Truncated tesseract


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.

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
Cells 24 8 3.8.8
16 3.3.3
Faces 88 64 {3}
24 {8}
Edges 128
Vertices 64
Vertex figure
Isosceles triangular pyramid
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

Construction

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:

Projections

A stereoscopic 3D projection of a truncated tesseract.

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

Images

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
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:

Truncated hypercubes
...
Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube

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

=
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 B4, [3,3,4], order 384
D4, [31,1,1], order 192
Properties convex, vertex-transitive
Uniform index 15 16 17
Net

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

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:

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 B4 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
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.

Stereographic projections

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

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

Bitruncated hypercubes
...
Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube

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

=
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 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

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√2 are given by all permutations, and sign combinations:

(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

Octahedron-first parallel projection into 3 dimensions, with octahedral cells highlighted

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

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

Projection of truncated 16-cell into 3 dimensions, centered on truncated tetrahedral cell, with hidden cells culled

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

Images

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
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}

Notes

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

References

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / E9 / E10 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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