B₄ polytope

Orthographic projections in the B₄ Coxeter plane

Tesseract

16-cell

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.

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. A_k has [k+1] symmetry, and B_k 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}

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.

Coordinates for uniform 4-polytopes in Tesseract/16-cell family

#	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!

• 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
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• 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)
  • Möller, Marco (2004). Vierdimensionale Archimedische Polytope (PDF) (Doctoral dissertation) (in German). University of Hamburg.
• Uniform Polytopes in Four Dimensions, George Olshevsky.
  • Convex uniform polychora based on the tesserract/16-cell, George Olshevsky.