices replaced by new faces. In geometry, a truncation is an operation in any dimension that cuts a polytope vertices, creating a new facet in place of each vertex. 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. ...
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 Truncated_cubic_honeycomb. ...
Image File history File links Truncated_cubic_honeycomb. ...
The truncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ...
Table of Geometry, from the 1728 Cyclopaedia. ...
Truncation in regular polyhedra and tilings
When the term applies to truncating platonic solids or regular tilings, usually "uniform truncation" is implied, which means to truncate until the original faces become regular polygons with double the sides. In geometry, a Platonic solid is a convex regular polyhedron. ...
This article needs to be cleaned up to conform to a higher standard of quality. ...
 Image File history File links Cube_truncation_sequence. ...
This sequence shows an example of the truncation of a cube, using four steps of a continuous truncating process between a full cube and a rectified cube. The final polyhedron is a cuboctahedron. A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ...
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. ...
A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...
The middle image is the uniform truncated cube. It is represented by an extended Schläfli symbol t0,1{p,q,...}. The truncated cube, or truncated hexahedron, is an Archimedean solid. ...
In mathematics, the Schläfli symbol is a simple notation that gives a summary of some important properties of a particular regular polytope. ...
Other truncations In quasiregular polyhedra, a truncation is a more qualitative term where some other adjustments are made to adjust truncated faces to become regular. These are sometimes called rhombitruncations.
For example, the truncated cuboctahedron is not really a truncation since the cut vertices of the cuboctahedron would form rectangular faces rather than squares, so a wider operation is needed to adjust the polyhedron to fit desired squares. A colored model Net (polyhedron) The truncated cuboctahedron is an Archimedean solid. ...
A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...
In the quasiregular duals, An alternate truncation operation only truncated alternate vertices. (This operation can also apply to any zonohedron which have even-sided faces.) A zonohedron is a convex polyhedron where every face is a polygon with point symmetry, or equivalently, symmetry under rotations through 180°. The regular polygons with such symmetry are those with an even number of sides, so the zonohedra with regular polygons for sides are easily enumerated: Of the Platonic...
Uniform polyhedron and tiling examples This table shows the truncation progression between the regular forms, with the rectified forms (full truncation) in the center. Comparable faces are colored red and yellow to show the continuum in the sequences. Image File history File links Download high-resolution version (800x800, 15 KB) [1] KaleidoTile Topology and and Geometry Software, Jeff Weeks Tetrahedron I, the creator of this work, hereby release it into the public domain. ...
A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
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The truncated tetrahedron is an Archimedean solid. ...
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The truncated tetrahedron is an Archimedean solid. ...
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A tetrahedron (plural: tetrahedra) is a polyhedron composed of four triangular faces, three of which meet at each vertex. ...
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A cube[1] is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. ...
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The truncated cube, or truncated hexahedron, is an Archimedean solid. ...
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A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...
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The truncated octahedron, also known as a Mecon, is an Archimedean solid. ...
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An octahedron (plural: octahedra) is a polyhedron with eight faces. ...
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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. ...
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The truncated dodecahedron is an Archimedean solid. ...
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An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ...
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A colored model Net (polyhedron) The truncated icosahedron is an Archimedean solid. ...
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An icosahedron noun (plural: -drons, -dra ) is a polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangles as faces. ...
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In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ...
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In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. ...
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In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. ...
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In geometry, the hexagonal tiling is a regular tiling of the Euclidean plane. ...
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In geometry, the triangular tiling is a regular tiling of the Euclidean plane. ...
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In geometry, the Square tiling is a regular tiling of the Euclidean plane. ...
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In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ...
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In geometry, the Square tiling is a regular tiling of the Euclidean plane. ...
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In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ...
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In geometry, the Square tiling is a regular tiling of the Euclidean plane. ...
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Prismatic polyhedron examples Image File history File links Download high-resolution version (636x622, 18 KB) I, the creator of this work, hereby release it into the public domain. ...
A dihedron is a type of polyhedron, made of two polygon faces which share the same set of edges. ...
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In geometry, the hexagonal prism is the fourth in an infinite set of prisms formed by square sides and two regular polygon caps. ...
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A tetragonal hosohedron, represented as a tessellation of 4 spherical lunes on a sphere. ...
rhombitruncated examples These forms start with a rectified regular form which is truncated. The vertices are order-4, and a true geometric truncation would create rectangular faces. The uniform rhombitruction requires adjustment to create square faces. 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. ...
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The truncated octahedron, also known as a Mecon, is an Archimedean solid. ...
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A cuboctahedron is a polyhedron with eight triangular faces and six square faces. ...
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A colored model Net (polyhedron) The truncated cuboctahedron is an Archimedean solid. ...
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An icosidodecahedron is a polyhedron with twenty triangular faces and twelve pentagonal faces. ...
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The truncated icosidodecahedron, or great rhombicosidodecahedron, is an Archimedean solid. ...
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In geometry, the trihexagonal tiling is a semiregular tiling of the Euclidean plane. ...
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In geometry, the Great rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. ...
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In geometry, the triheptagonal tiling is a semiregular tiling of the hyperbolic plane. ...
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In geometry, the Great rhombitrihexagonal tiling (or Omnitruncated trihexagonal tiling) is a semiregular tiling of the Euclidean plane. ...
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In geometry, the Square tiling is a regular tiling of the Euclidean plane. ...
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In geometry, the truncated square tiling is a semiregular tiling of the Euclidean plane. ...
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In geometry, the order-5 square tiling is a regular tiling of the hyperbolic plane. ...
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Truncation in polychora and honeycomb tessellation A regular polychoron or tessellation {p,q,r}, truncated becomes a uniform polychoron or tessellation with 2 cells: truncated {p,q}, and {q,r} cells are created on the truncated section.
See: uniform polychoron and convex uniform honeycomb. 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, a convex uniform honeycomb is a uniform space-filling tessellation in three-dimensional Euclidean space with non-overlapping convex uniform polyhedral cells. ...
| Family | Parent | Truncation | Rectification
(birectified dual) | Bitruncation
(bitruncated dual) | Birectification
(rectified dual) | Tritruncation
(truncated dual) | Tetrarectification
(dual) | | [3,3,3] |
5-cell |
truncated 5-cell |
rectified 5-cell |
bitruncated 5-cell |
rectified 5-cell |
truncated 5-cell |
5-cell (self-dual) | | [4,3,3] |
Tesseract |
truncated tesseract |
rectified tesseract |
bitruncated tesseract
(bitruncated 16-cell) |
rectified 16-cell
(Same as 24-cell) |
truncated 16-cell
(Same as 24-cell) |
16-cell (self-dual) | | [3,4,3] |
24-cell |
truncated 24-cell |
rectified 24-cell |
bitruncated 24-cell |
rectified 24-cell |
truncated 24-cell |
24-cell (self-dual) | | [5,3,3] |
120-cell |
truncated 120-cell |
rectified 120-cell |
bitruncated 120-cell
(bitruncated 600-cell) |
rectified 600-cell |
truncated 600-cell |
600-cell | | [4,3,4] |
cubic |
truncated cubic |
Rectified cubic |
bitruncated cubic |
Rectified cubic |
truncated cubic |
cubic (self-dual) | | [5,3,4] |
dodecahedral | truncated dodecahedral | (No image)
rectified dodecahedral | bitruncated cubic
(bitruncated dodecahedral) | (No image)
rectified cubic | Truncated cubic |
cubic | Image File history File links Download high-resolution version (1000x1000, 150 KB) Stereographic projection of the 5-cell, a 4-dimensional 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 Truncated_simplex_stereographic. ...
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 Download high-resolution version (558x602, 166 KB) Stereographic projection of a uniform polychoron, created by me, Anton Sherwood, using Fritz Obermeyers Jenn I, the creator of this work, hereby release it into the public domain. ...
In geometry, the Rectified 5-cell is a polychoron composed of 5 regular tetrahedra and 5 regular octahedra cells. ...
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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 Download high-resolution version (558x602, 166 KB) Stereographic projection of a uniform polychoron, created by me, Anton Sherwood, using Fritz Obermeyers Jenn I, the creator of this work, hereby release it into the public domain. ...
In geometry, the Rectified 5-cell is a polychoron composed of 5 regular tetrahedra and 5 regular octahedra cells. ...
Image File history File links Truncated_simplex_stereographic. ...
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 Download high-resolution version (1000x1000, 150 KB) Stereographic projection of the 5-cell, a 4-dimensional 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 Download high-resolution version (1000x1000, 230 KB) Summary Stereographic projection of the 8-cell, a 4-dimensional polytope. ...
Stereographic projection In geometry, the tesseract 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. ...
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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. ...
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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 Bitrunc_tessa_schlegel. ...
In geometry, the bitruncated tesseract is a uniform polychoron. ...
Image File history File links Download high-resolution version (1100x1000, 320 KB) Stereographic projection of the 24-cell, a 4-dimensional 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 Download high-resolution version (1100x1000, 320 KB) Stereographic projection of the 24-cell, a 4-dimensional 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 Download high-resolution version (1100x1000, 214 KB) Stereographic projection of the 16-cell, a 4-dimensional polytope. ...
(Redirected from 16-cell) In geometry, a cross-polytope, or orthoplex, is a regular, convex polytope that exists in any number of dimensions. ...
Image File history File links Download high-resolution version (1100x1000, 320 KB) Stereographic projection of the 24-cell, a 4-dimensional 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 Truncated_xylotetron_stereographic_oblique. ...
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 Rectified_24cell. ...
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 Bitruncated_xylotetron_stereographic_close-up. ...
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 Rectified_24cell. ...
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 Truncated_xylotetron_stereographic_oblique. ...
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 Download high-resolution version (1100x1000, 320 KB) Stereographic projection of the 24-cell, a 4-dimensional 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 Download high-resolution version (1000x1000, 429 KB) Stereographic projection of the 120-cell, a 4-dimensional 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 Download high-resolution version (1300x1150, 1037 KB) Stereographic projection of the truncated 120-cell, a 4-dimensional polytope. ...
In geometry, the truncated 120-cell is a uniform polychoron. ...
Image File history File links Download high-resolution version (1100x1100, 972 KB) Stereographic projection of the rectified 120-cell, a 4-dimensional polytope. ...
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 Bitruncated_cosmotetron_stereographic_close-up. ...
In geometry, the bitruncated 120-cell is a uniform polychoron. ...
Image File history File links Download high-resolution version (1150x1150, 968 KB) Stereographic projection of the rectified 600-cell, a 4-dimensional polytope. ...
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 (1100x1100, 682 KB) Stereographic projection of the truncated 600-cell, a 4-dimensional polytope. ...
Central part of Schlegel diagram and some of 120 red icosahedra. ...
Image File history File links Download high-resolution version (1100x1100, 413 KB) Stereographic projection of the 600-cell, a 4-dimensional polytope. ...
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 (873x866, 24 KB) I, the creator of this work, hereby release it into the public domain. ...
The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ...
Image File history File links Truncated_cubic_honeycomb. ...
The truncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ...
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The rectified cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ...
Image File history File links Download high-resolution version (868x848, 43 KB) Image for Bitruncated_cubic_honeycomb - 2 colored truncated octahedron cells for original cells and vertices. ...
The bitruncated cubic honeycomb is a tessellation (or honeycomb) in Euclidean 3-space made up of truncated octahedra. ...
Image File history File links Rectified_cubic_honeycomb. ...
The rectified cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ...
Image File history File links Truncated_cubic_honeycomb. ...
The truncated cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. ...
Image File history File links Download high-resolution version (873x866, 24 KB) I, the creator of this work, hereby release it into the public domain. ...
The cubic honeycomb is the only regular tessellation (or honeycomb) in Euclidean 3-space. ...
Image File history File links Download high resolution version (902x902, 1341 KB) Summary View of Hyperbolic small dodecahedral honeycomb generated by software: [1] Curved Spaces v1. ...
The order-4 dodecahedral honeycomb is one of four regular space-filling tessellation (or honeycombs) in hyperbolic 3-space. ...
Image File history File links Download high-resolution version (800x800, 942 KB) File links The following pages on the English Wikipedia link to this file (pages on other projects are not listed): List of regular polytopes Talk:Hyperbolic geometry Rectification (geometry) Order-5 cubic honeycomb Truncation (geometry) User:Rocchini ...
The order-5 cubic honeycomb is one of four space-filling tessellations (or honeycombs) in hyperbolic 3-space. ...
See also A uniform polyhedron is a polyhedron with regular polygons as faces and identical vertices. ...
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. ...
A bitruncated cubic honeycomb - Cubic cells become orange truncated octahedra, and vertices are replaced by blue truncated octahedra. ...
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. ...
Two snub cubes from great rhombicuboctahedron See that red and green dots are placed at alternate vertices. ...
References H.S.M. Coxeter. ...
Stereographic projection of the 120-cell, a 4-dimensional regular polytope. ...
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