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 polychoron (4-dimensional uniform polytope) formed as the truncation of the regular tesseract.

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

Contents

Truncated tesseract

Trucated tesseract

Schlegel diagram
(tetrahedron cells visible)
Type Uniform polychoron
Schläfli symbol t0,1{4,3,3}
Coxeter-Dynkin 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 A4, [4,3,3]
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 1/(\sqrt{2}%2B2) 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:

\left(\pm1,\ \pm(1%2B\sqrt{2}),\ \pm(1%2B\sqrt{2}),\ \pm(1%2B\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:

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.

Bitruncated tesseract

Bitruncated tesseract

Two Schlegel diagrams, centered on truncated tetrahedral or truncated octahedral cells, with alternate cell types hidden.
Type Uniform polychoron
Schläfli symbol t1,2{4,3,3}
t0,1,2{31,1,1}
 
Coxeter-Dynkin 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]
D4, [31,1,1]
Properties convex, vertex-transitive
Uniform index 15 16 17

The bitruncated tesseract (also called a bitruncated 16-cell) is constructed by a bitruncation operation applied to the tesseract.

Alternate names

Construction

A tesseract is bitruncated by truncating its cells beyond their mid-points, 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:

\left(0,\ \pm\sqrt{2},\ \pm2\sqrt{2},\ \pm2\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 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

Truncated 16-cell

Truncated 16-cell

Schlegel diagram
(octahedron cells visible)
Type Uniform polychoron
Schläfli symbol t0,1{4,3,3}
t0,1{31,1,1}
Coxeter-Dynkin 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 BC4 [3,3,4]
D4 [31,1,1]
Properties convex
Uniform index 16 17 18

The truncated 16-cell which is bounded by 24 cells: 8 regular octahedra, and 16 truncated tetrahedra.

It is related to, but not to be confused with, the 24-cell, which is a regular polychoron 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

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

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)

Related uniform polytopes

Name tesseract rectified
tesseract
truncated
tesseract
cantellated
tesseract
runcinated
tesseract
bitruncated
tesseract
cantitruncated
tesseract
runcitruncated
tesseract
omnitruncated
tesseract
Coxeter-Dynkin
diagram
Schläfli
symbol
{4,3,3} t1{4,3,3} t0,1{4,3,3} t0,2{4,3,3} t0,3{4,3,3} t1,2{4,3,3} t0,1,2{4,3,3} t0,1,3{4,3,3} t0,1,2,3{4,3,3}
Schlegel
diagram
B4 Coxeter plane graph
 
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-Dynkin
diagram
Schläfli
symbol
{3,3,4} t1{3,3,4} t0,1{3,3,4} t0,2{3,3,4} t0,3{3,3,4} t1,2{3,3,4} t0,1,2{3,3,4} t0,1,3{3,3,4} t0,1,2,3{3,3,4}
Schlegel
diagram
B4 Coxeter plane graph

Notes

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

References

External links