24-cell

In 4-dimensional geometry, there are 9 uniform 4-polytopes with F₄ symmetry, and one chiral half symmetry, the snub 24-cell. There is one self-dual regular form, the 24-cell with 24 vertices.

Visualization

Each can be visualized as symmetric orthographic projections in Coxeter planes of the F₄ Coxeter group, and other subgroups.

The 3D picture are drawn as Schlegel diagram projections, centered on the cell at pos. 3, with a consistent orientation, and the 5 cells at position 0 are shown solid.

F4, [3,4,3] symmetry polytopes

#	Name Coxeter diagram Schläfli symbol	Graph				Schlegel diagram		Net
		F4 [12]	B4 [8]	B3 [6]		B2 [4]	Octahedron centered		Dual octahedron centered
1	24-cell (rectified 16-cell) = {3,4,3} = r{3,3,4}
2	rectified 24-cell (cantellated 16-cell) = r{3,4,3} = rr{3,3,4}
3	truncated 24-cell (cantitruncated 16-cell) = t{3,4,3} = tr{3,3,4}
4	cantellated 24-cell rr{3,4,3}
5	cantitruncated 24-cell tr{3,4,3}
6	runcitruncated 24-cell t0,1,3{3,4,3}

[[3,3,3]] extended symmetries of F4

#	Name Coxeter diagram Schläfli symbol	Graph				Schlegel diagram	Net
		F4 [[12]] = [24]	B4 [8]	B3 [6]	B2 [[4]] = [8]	Octahedron centered
7	*runcinated 24-cell t0,3{3,4,3}
8	*bitruncated 24-cell 2t{3,4,3}
9	*omnitruncated 24-cell t0,1,2,3{3,4,3}

[3⁺,4,3] half symmetries of F4

#	Name Coxeter diagram Schläfli symbol	Graph				Schlegel diagram		Orthogonal Projection	Net
		F4 [12]+	B4 [8]	B3 [6]+	B2 [4]	Octahedron centered	Dual octahedron centered	Octahedron centered
10	snub 24-cell s{3,4,3}
11 Nonuniform	runcic snub 24-cell s3{3,4,3}

Coordinates

Vertex coordinates for all 15 forms are given below, including dual configurations from the two regular 24-cells. (The dual configurations are named in bold.) Active rings in the first and second nodes generate points in the first column. Active rings in the third and fourth nodes generate the points in the second column. The sum of each of these points are then permutated by coordinate positions, and sign combinations. This generates all vertex coordinates. Edge lengths are 2.

The only exception is the snub 24-cell, which is generated by half of the coordinate permutations, only an even number of coordinate swaps. φ=(√5+1)/2.

24-cell family coordinates

#	Base point(s) t(0,1)	Base point(s) t(2,3)	Schläfli symbol	Name	Coxeter diagram
1		(0,0,1,1)√2	{3,4,3}	24-cell
2		(0,1,1,2)√2	r{3,4,3}	rectified 24-cell
3		(0,1,2,3)√2	t{3,4,3}	truncated 24-cell
10		(0,1,φ,φ+1)√2	s{3,4,3}	snub 24-cell
2	(0,2,2,2) (1,1,1,3)		r{3,4,3}	rectified 24-cell
4	(0,2,2,2) + (1,1,1,3) +	(0,0,1,1)√2 "	rr{3,4,3}	cantellated 24-cell
8	(0,2,2,2) + (1,1,1,3) +	(0,1,1,2)√2 "	2t{3,4,3}	bitruncated 24-cell
5	(0,2,2,2) + (1,1,1,3) +	(0,1,2,3)√2 "	tr{3,4,3}	cantitruncated 24-cell
1	(0,0,0,2) (1,1,1,1)		{3,4,3}	24-cell
7	(0,0,0,2) + (1,1,1,1) +	(0,0,1,1)√2 "	t0,3{3,4,3}	runcinated 24-cell
4	(0,0,0,2) + (1,1,1,1) +	(0,1,1,2)√2 "	t1,3{3,4,3}	cantellated 24-cell
6	(0,0,0,2) + (1,1,1,1) +	(0,1,2,3)√2 "	t0,1,3{3,4,3}	runcitruncated 24-cell
3	(1,1,1,5) (1,3,3,3) (2,2,2,4)		t{3,4,3}	truncated 24-cell
6	(1,1,1,5) + (1,3,3,3) + (2,2,2,4) +	(0,0,1,1)√2 " "	t0,2,3{3,4,3}	runcitruncated 24-cell
5	(1,1,1,5) + (1,3,3,3) + (2,2,2,4) +	(0,1,1,2)√2 " "	tr{3,4,3}	cantitruncated 24-cell
9	(1,1,1,5) + (1,3,3,3) + (2,2,2,4) +	(0,1,2,3)√2 " "	t0,1,2,3{3,4,3}	Omnitruncated 24-cell

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)
  • 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 24-cell, George Olshevsky.