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
= {4,3,3}
2 rectified 8-cell
= r{4,3,3}
3 16-cell
= {3,3,4}
4 truncated 8-cell
= t{4,3,3}
5 cantellated 8-cell
= rr{4,3,3}
6 runcinated 8-cell
(also runcinated 16-cell)
= t03{4,3,3}
7 bitruncated 8-cell
(also bitruncated 16-cell)
= 2t{4,3,3}
8 truncated 16-cell
= t{3,3,4}
9 cantitruncated 8-cell
= tr{3,3,4}
10 runcitruncated 8-cell
= t013{4,3,3}
11 runcitruncated 16-cell
= t013{3,3,4}
12 omnitruncated 8-cell
(also omnitruncated 16-cell)
= t0123{4,3,3}
# 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)
=
r{3,3,4} = {3,4,3}
14 *cantellated 16-cell
(Same as rectified 24-cell)
=
rr{3,3,4} = r{3,4,3}
15 *cantitruncated 16-cell
(Same as truncated 24-cell)
=
tr{3,3,4} = t{3,4,3}
# 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)
=
sr{3,3,4} = s{3,4,3}

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 8 24-34!/3!
1 (1,1,1,1) Tesseract 16 244!/4!
13 (0,0,1,1)√2 Rectified 16-cell (24-cell) 24 24-24!/(2!2!)
2 (0,1,1,1)√2 Rectified tesseract 32 244!/(3!2!)
8 (0,0,1,2)√2 Truncated 16-cell 48 24-24!/2!
6 (1,1,1,1) + (0,0,0,1)√2 Runcinated tesseract 64 244!/3!
4 (1,1,1,1) + (0,1,1,1)√2 Truncated tesseract 64 244!/3!
14 (0,1,1,2)√2 Cantellated 16-cell (rectified 24-cell) 96 244!/(2!2!)
7 (0,1,2,2)√2 Bitruncated 16-cell 96 244!/(2!2!)
5 (1,1,1,1) + (0,0,1,1)√2 Cantellated tesseract 96 244!/(2!2!)
15 (0,1,2,3)√2 cantitruncated 16-cell (truncated 24-cell) 192 244!/2!
11 (1,1,1,1) + (0,0,1,2)√2 Runcitruncated 16-cell 192 244!/2!
10 (1,1,1,1) + (0,1,1,2)√2 Runcitruncated tesseract 192 244!/2!
9 (1,1,1,1) + (0,1,2,2)√2 Cantitruncated tesseract 192 244!/2!
12 (1,1,1,1) + (0,1,2,3)√2 Omnitruncated 16-cell 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

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.

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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