User:Tomruen/tempaaa

Summary of constructions by extended symmetry

The 46 uniform polychora constructed from the A₄, BC₄, F₄, H₄ symmetry are given in this table by their full extended symmetry and Coxeter diagrams. Alternations are grouped by their chiral symmetry. All alternations are given, although the snub 24-cell, with its 3 family of constructions is the only one that is uniform. Counts in parenthesis are either repeats or nonuniform. The Coxeter diagrams are given with subscript indices 1 through 46.

Coxeter group	Extended symmetry	Polychora		Chiral extended symmetry	Alternation honeycombs
[3,3,3]	[3,3,3] (order 120)	6	1 | 2 | 3 4 | 7 | 8
	[2+[3,3,3]] (order 240)	3	5| 6 | 9	[2+[3,3,3]]+ (order 120)	(1)	-
[3,31,1]	[3,31,1] (order 192)	0	(none)
	[1[3,31,1]]=[4,3,3] = (order 384)	(4)	12 | 17 | 11 | 16
	[3[31,1,1]]=[3,4,3] = (order 1152)	(3)	22 | 23 | 24	[3[3,31,1]]+ =[3,4,3]+ (order 576)	(1)	31
[4,3,3]	[3[1+,4,3,3]]=[3,4,3] = (order 1152)	(3)	22 | 23 | 24
	[4,3,3] (order 384)	12	10 | 11 | 12 | 13 | 14 15 | 16 | 17 | 18 | 19 20 | 21	[1+,4,3,3]+ (order 96)	(2)	12 | 31
				[4,3,3]+ (order 192)	(1)	-
[3,4,3]	[3,4,3] (order 1152)	6	22 | 23 | 24 25 | 28 | 29	[2+[3+,4,3+]] (order 576)	1	31
	[2+[3,4,3]] (order 2304)	3	26 | 27 | 30	[2+[3,4,3]]+ (order 1152)	(1)	-
[5,3,3]	[5,3,3] (order 14400)	15	32 | 33 | 34 | 35 | 36 37 | 38 | 39 | 40 | 41 42 | 43 | 44 | 45 | 46	[5,3,3]+ (order 7200)	(1)	-
[4,2,4]	[4,2,4] (order 64)	0	(none)	[(4,2,4]+ (order 32)	0	(none)
	[2+[4,2,4]] (order 128)	0	(none)	[2+[(4,2+,4,2+)]] (order 64)	0	(none)
	[3[1+,4,2,4,1+]]=[4,3,2] = (order 96)	(1)		[3[1+,4,2,4,1+]]+=[4,3,2]+ (order 48)	(1)
	[(3,3)[1+,4,2,4,1+]]=[4,3,3] = (order 384)	(1)	10	[(3,3)[1+,4,2,4,1+]]+=[4,3,3]+ (order 192)	(1)	12
	[[4],2,4]=[8,2,4] = (order 128)	(1)		[[4],2,4]+=[8,2,4]+ (order 64)	(1)
	[2+[[4],2,[4]]]=[2+[8,2,8]] = (order 512)	(1)		[2+[[4],2,[4]]]+=[2+[8,2,8]]+ (order 256)	(1)