World Library  
Flag as Inappropriate
Email this Article

Bitruncated tesseract

Article Id: WHEBN0006397361
Reproduction Date:

Title: Bitruncated tesseract  
Author: World Heritage Encyclopedia
Language: English
Subject: Runcinated tesseract, Truncated tesseract
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

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
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.

Truncated tesseract

Truncated tesseract
tetrahedron cells visible)
Type Uniform polychoron
Schläfli symbol t{4,3,3}
Coxeter-Dynkin diagrams
Cells 24 8
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.

Alternate names

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

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

net stereographic projection
into 3-space.

Related polytopes

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

Bitruncated tesseract

Bitruncated tesseract
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
Faces 120 32 {3}
24 {4}
64 {6}
Edges 192
Vertices 96
Vertex figure 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 or bitruncated 16-cell 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 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

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
300px
Colored transparently with pink triangles, blue squares, and gray hexagons

Related polytopes

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

Truncated 16-cell

Truncated 16-cell
Cantic tesseract
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
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 or 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.

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:

  • 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

Net 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}
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 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}
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 Coxeter plane graph

Notes

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, editied by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995, [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)
  • George Olshevsky.
  • o3o3o4o - tat, o3x3x4o - tah, x3x3o4o - thex

External links

  • Stella4D software
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.