Tesseract | Truncated tesseract | Rectified tesseract | Bitruncated tesseract |

Schlegel diagrams centered on [4,3] (cells visible at [3,3]) | |||

16-cell | Truncated 16-cell | Rectified 16-cell (24-cell) | Bitruncated tesseract |

Schlegel diagrams centered on [3,3] (cells visible at [4,3]) |

In geometry, a **truncated tesseract** is a uniform 4-polytope formed as the truncation of the regular tesseract.

- Truncated tesseract
- Alternate names
- Construction
- Projections
- Images
- Related polytopes
- Bitruncated tesseract
- Alternate names 2
- Construction 2
- Structure
- Projections 2
- Stereographic projections
- Related polytopes 2
- Truncated 16-cell
- Alternate names 3
- Construction 3
- Structure 2
- Projections 3
- Images 2
- Related polytopes 3
- Related uniform polytopes
- Related uniform polytopes in demitesseract symmetry
- Related uniform polytopes in tesseract symmetry
- Notes
- References
- External links

There are three truncations, including a bitruncation, and a tritruncation, which creates the *truncated 16-cell*.

Truncated tesseract | ||
---|---|---|

Schlegel diagram (tetrahedron cells visible) | ||

Type | Uniform 4-polytope | |

Schläfli symbol | t{4,3,3} | |

Coxeter diagrams | ||

Cells | 24 | 8 3.8.8 16 3.3.3 |

Faces | 88 | 64 {3} 24 {8} |

Edges | 128 | |

Vertices | 64 | |

Vertex figure | ( )v{3} | |

Dual | Tetrakis 16-cell | |

Symmetry group | B_{4}, [4,3,3], order 384 | |

Properties | convex | |

Uniform index | 12 13 14 |

The **truncated tesseract** is bounded by 24 cells: 8 truncated cubes, and 16 tetrahedra.

- Truncated tesseract (Norman W. Johnson)
- Truncated tesseract (Acronym tat) (George Olshevsky, and Jonathan Bowers)
^{ [1] }

The truncated tesseract may be constructed by truncating the vertices of the tesseract at of the edge length. A regular tetrahedron is formed at each truncated vertex.

The Cartesian coordinates of the vertices of a truncated tesseract having edge length 2 is given by all permutations of:

In the truncated cube first parallel projection of the truncated tesseract into 3-dimensional space, the image is laid out as follows:

- The projection envelope is a cube.
- Two of the truncated cube cells project onto a truncated cube inscribed in the cubical envelope.
- The other 6 truncated cubes project onto the square faces of the envelope.
- The 8 tetrahedral volumes between the envelope and the triangular faces of the central truncated cube are the images of the 16 tetrahedra, a pair of cells to each image.

Coxeter plane | B_{4} | B_{3} / D_{4} / A_{2} | B_{2} / D_{3} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [8] | [6] | [4] |

Coxeter plane | F_{4} | A_{3} | |

Graph | |||

Dihedral symmetry | [12/3] | [4] |

A polyhedral net | Truncated tesseract projected onto the 3-sphere with a stereographic projection into 3-space. |

The * truncated tesseract *, is third in a sequence of truncated hypercubes:

Image | ... | |||||||
---|---|---|---|---|---|---|---|---|

Name | Octagon | Truncated cube | Truncated tesseract | Truncated 5-cube | Truncated 6-cube | Truncated 7-cube | Truncated 8-cube | |

Coxeter diagram | ||||||||

Vertex figure | ( )v( ) | ( )v{ } | ( )v{3} | ( )v{3,3} | ( )v{3,3,3} | ( )v{3,3,3,3} | ( )v{3,3,3,3,3} |

Bitruncated tesseract | ||
---|---|---|

Two Schlegel diagrams, centered on truncated tetrahedral or truncated octahedral cells, with alternate cell types hidden. | ||

Type | Uniform 4-polytope | |

Schläfli symbol | 2t{4,3,3} 2t{3,3 ^{1,1}}h _{2,3}{4,3,3} | |

Coxeter diagrams | = | |

Cells | 24 | 8 4.6.6 16 3.6.6 |

Faces | 120 | 32 {3} 24 {4} 64 {6} |

Edges | 192 | |

Vertices | 96 | |

Vertex figure | Digonal disphenoid | |

Symmetry group | B_{4}, [3,3,4], order 384D _{4}, [3^{1,1,1}], order 192 | |

Properties | convex, vertex-transitive | |

Uniform index | 15 16 17 |

The **bitruncated tesseract**, **bitruncated 16-cell**, or **tesseractihexadecachoron** is constructed by a bitruncation operation applied to the tesseract. It can also be called a **runcicantic tesseract** with half the vertices of a runcicantellated tesseract with a construction.

- Bitruncated tesseract/Runcicantic tesseract (Norman W. Johnson)
- Tesseractihexadecachoron (Acronym tah) (George Olshevsky, and Jonathan Bowers)
^{ [2] }

A tesseract is bitruncated by truncating its cells beyond their midpoints, turning the eight cubes into eight truncated octahedra. These still share their square faces, but the hexagonal faces form truncated tetrahedra which share their triangular faces with each other.

The Cartesian coordinates of the vertices of a bitruncated tesseract having edge length 2 is given by all permutations of:

The truncated octahedra are connected to each other via their square faces, and to the truncated tetrahedra via their hexagonal faces. The truncated tetrahedra are connected to each other via their triangular faces.

Coxeter plane | B_{4} | B_{3} / D_{4} / A_{2} | B_{2} / D_{3} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [8] | [6] | [4] |

Coxeter plane | F_{4} | A_{3} | |

Graph | |||

Dihedral symmetry | [12/3] | [4] |

The truncated-octahedron-first projection of the bitruncated tesseract into 3D space has a truncated cubical envelope. Two of the truncated octahedral cells project onto a truncated octahedron inscribed in this envelope, with the square faces touching the centers of the octahedral faces. The 6 octahedral faces are the images of the remaining 6 truncated octahedral cells. The remaining gap between the inscribed truncated octahedron and the envelope are filled by 8 flattened truncated tetrahedra, each of which is the image of a pair of truncated tetrahedral cells.

Colored transparently with pink triangles, blue squares, and gray hexagons |

The * bitruncated tesseract * is second in a sequence of bitruncated hypercubes:

Image | ... | ||||||
---|---|---|---|---|---|---|---|

Name | Bitruncated cube | Bitruncated tesseract | Bitruncated 5-cube | Bitruncated 6-cube | Bitruncated 7-cube | Bitruncated 8-cube | |

Coxeter | |||||||

Vertex figure | ( )v{ } | { }v{ } | { }v{3} | { }v{3,3} | { }v{3,3,3} | { }v{3,3,3,3} |

Truncated 16-cellCantic tesseract | ||
---|---|---|

Schlegel diagram (octahedron cells visible) | ||

Type | Uniform 4-polytope | |

Schläfli symbol | t{4,3,3} t{3,3 ^{1,1}}h _{2}{4,3,3} | |

Coxeter diagrams | = | |

Cells | 24 | 8 3.3.3.3 16 3.6.6 |

Faces | 96 | 64 {3} 32 {6} |

Edges | 120 | |

Vertices | 48 | |

Vertex figure | square pyramid | |

Dual | Hexakis tesseract | |

Coxeter groups | B_{4} [3,3,4], order 384D _{4} [3^{1,1,1}], order 192 | |

Properties | convex | |

Uniform index | 16 17 18 |

The **truncated 16-cell**, **truncated hexadecachoron**, **cantic tesseract** which is bounded by 24 cells: 8 regular octahedra, and 16 truncated tetrahedra. It has half the vertices of a cantellated tesseract with construction .

It is related to, but not to be confused with, the 24-cell, which is a regular 4-polytope bounded by 24 regular octahedra.

- Truncated 16-cell/Cantic tesseract (Norman W. Johnson)
- Truncated hexadecachoron (Acronym thex) (George Olshevsky, and Jonathan Bowers)
^{ [3] }

The truncated 16-cell may be constructed from the 16-cell by truncating its vertices at 1/3 of the edge length. This results in the 16 truncated tetrahedral cells, and introduces the 8 octahedra (vertex figures).

(Truncating a 16-cell at 1/2 of the edge length results in the 24-cell, which has a greater degree of symmetry because the truncated cells become identical with the vertex figures.)

The Cartesian coordinates of the vertices of a truncated 16-cell having edge length √2 are given by all permutations, and sign combinations of

- (0,0,1,2)

An alternate construction begins with a demitesseract with vertex coordinates (±3,±3,±3,±3), having an even number of each sign, and truncates it to obtain the permutations of

- (1,1,3,3), with an even number of each sign.

The truncated tetrahedra are joined to each other via their hexagonal faces. The octahedra are joined to the truncated tetrahedra via their triangular faces.

The octahedron-first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure:

- The projection envelope is a truncated octahedron.
- The 6 square faces of the envelope are the images of 6 of the octahedral cells.
- An octahedron lies at the center of the envelope, joined to the center of the 6 square faces by 6 edges. This is the image of the other 2 octahedral cells.
- The remaining space between the envelope and the central octahedron is filled by 8 truncated tetrahedra (distorted by projection). These are the images of the 16 truncated tetrahedral cells, a pair of cells to each image.

This layout of cells in projection is analogous to the layout of faces in the projection of the truncated octahedron into 2-dimensional space. Hence, the truncated 16-cell may be thought of as the 4-dimensional analogue of the truncated octahedron.

The truncated tetrahedron first parallel projection of the truncated 16-cell into 3-dimensional space has the following structure:

- The projection envelope is a truncated cube.
- The nearest truncated tetrahedron to the 4D viewpoint projects to the center of the envelope, with its triangular faces joined to 4 octahedral volumes that connect it to 4 of the triangular faces of the envelope.
- The remaining space in the envelope is filled by 4 other truncated tetrahedra.
- These volumes are the images of the cells lying on the near side of the truncated 16-cell; the other cells project onto the same layout except in the dual configuration.
- The six octagonal faces of the projection envelope are the images of the remaining 6 truncated tetrahedral cells.

Coxeter plane | B_{4} | B_{3} / D_{4} / A_{2} | B_{2} / D_{3} |
---|---|---|---|

Graph | |||

Dihedral symmetry | [8] | [6] | [4] |

Coxeter plane | F_{4} | A_{3} | |

Graph | |||

Dihedral symmetry | [12/3] | [4] |

Net | Stereographic projection (centered on truncated tetrahedron) |

A truncated 16-cell, as a cantic 4-cube, is related to the dimensional family of cantic n-cubes:

n | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|

Symmetry [1 ^{+},4,3^{n-2}] | [1^{+},4,3]= [3,3] | [1^{+},4,3^{2}]= [3,3 ^{1,1}] | [1^{+},4,3^{3}]= [3,3 ^{2,1}] | [1^{+},4,3^{4}]= [3,3 ^{3,1}] | [1^{+},4,3^{5}]= [3,3 ^{4,1}] | [1^{+},4,3^{6}]= [3,3 ^{5,1}] |

Cantic figure | ||||||

Coxeter | = | = | = | = | = | = |

Schläfli | h_{2}{4,3} | h_{2}{4,3^{2}} | h_{2}{4,3^{3}} | h_{2}{4,3^{4}} | h_{2}{4,3^{5}} | h_{2}{4,3^{6}} |

D_{4} uniform polychora | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

{3,3^{1,1}} h{4,3,3} | 2r{3,3^{1,1}} h _{3}{4,3,3} | t{3,3^{1,1}} h _{2}{4,3,3} | 2t{3,3^{1,1}} h _{2,3}{4,3,3} | r{3,3^{1,1}} {3 ^{1,1,1}}={3,4,3} | rr{3,3^{1,1}} r{3 ^{1,1,1}}=r{3,4,3} | tr{3,3^{1,1}} t{3 ^{1,1,1}}=t{3,4,3} | sr{3,3^{1,1}} s{3 ^{1,1,1}}=s{3,4,3} |

B4 symmetry polytopes | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Name | tesseract | rectified tesseract | truncated tesseract | cantellated tesseract | runcinated tesseract | bitruncated tesseract | cantitruncated tesseract | runcitruncated tesseract | omnitruncated tesseract | ||

Coxeter diagram | = | = | |||||||||

Schläfli symbol | {4,3,3} | t_{1}{4,3,3}r{4,3,3} | t_{0,1}{4,3,3}t{4,3,3} | t_{0,2}{4,3,3}rr{4,3,3} | t_{0,3}{4,3,3} | t_{1,2}{4,3,3}2t{4,3,3} | t_{0,1,2}{4,3,3}tr{4,3,3} | t_{0,1,3}{4,3,3} | t_{0,1,2,3}{4,3,3} | ||

Schlegel diagram | |||||||||||

B_{4} | |||||||||||

Name | 16-cell | rectified 16-cell | truncated 16-cell | cantellated 16-cell | runcinated 16-cell | bitruncated 16-cell | cantitruncated 16-cell | runcitruncated 16-cell | omnitruncated 16-cell | ||

Coxeter diagram | = | = | = | = | = | = | |||||

Schläfli symbol | {3,3,4} | t_{1}{3,3,4}r{3,3,4} | t_{0,1}{3,3,4}t{3,3,4} | t_{0,2}{3,3,4}rr{3,3,4} | t_{0,3}{3,3,4} | t_{1,2}{3,3,4}2t{3,3,4} | t_{0,1,2}{3,3,4}tr{3,3,4} | t_{0,1,3}{3,3,4} | t_{0,1,2,3}{3,3,4} | ||

Schlegel diagram | |||||||||||

B_{4} |

A **cuboctahedron** is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is the only radially equilateral convex polyhedron.

In geometry, an **octahedron** is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the **regular** octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

In four-dimensional geometry, the **24-cell** is the convex regular 4-polytope with Schläfli symbol {3,4,3}. It is also called **C _{24}**, or the

In geometry, the **600-cell** is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol {3,3,5}. It is also known as the **C _{600}**,

In four-dimensional geometry, a **16-cell** is a regular convex 4-polytope. It is one of the six regular convex 4-polytopes first described by the Swiss mathematician Ludwig Schläfli in the mid-19th century. It is also called **C _{16}**,

In four-dimensional geometry, a **runcinated 5-cell** is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

In four-dimensional geometry, a **runcinated tesseract** is a convex uniform 4-polytope, being a runcination of the regular tesseract.

In four-dimensional geometry, the **rectified 5-cell** is a uniform 4-polytope composed of 5 regular tetrahedral and 5 regular octahedral cells. Each edge has one tetrahedron and two octahedra. Each vertex has two tetrahedra and three octahedra. In total it has 30 triangle faces, 30 edges, and 10 vertices. Each vertex is surrounded by 3 octahedra and 2 tetrahedra; the vertex figure is a triangular prism.

In geometry, the **snub 24-cell** or **snub disicositetrachoron** is a convex uniform 4-polytope composed of 120 regular tetrahedral and 24 icosahedral cells. Five tetrahedra and three icosahedra meet at each vertex. In total it has 480 triangular faces, 432 edges, and 96 vertices. One can build it from the 600-cell by diminishing a select subset of icosahedral pyramids and leaving only their icosahedral bases, thereby removing 480 tetrahedra and replacing them with 24 icosahedra.

In four-dimensional geometry, a **cantellated tesseract** is a convex uniform 4-polytope, being a cantellation of the regular tesseract.

The **cubic honeycomb** or **cubic cellulation** is the only proper regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a **cubille**.

The **tetrahedral-octahedral honeycomb**, **alternated cubic honeycomb** is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

The **bitruncated cubic honeycomb** is a space-filling tessellation in Euclidean 3-space made up of truncated octahedra. It has 4 truncated octahedra around each vertex. Being composed entirely of truncated octahedra, it is cell-transitive. It is also edge-transitive, with 2 hexagons and one square on each edge, and vertex-transitive. It is one of 28 uniform honeycombs.

In geometry, a **truncated 24-cell** is a uniform 4-polytope formed as the truncation of the regular 24-cell.

In geometry, the **rectified tesseract**, **rectified 8-cell** is a uniform 4-polytope bounded by 24 cells: 8 cuboctahedra, and 16 tetrahedra. It has half the vertices of a runcinated tesseract, with its construction, called a **runcic tesseract**.

In geometry, a **truncated 5-cell** is a uniform 4-polytope formed as the truncation of the regular 5-cell.

In four-dimensional geometry, a **cantellated 5-cell** is a convex uniform 4-polytope, being a cantellation of the regular 5-cell.

In four-dimensional geometry, a **runcinated 24-cell** is a convex uniform 4-polytope, being a runcination of the regular 24-cell.

In five-dimensional geometry, a **rectified 5-simplex** is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In 4-dimensional geometry, a **truncated octahedral prism** or **omnitruncated tetrahedral prism** is a convex uniform 4-polytope. This 4-polytope has 16 cells It has 64 faces, and 96 edges and 48 vertices.

- T. Gosset:
*On the Regular and Semi-Regular Figures in Space of n Dimensions*, Messenger of Mathematics, Macmillan, 1900 - H.S.M. Coxeter:
- Coxeter,
*Regular Polytopes*, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) - H.S.M. Coxeter,
*Regular Polytopes*, 3rd Edition, Dover New York, 1973, p. 296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) **Kaleidoscopes: Selected Writings of H.S.M. Coxeter**, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, ISBN 978-0-471-01003-6- (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]

- (Paper 22) H.S.M. Coxeter,

- Coxeter,
- John H. Conway, Heidi Burgiel, Chaim Goodman-Strass,
*The Symmetries of Things*2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_{n1}) - Norman Johnson
*Uniform Polytopes*, Manuscript (1991)- N.W. Johnson:
*The Theory of Uniform Polytopes and Honeycombs*, Ph.D. (1966)

- N.W. Johnson:
- 2. Convex uniform polychora based on the tesseract (8-cell) and hexadecachoron (16-cell) - Models 13, 16, 17 , George Olshevsky.
- Klitzing, Richard. "4D uniform polytopes (polychora)". o3o3o4o - tat, o3x3x4o - tah, x3x3o4o - thex

- Paper model of truncated tesseract created using nets generated by Stella4D software

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Images, videos and audio are available under their respective licenses.