In 4-dimensional geometry, there are 7 uniform 4-polytopes with reflections of D₄ symmetry, all are shared with higher symmetry constructions in the B₄ or F₄ symmetry families. there is also one half symmetry alternation, the snub 24-cell.

Visualizations

Each can be visualized as symmetric orthographic projections in Coxeter planes of the D₄ Coxeter group, and other subgroups. The B₄ coxeter planes are also displayed, while D₄ polytopes only have half the symmetry. They can also be shown in perspective projections of Schlegel diagrams, centered on different cells.

D₄ polytopes related to B₄

index	Name Coxeter diagram = =	Coxeter plane projections			Schlegel diagrams		Net
		B4 [8]	D4, B3 [6]	D3, B2 [4]	Cube centered	Tetrahedron centered
1	demitesseract (Same as 16-cell) = = h{4,3,3} = = {3,3,4} {3,31,1}
2	cantic tesseract (Same as truncated 16-cell) = = h2{4,3,3} = = t{3,3,4} t{3,31,1}
3	runcic tesseract birectified 16-cell (Same as rectified tesseract) = = h3{4,3,3} = = r{4,3,3} 2r{3,31,1}
4	runcicantic tesseract bitruncated 16-cell (Same as bitruncated tesseract) = = h2,3{4,3,3} = = 2t{4,3,3} 2t{3,31,1}

D₄ polytopes related to F₄ and B₄

index	Name Coxeter diagram = =	Coxeter plane projections				Schlegel diagrams		Parallel 3D	Net
		F4 [12]	B4 [8]	D4, B3 [6]	D3, B2 [2]	Cube centered	Tetrahedron centered	D4 [6]
5	rectified 16-cell (Same as 24-cell) = = {31,1,1} = r{3,3,4} = {3,4,3}
6	cantellated 16-cell (Same as rectified 24-cell) = = r{31,1,1} = rr{3,3,4} = r{3,4,3}
7	cantitruncated 16-cell (Same as truncated 24-cell) = = t{31,1,1} = tr{3,31,1} = tr{3,3,4} = t{3,4,3}
8	(Same as snub 24-cell) = = s{31,1,1} = sr{3,31,1} = sr{3,3,4} = s{3,4,3}

Coordinates

The base point can generate the coordinates of the polytope by taking all coordinate permutations and sign combinations. The edges' length will be √2. Some polytopes have two possible generator points. Points are prefixed by Even to imply only an even count of sign permutations should be included.

#	Name(s)	Base point	Johnson	Coxeter diagrams
				D4	B4	F4
1	hγ4	Even (1,1,1,1)	demitesseract
3	h3γ4	Even (1,1,1,3)	runcic tesseract
2	h2γ4	Even (1,1,3,3)	cantic tesseract
4	h2,3γ4	Even (1,3,3,3)	runcicantic tesseract
1	t3γ4 = β4	(0,0,0,2)	16-cell
5	t2γ4 = t1β4	(0,0,2,2)	rectified 16-cell
2	t2,3γ4 = t0,1β4	(0,0,2,4)	truncated 16-cell
6	t1γ4 = t2β4	(0,2,2,2)	cantellated 16-cell
9	t1,3γ4 = t0,2β4	(0,2,2,4)	cantellated 16-cell
7	t1,2,3γ = t0,1,2β4	(0,2,4,6)	cantitruncated 16-cell
8	s{31,1,1}	(0,1,φ,φ+1)/√2	Snub 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 tesseract/16-cell, George Olshevsky.
  • Convex uniform polychora based on the 24-cell, George Olshevsky.
  • Uniform polychora derived from B4 (D4), George Olshevsky.

D4 uniform polychora
{3,31,1} h{4,3,3}	2r{3,31,1} h3{4,3,3}	t{3,31,1} h2{4,3,3}	2t{3,31,1} h2,3{4,3,3}	r{3,31,1} {31,1,1}={3,4,3}	rr{3,31,1} r{31,1,1}=r{3,4,3}	tr{3,31,1} t{31,1,1}=t{3,4,3}	sr{3,31,1} s{31,1,1}=s{3,4,3}