Truncated tesseract

(Redirected from Bitruncated 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 truncations, including a bitruncation, and a tritruncation, which creates the truncated 16-cell.

Truncated tesseract

Schlegel diagram
(tetrahedron cells visible)
Type Uniform polychoron
Schläfli symbol t{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 BC4, [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.

Construction

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

$\left(\pm1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)$

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:

• 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 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.

Related polytopes

The truncated tesseract, is third in a sequence of 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 polychoron
Schläfli symbol 2t{4,3,3}
2t{3,31,1}
h2,3{4,3,3}
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 BC4, [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.

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:

$\left(0,\ \pm\sqrt{2},\ \pm2\sqrt{2},\ \pm2\sqrt{2}\right)$

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.

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

Related polytopes

The bitruncated tesseract is second in a sequence of 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 polychoron
Schläfli symbol t{4,3,3}
t{3,31,1}
h2{4,3,3}
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], 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 polychoron bounded by 24 regular octahedra.

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:

• 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

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:

• 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 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

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

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 [1]
• (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.
• Richard Klitzing, 4D, uniform polytopes (polychora) o3o3o4o - tat, o3x3x4o - tah, x3x3o4o - thex