Truncated tesseract

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Schlegel wireframe 8-cell.png
Tesseract
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid truncated tesseract.png
Truncated tesseract
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid rectified 8-cell.png
Rectified tesseract
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schlegel half-solid bitruncated 8-cell.png
Bitruncated tesseract
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel diagrams centered on [4,3] (cells visible at [3,3])
Schlegel wireframe 16-cell.png
16-cell
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png
Schlegel half-solid truncated 16-cell.png
Truncated 16-cell
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Schlegel half-solid rectified 16-cell.png
Rectified 16-cell
(24-cell)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Schlegel half-solid bitruncated 16-cell.png
Bitruncated tesseract
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
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[edit]

Truncated tesseract
Schlegel half-solid truncated tesseract.png
Schlegel diagram
(tetrahedron cells visible)
Type Uniform 4-polytope
Schläfli symbol t{4,3,3}
Coxeter diagrams CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Cells 24 8 3.8.8 Truncated hexahedron.png
16 3.3.3 Tetrahedron.png
Faces 88 64 {3}
24 {8}
Edges 128
Vertices 64
Vertex figure Truncated 8-cell verf.png
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[edit]

Construction[edit]

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[edit]

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[edit]

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph 4-cube t01.svg 4-cube t01 B3.svg 4-cube t01 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph 4-cube t01 F4.svg 4-cube t01 A3.svg
Dihedral symmetry [12/3] [4]
Truncated tesseract net.png
A polyhedral net
Truncated tesseract stereographic (tC).png
Truncated tesseract
projected onto the 3-sphere
with a stereographic projection
into 3-space.

Related polytopes[edit]

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

Truncated hypercubes
Regular polygon 8 annotated.svg 3-cube t01.svgTruncated hexahedron.png 4-cube t01.svgSchlegel half-solid truncated tesseract.png 5-cube t01.svg5-cube t01 A3.svg 6-cube t01.svg6-cube t01 A5.svg 7-cube t01.svg7-cube t01 A5.svg 8-cube t01.svg8-cube t01 A7.svg ...
Octagon Truncated cube Truncated tesseract Truncated 5-cube Truncated 6-cube Truncated 7-cube Truncated 8-cube
CDel node 1.pngCDel 4.pngCDel node 1.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Bitruncated tesseract[edit]

Bitruncated tesseract
Schlegel half-solid bitruncated 16-cell.pngSchlegel half-solid bitruncated 8-cell.png
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 CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel nodes 11.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png
Cells 24 8 4.6.6 Truncated octahedron.png
16 3.6.6 Truncated tetrahedron.png
Faces 120 32 {3}
24 {4}
64 {6}
Edges 192
Vertices 96
Vertex figure Bitruncated 8-cell verf.pngCantitruncated demitesseract verf.png
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 CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png construction.

Alternate names[edit]

Construction[edit]

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[edit]

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[edit]

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph 4-cube t12.svg 4-cube t12 B3.svg 4-cube t12 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph 4-cube t12 F4.svg 4-cube t12 A3.svg
Dihedral symmetry [12/3] [4]

Stereographic projections[edit]

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
Bitruncated tesseract stereographic (tT).png Bitruncated tesseract stereographic.png Bitrunc tessa schlegel.png
Colored transparently with pink triangles, blue squares, and gray hexagons

Related polytopes[edit]

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

Bitruncated hypercubes
3-cube t12.svgTruncated octahedron.png 4-cube t12.svgSchlegel half-solid bitruncated 8-cell.png 5-cube t12.svg5-cube t12 A3.svg 6-cube t12.svg6-cube t12 A5.svg 7-cube t12.svg7-cube t12 A5.svg 8-cube t12.svg8-cube t12 A7.svg ...
Bitruncated cube Bitruncated tesseract Bitruncated 5-cube Bitruncated 6-cube Bitruncated 7-cube Bitruncated 8-cube
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

Truncated 16-cell[edit]

Truncated 16-cell
Cantic tesseract
Schlegel half-solid truncated 16-cell.png
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 CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel split1.pngCDel nodes.png
CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
Cells 24 8 3.3.3.3 Octahedron.png
16 3.6.6 Truncated tetrahedron.png
Faces 96 64 {3}
32 {6}
Edges 120
Vertices 48
Vertex figure Truncated 16-cell verf.pngTruncated demitesseract verf.png
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 CDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png.

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[edit]

Construction[edit]

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[edit]

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[edit]

Centered on octahedron[edit]

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[edit]

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[edit]

orthographic projections
Coxeter plane B4 B3 / D4 / A2 B2 / D3
Graph 4-cube t23.svg 4-cube t23 B3.svg 4-cube t23 B2.svg
Dihedral symmetry [8] [6] [4]
Coxeter plane F4 A3
Graph 4-cube t23 F4.svg 4-cube t23 A3.svg
Dihedral symmetry [12/3] [4]
Truncated 16-cell net.png
Net
Truncated cross stereographic close-up.png
Stereographic projection
(centered on truncated tetrahedron)

Related polytopes[edit]

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
n 3 4 5 6 7 8
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
Cantic cube.png Schlegel half-solid truncated 16-cell.png Truncated 5-demicube D5.svg Truncated 6-demicube D6.svg Truncated 7-demicube D7.svg Truncated 8-demicube D8.svg
Coxeter CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes 10ru.pngCDel split2.pngCDel node 1.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
= CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
= CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
= CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
= CDel nodes 10ru.pngCDel split2.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Schläfli h2{4,3} h2{4,32} h2{4,33} h2{4,34} h2{4,35} h2{4,36}

Related uniform polytopes[edit]

Related uniform polytopes in demitesseract symmetry[edit]

Related uniform polytopes in tesseract symmetry[edit]

Notes[edit]

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

References[edit]

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

External links[edit]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 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