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In 4-dimensional geometry, there are 15 uniform 4-polytopes with B4 symmetry. There are two regular forms, the tesseract, and 16-cell with 16 and 8 vertices respectively.

Visualizations

They can be visualized as symmetric orthographic projections in Coxeter planes of the B5 Coxeter group, and other subgroups.

Symmetric orthographic projections of these 32 polytopes can be made in the B5, B4, B3, B2, A3, Coxeter planes. Ak has [k+1] symmetry, and Bk has [2k] symmetry.

These 32 polytopes are each shown in these 5 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

The pictures are drawn as Schlegel diagram perspective projections, centered on the cell at pos. 3, with a consistent orientation, and the 16 cells at position 0 are shown solid, alternately colored.

# Name Coxeter plane projections Schlegel
diagrams
Net
B4
[8]
B3
[6]
B2
[4]
A3
[4]
Cube
centered
Tetrahedron
centered
1 8-cell or tesseract
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = {4,3,3}
4-cube t0.svg 4-cube t0 B3.svg 4-cube t0 B2.svg 4-cube t0 A3.svg Schlegel wireframe 8-cell.png 8-cell net.png
2 rectified 8-cell
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = r{4,3,3}
4-cube t1.svg 4-cube t1 B3.svg 4-cube t1 B2.svg 4-cube t1 A3.svg Schlegel half-solid rectified 8-cell.png Rectified tesseract net.png
3 16-cell
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = {3,3,4}
4-cube t3.svg 4-demicube t0 D4.svg 4-cube t3 B2.svg 4-cube t3 A3.svg Schlegel wireframe 16-cell.png 16-cell net.png
4 truncated 8-cell
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = t{4,3,3}
4-cube t01.svg 4-cube t01 B3.svg 4-cube t01 B2.svg 4-cube t01 A3.svg Schlegel half-solid truncated tesseract.png Truncated tesseract net.png
5 cantellated 8-cell
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = rr{4,3,3}
4-cube t02.svg 24-cell t03 B3.svg 4-cube t02 B2.svg 4-cube t02 A3.svg Schlegel half-solid cantellated 8-cell.png Small rhombated tesseract net.png
6 runcinated 8-cell
(also runcinated 16-cell)
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = t03{4,3,3}
4-cube t03.svg 4-cube t03 B3.svg 4-cube t03 B2.svg 4-cube t03 A3.svg Schlegel half-solid runcinated 8-cell.png Schlegel half-solid runcinated 16-cell.png Small disprismatotesseractihexadecachoron net.png
7 bitruncated 8-cell
(also bitruncated 16-cell)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = 2t{4,3,3}
4-cube t12.svg 4-cube t12 B3.svg 4-cube t12 B2.svg 4-cube t12 A3.svg Schlegel half-solid bitruncated 8-cell.png Schlegel half-solid bitruncated 16-cell.png Tesseractihexadecachoron net.png
8 truncated 16-cell
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = t{3,3,4}
4-cube t23.svg 4-cube t23 B3.svg 4-cube t23 B2.svg 4-cube t23 A3.svg Schlegel half-solid truncated 16-cell.png Truncated hexadecachoron net.png
9 cantitruncated 8-cell
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = tr{3,3,4}
4-cube t012.svg 4-cube t012 B3.svg 4-cube t012 B2.svg 4-cube t012 A3.svg Schlegel half-solid cantitruncated 8-cell.png Great rhombated tesseract net.png
10 runcitruncated 8-cell
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = t013{4,3,3}
4-cube t013.svg 24-cell t02 B3.svg 4-cube t013 B2.svg 4-cube t013 A3.svg Schlegel half-solid runcitruncated 8-cell.png Prismatorhombated hexadecachoron net.png
11 runcitruncated 16-cell
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = t013{3,3,4}
4-cube t023.svg 4-cube t023 B3.svg 4-cube t023 B2.svg 4-cube t023 A3.svg Schlegel half-solid runcitruncated 16-cell.png Prismatorhombated tesseract net.png
12 omnitruncated 8-cell
(also omnitruncated 16-cell)
CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = t0123{4,3,3}
4-cube t0123.svg 24-cell t023 B3.svg 4-cube t0123 B2.svg 4-cube t0123 A3.svg Schlegel half-solid omnitruncated 8-cell.png Schlegel half-solid omnitruncated 16-cell.png Great disprismatotesseractihexadecachoron net.png
# Name Coxeter plane projections Schlegel
diagrams
Net
F4
[12]
B4
[8]
B3
[6]
B2
[4]
A3
[4]
Cube
centered
Tetrahedron
centered
13 *rectified 16-cell
(Same as 24-cell)
CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node 1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
r{3,3,4} = {3,4,3}
24-cell t3 F4.svg 24-cell t0 B4.svg 24-cell t3 B3.svg 24-cell t3 B2.svg 24-cell t0 B2.svg Schlegel half-solid rectified 16-cell.png 24-cell net.png
14 *cantellated 16-cell
(Same as rectified 24-cell)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png = CDel node.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
rr{3,3,4} = r{3,4,3}
24-cell t2 F4.svg 24-cell t1 B4.svg 24-cell t2 B3.svg 24-cell t2 B2.svg 24-cell t1 B2.svg Schlegel half-solid cantellated 16-cell.png Rectified icositetrachoron net.png
15 *cantitruncated 16-cell
(Same as truncated 24-cell)
CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png = CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
tr{3,3,4} = t{3,4,3}
24-cell t23 F4.svg 4-cube t123.svg 24-cell t23 B3.svg 4-cube t123 B2.svg 24-cell t01 B2.svg Schlegel half-solid cantitruncated 16-cell.png Truncated icositetrachoron net.png
# Name Coxeter plane projections Schlegel
diagrams
Net
F4
[12]
B4
[8]
B3
[6]
B2
[4]
A3
[4]
Cube
centered
Tetrahedron
centered
16 alternated cantitruncated 16-cell
(Same as the snub 24-cell)
CDel node.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png = CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
sr{3,3,4} = s{3,4,3}
24-cell h01 F4.svg 24-cell h01 B4.svg 24-cell h01 B3.svg 24-cell h01 B2.svg Schlegel half-solid alternated cantitruncated 16-cell.png Snub disicositetrachoron net.png

Coordinates

The tesseractic family of 4-polytopes are given by the convex hulls of the base points listed in the following table, with all permutations of coordinates and sign taken. Each base point generates a distinct uniform 4-polytopes. All coordinates correspond with uniform 4-polytopes of edge length 2.

# Base point Name Coxeter diagram Vertices
3 (0,0,0,1)√2 16-cell CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 8 24-34!/3!
1 (1,1,1,1) Tesseract CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 16 244!/4!
13 (0,0,1,1)√2 Rectified 16-cell (24-cell) CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 24 24-24!/(2!2!)
2 (0,1,1,1)√2 Rectified tesseract CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 32 244!/(3!2!)
8 (0,0,1,2)√2 Truncated 16-cell CDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png 48 24-24!/2!
6 (1,1,1,1) + (0,0,0,1)√2 Runcinated tesseract CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 64 244!/3!
4 (1,1,1,1) + (0,1,1,1)√2 Truncated tesseract CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png 64 244!/3!
14 (0,1,1,2)√2 Cantellated 16-cell (rectified 24-cell) CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 96 244!/(2!2!)
7 (0,1,2,2)√2 Bitruncated 16-cell CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 96 244!/(2!2!)
5 (1,1,1,1) + (0,0,1,1)√2 Cantellated tesseract CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 96 244!/(2!2!)
15 (0,1,2,3)√2 cantitruncated 16-cell (truncated 24-cell) CDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png 192 244!/2!
11 (1,1,1,1) + (0,0,1,2)√2 Runcitruncated 16-cell CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png 192 244!/2!
10 (1,1,1,1) + (0,1,1,2)√2 Runcitruncated tesseract CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png 192 244!/2!
9 (1,1,1,1) + (0,1,2,2)√2 Cantitruncated tesseract CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png 192 244!/2!
12 (1,1,1,1) + (0,1,2,3)√2 Omnitruncated 16-cell CDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png 384 244!

References

J.H. Conway and M.J.T. Guy: Four-Dimensional Archimedean Polytopes, Proceedings of the Colloquium on Convexity at Copenhagen, page 38 und 39, 1965
John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26)
H.S.M. Coxeter:
H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
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 Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter
(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]
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

External links

Klitzing, Richard. "4D uniform 4-polytopes".
Uniform, convex polytopes in four dimensions:, Marco Möller (in German)
2004 Dissertation Four-dimensional Archimedean polytopes (in German)
Uniform Polytopes in Four Dimensions, George Olshevsky.
Convex uniform polychora based on the tesserract/16-cell, George Olshevsky.

vte

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 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

Mathematics Encyclopedia

Hellenica World - Scientific Library

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