# Truncated tesseract

Last updated
 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.

## Contents

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

## Truncated tesseract

Truncated tesseract

Schlegel diagram
(tetrahedron cells visible)
Type Uniform 4-polytope
Schläfli symbol t{4,3,3}
Coxeter diagrams
Cells248 3.8.8
16 3.3.3
Faces8864 {3}
24 {8}
Edges128
Vertices64
Vertex figure
( )v{3}
Dual Tetrakis 16-cell
Symmetry group B4, [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.

### Alternate names

• Truncated tesseract (Norman W. Johnson)
• Truncated tesseract (Acronym tat) (George Olshevsky, and Jonathan Bowers) [1]

### Construction

The truncated tesseract may be constructed by truncating the vertices of the tesseract at ${\displaystyle 1/({\sqrt {2}}+2)}$ 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:

${\displaystyle \left(\pm 1,\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}}),\ \pm (1+{\sqrt {2}})\right)}$

### Projections

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.

### Images

orthographic projections
Coxeter plane B4B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry [8][6][4]
Coxeter planeF4A3
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 Coxeter diagram Vertex figure ... Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube ( )v( ) ( )v{ } ( )v{3} ( )v{3,3} ( )v{3,3,3} ( )v{3,3,3,3} ( )v{3,3,3,3,3}

## Bitruncated tesseract

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{3,31,1}
h2,3{4,3,3}
Coxeter diagrams

=
Cells248 4.6.6
16 3.6.6
Faces12032 {3}
24 {4}
64 {6}
Edges192
Vertices96
Vertex figure
Digonal disphenoid
Symmetry group B4, [3,3,4], order 384
D4, [31,1,1], 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.

### Alternate names

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

### Construction

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:

${\displaystyle \left(0,\ \pm {\sqrt {2}},\ \pm 2{\sqrt {2}},\ \pm 2{\sqrt {2}}\right)}$

### Structure

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.

### Projections

orthographic projections
Coxeter plane B4B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry [8][6][4]
Coxeter planeF4A3
Graph
Dihedral symmetry[12/3][4]

### Stereographic projections

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 Coxeter Vertex figure ... Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube ( )v{ } { }v{ } { }v{3} { }v{3,3} { }v{3,3,3} { }v{3,3,3,3}

## Truncated 16-cell

Truncated 16-cell
Cantic tesseract

Schlegel diagram
(octahedron cells visible)
Type Uniform 4-polytope
Schläfli symbol t{4,3,3}
t{3,31,1}
h2{4,3,3}
Coxeter diagrams

=
Cells248 3.3.3.3
16 3.6.6
Faces9664 {3}
32 {6}
Edges120
Vertices48
Vertex figure
square pyramid
Dual Hexakis tesseract
Coxeter groups B4 [3,3,4], order 384
D4 [31,1,1], 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.

### Alternate names

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

### Construction

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.

### Structure

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

### Projections

#### Centered on octahedron

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.

#### Centered on truncated tetrahedron

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.

### Images

orthographic projections
Coxeter plane B4B3 / D4 / A2B2 / D3
Graph
Dihedral symmetry [8][6][4]
Coxeter planeF4A3
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:

Dimensional family of cantic n-cubes
n345678
Symmetry
[1+,4,3n-2]
[1+,4,3]
= [3,3]
[1+,4,32]
= [3,31,1]
[1+,4,33]
= [3,32,1]
[1+,4,34]
= [3,33,1]
[1+,4,35]
= [3,34,1]
[1+,4,36]
= [3,35,1]
Cantic
figure
Coxeter
=

=

=

=

=

=
Schläfli h2{4,3} h2{4,32} h2{4,33} h2{4,34} h2{4,35} h2{4,36}
D4 uniform polychora

{3,31,1}
h{4,3,3}
2r{3,31,1}
h3{4,3,3}
t{3,31,1}
h2{4,3,3}
2t{3,31,1}
h2,3{4,3,3}
r{3,31,1}
{31,1,1}={3,4,3}
rr{3,31,1}
r{31,1,1}=r{3,4,3}
tr{3,31,1}
t{31,1,1}=t{3,4,3}
sr{3,31,1}
s{31,1,1}=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}t1{4,3,3}
r{4,3,3}
t0,1{4,3,3}
t{4,3,3}
t0,2{4,3,3}
rr{4,3,3}
t0,3{4,3,3}t1,2{4,3,3}
2t{4,3,3}
t0,1,2{4,3,3}
tr{4,3,3}
t0,1,3{4,3,3}t0,1,2,3{4,3,3}
Schlegel
diagram
B4

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}t1{3,3,4}
r{3,3,4}
t0,1{3,3,4}
t{3,3,4}
t0,2{3,3,4}
rr{3,3,4}
t0,3{3,3,4}t1,2{3,3,4}
2t{3,3,4}
t0,1,2{3,3,4}
tr{3,3,4}
t0,1,3{3,3,4}t0,1,2,3{3,3,4}
Schlegel
diagram
B4

## Notes

1. Klitzing, (o3o3o4o - tat)
2. Klitzing, (o3x3x4o - tah)
3. Klitzing, (x3x3o4o - thex)

## Related Research Articles

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 C24, or the icositetrachoron, octaplex (short for "octahedral complex"), icosatetrahedroid, octacube, hyper-diamond or polyoctahedron, being constructed of octahedral cells.

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 C600, hexacosichoron and hexacosihedroid. It is also called a tetraplex (abbreviated from "tetrahedral complex") and a polytetrahedron, being bounded by tetrahedral cells.

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 C16, hexadecachoron, or hexdecahedroid.

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.

## References

• 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]
• John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN   978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. (1966)
• 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
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds