Равномерный 8-многогранник

Графы трех правильных и связанных однородных многогранников .

8-симплекс				Выпрямленный 8-симплекс				Усеченный 8-симплекс
Сочлененный 8-симплекс				Ранцинированный 8-симплекс				Стерический 8-симплекс
Пятеричный 8-симплекс				Шестигранный 8-симплекс				Гептеллированный 8-симплекс
8-ортоплекс				Выпрямленный 8-ортоплекс				Усеченный 8-ортоплекс
Сочлененный 8-ортоплекс						Ранцинированный 8-ортоплекс
Шестигранный 8-ортоплекс						Согнутый 8-куб
Ранцинированный 8-кубовый				Стерилизованный 8-кубовый				Пятеричный 8-куб
Шестигранный 8-кубовый						Гептеллированный 8-куб
8-кубовый				Ректифицированный 8-куб				Усеченный 8-куб
8-демикуб				Усеченный 8-микуб				Согнутый 8-ми куб
Ранцинированный 8-ми куб						Стерилизованный 8-ми куб
Пятеричный 8-ми куб						Шестигранный 8-микуб
4 21				1 42				2 41

В восьмимерной геометрии многогранник восьмимерный многогранник или 8-мерный — это многогранник , содержащий 7-мерные грани. Каждый 6-многогранника гребень разделяется ровно двумя 7-многогранника гранями .

Однородный 8-многогранник многогранник — это вершинно-транзитивный , построенный из однородных 7-многогранников .

Правильные 8-многогранники

Правильные 8-многогранники могут быть представлены символом Шлефли {p,q,r,s,t,u,v} с 7-многогранника v {p,q,r,s,t,u} гранями вокруг каждой вершины .

ровно три Таких выпуклых правильных 8-многогранников :

1. {3,3,3,3,3,3,3} - 8-симплекс
2. {4,3,3,3,3,3,3} - 8-куб
3. {3,3,3,3,3,3,4} - 8-ортоплекс

Невыпуклых правильных 8-многогранников не существует.

Характеристики

Топология любого данного 8-многогранника определяется его числами Бетти и коэффициентами кручения .

Значение характеристики Эйлера , используемой для характеристики многогранников, не дает полезного обобщения на более высокие измерения и равно нулю для всех 8-многогранников, независимо от их базовой топологии. Эта неадекватность характеристики Эйлера для надежного различения различных топологий в более высоких измерениях привела к открытию более сложных чисел Бетти.

Точно так же понятия ориентируемости многогранника недостаточно для характеристики скручивания поверхности тороидальных многогранников, и это привело к использованию коэффициентов кручения.

Равномерные 8-многогранники по фундаментальным группам Кокстера

Однородные 8-многогранники с отражательной симметрией могут быть порождены этими четырьмя группами Кокстера, представленными перестановками колец диаграмм Кокстера-Дынкина :

#	Группа Коксетера			Формы
1	А 8	[3 7 ]		135
2	г. до н.э. 8	[4,3 6 ]		255
3	Д 8	[3 5,1,1 ]		191 (64 уникальных)
4	EЕ8	[3 4,2,1 ]		255

Избранные правильные и однородные 8-многогранники из каждого семейства включают:

1. Семейство симплекс : А ₈ [3 ⁷ ] -
   • 135 однородных 8-многогранников как перестановок колец групповой диаграммы, включая один регулярный:
     1. {3 ⁷ } - 8-симплекс или эннеа-9-топ или эннеазеттон -
2. гиперкуба / ортоплекса Семейство _{: B 8} [4,3 ⁶ ] -
   • 255 однородных 8-многогранников как перестановок колец групповой диаграммы, включая два правильных:
     1. {4,3 ⁶ } - 8-куб или октеракт -
     2. {3 ⁶ ,4} - 8-ортоплекс или октакрокс -
3. Демигиперкуб Д _{8 : [3} Семейство ^5,1,1 ] -
   • 191 однородный 8-многогранник как перестановка колец в групповой диаграмме, в том числе:
     1. {3,3 ^5,1 } — 8-демикуб или демиоктеракт , 1 ₅₁ — ; также как h{4,3 ⁶ } .
     2. {3,3,3,3,3,3 ^1,1 } - 8-ортоплекс , 5 ₁₁ -
4. Семейство E-многогранников Семейство E ₈ : [3 ^4,1,1 ] -
   • 255 однородных 8-многогранников как перестановок колец групповой диаграммы, в том числе:
     1. {3,3,3,3,3 ^2,1 } — Торольда Госсета полурегулярное 4 ₂₁ ,
     2. {3,3 ^4,2 } — униформа 1 ₄₂ , ,
     3. {3,3,3 ^4,1 } — униформа 2 ₄₁ ,

Равномерные призматические формы

Существует множество однородных призматических семейств, в том числе:

showСемейства однородных 8-многогранных призм
#	Coxeter group		Coxeter-Dynkin diagram
7+1
1	A7A1	[3,3,3,3,3,3]×[ ]
2	B7A1	[4,3,3,3,3,3]×[ ]
3	D7A1	[34,1,1]×[ ]
4	E7A1	[33,2,1]×[ ]
6+2
1	A6I2(p)	[3,3,3,3,3]×[p]
2	B6I2(p)	[4,3,3,3,3]×[p]
3	D6I2(p)	[33,1,1]×[p]
4	E6I2(p)	[3,3,3,3,3]×[p]
6+1+1
1	A6A1A1	[3,3,3,3,3]×[ ]x[ ]
2	B6A1A1	[4,3,3,3,3]×[ ]x[ ]
3	D6A1A1	[33,1,1]×[ ]x[ ]
4	E6A1A1	[3,3,3,3,3]×[ ]x[ ]
5+3
1	A5A3	[34]×[3,3]
2	B5A3	[4,33]×[3,3]
3	D5A3	[32,1,1]×[3,3]
4	A5B3	[34]×[4,3]
5	B5B3	[4,33]×[4,3]
6	D5B3	[32,1,1]×[4,3]
7	A5H3	[34]×[5,3]
8	B5H3	[4,33]×[5,3]
9	D5H3	[32,1,1]×[5,3]
5+2+1
1	A5I2(p)A1	[3,3,3]×[p]×[ ]
2	B5I2(p)A1	[4,3,3]×[p]×[ ]
3	D5I2(p)A1	[32,1,1]×[p]×[ ]
5+1+1+1
1	A5A1A1A1	[3,3,3]×[ ]×[ ]×[ ]
2	B5A1A1A1	[4,3,3]×[ ]×[ ]×[ ]
3	D5A1A1A1	[32,1,1]×[ ]×[ ]×[ ]
4+4
1	A4A4	[3,3,3]×[3,3,3]
2	B4A4	[4,3,3]×[3,3,3]
3	D4A4	[31,1,1]×[3,3,3]
4	F4A4	[3,4,3]×[3,3,3]
5	H4A4	[5,3,3]×[3,3,3]
6	B4B4	[4,3,3]×[4,3,3]
7	D4B4	[31,1,1]×[4,3,3]
8	F4B4	[3,4,3]×[4,3,3]
9	H4B4	[5,3,3]×[4,3,3]
10	D4D4	[31,1,1]×[31,1,1]
11	F4D4	[3,4,3]×[31,1,1]
12	H4D4	[5,3,3]×[31,1,1]
13	F4×F4	[3,4,3]×[3,4,3]
14	H4×F4	[5,3,3]×[3,4,3]
15	H4H4	[5,3,3]×[5,3,3]
4+3+1
1	A4A3A1	[3,3,3]×[3,3]×[ ]
2	A4B3A1	[3,3,3]×[4,3]×[ ]
3	A4H3A1	[3,3,3]×[5,3]×[ ]
4	B4A3A1	[4,3,3]×[3,3]×[ ]
5	B4B3A1	[4,3,3]×[4,3]×[ ]
6	B4H3A1	[4,3,3]×[5,3]×[ ]
7	H4A3A1	[5,3,3]×[3,3]×[ ]
8	H4B3A1	[5,3,3]×[4,3]×[ ]
9	H4H3A1	[5,3,3]×[5,3]×[ ]
10	F4A3A1	[3,4,3]×[3,3]×[ ]
11	F4B3A1	[3,4,3]×[4,3]×[ ]
12	F4H3A1	[3,4,3]×[5,3]×[ ]
13	D4A3A1	[31,1,1]×[3,3]×[ ]
14	D4B3A1	[31,1,1]×[4,3]×[ ]
15	D4H3A1	[31,1,1]×[5,3]×[ ]
4+2+2
...
4+2+1+1
...
4+1+1+1+1
...
3+3+2
1	A3A3I2(p)	[3,3]×[3,3]×[p]
2	B3A3I2(p)	[4,3]×[3,3]×[p]
3	H3A3I2(p)	[5,3]×[3,3]×[p]
4	B3B3I2(p)	[4,3]×[4,3]×[p]
5	H3B3I2(p)	[5,3]×[4,3]×[p]
6	H3H3I2(p)	[5,3]×[5,3]×[p]
3+3+1+1
1	A32A12	[3,3]×[3,3]×[ ]×[ ]
2	B3A3A12	[4,3]×[3,3]×[ ]×[ ]
3	H3A3A12	[5,3]×[3,3]×[ ]×[ ]
4	B3B3A12	[4,3]×[4,3]×[ ]×[ ]
5	H3B3A12	[5,3]×[4,3]×[ ]×[ ]
6	H3H3A12	[5,3]×[5,3]×[ ]×[ ]
3+2+2+1
1	A3I2(p)I2(q)A1	[3,3]×[p]×[q]×[ ]
2	B3I2(p)I2(q)A1	[4,3]×[p]×[q]×[ ]
3	H3I2(p)I2(q)A1	[5,3]×[p]×[q]×[ ]
3+2+1+1+1
1	A3I2(p)A13	[3,3]×[p]×[ ]x[ ]×[ ]
2	B3I2(p)A13	[4,3]×[p]×[ ]x[ ]×[ ]
3	H3I2(p)A13	[5,3]×[p]×[ ]x[ ]×[ ]
3+1+1+1+1+1
1	A3A15	[3,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2	B3A15	[4,3]×[ ]x[ ]×[ ]x[ ]×[ ]
3	H3A15	[5,3]×[ ]x[ ]×[ ]x[ ]×[ ]
2+2+2+2
1	I2(p)I2(q)I2(r)I2(s)	[p]×[q]×[r]×[s]
2+2+2+1+1
1	I2(p)I2(q)I2(r)A12	[p]×[q]×[r]×[ ]×[ ]
2+2+1+1+1+1
2	I2(p)I2(q)A14	[p]×[q]×[ ]×[ ]×[ ]×[ ]
2+1+1+1+1+1+1
1	I2(p)A16	[p]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]
1+1+1+1+1+1+1+1
1	A18	[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]×[ ]

А ₈ Семья

Семейство A ₈ имеет симметрию порядка 362880 (9- факториал ).

Существует 135 форм, основанных на всех перестановках диаграмм Кокстера-Динкина с одним или несколькими кольцами. (128+8-1 случаев) Все они перечислены ниже. Названия аббревиатур в стиле Бауэрса приведены в скобках для перекрестных ссылок.

См. также список 8-симплексных многогранников для симметричных плоских графов Кокстера этих многогранников.

show8 многогранников однородных
#	Coxeter-Dynkin diagram	Truncation indices	Johnson name	Basepoint	Element counts
					7	6	5	4	3	2	1	0
1		t0	8-simplex (ene)	(0,0,0,0,0,0,0,0,1)	9	36	84	126	126	84	36	9
2		t1	Rectified 8-simplex (rene)	(0,0,0,0,0,0,0,1,1)	18	108	336	630	576	588	252	36
3		t2	Birectified 8-simplex (bene)	(0,0,0,0,0,0,1,1,1)	18	144	588	1386	2016	1764	756	84
4		t3	Trirectified 8-simplex (trene)	(0,0,0,0,0,1,1,1,1)							1260	126
5		t0,1	Truncated 8-simplex (tene)	(0,0,0,0,0,0,0,1,2)							288	72
6		t0,2	Cantellated 8-simplex	(0,0,0,0,0,0,1,1,2)							1764	252
7		t1,2	Bitruncated 8-simplex	(0,0,0,0,0,0,1,2,2)							1008	252
8		t0,3	Runcinated 8-simplex	(0,0,0,0,0,1,1,1,2)							4536	504
9		t1,3	Bicantellated 8-simplex	(0,0,0,0,0,1,1,2,2)							5292	756
10		t2,3	Tritruncated 8-simplex	(0,0,0,0,0,1,2,2,2)							2016	504
11		t0,4	Stericated 8-simplex	(0,0,0,0,1,1,1,1,2)							6300	630
12		t1,4	Biruncinated 8-simplex	(0,0,0,0,1,1,1,2,2)							11340	1260
13		t2,4	Tricantellated 8-simplex	(0,0,0,0,1,1,2,2,2)							8820	1260
14		t3,4	Quadritruncated 8-simplex	(0,0,0,0,1,2,2,2,2)							2520	630
15		t0,5	Pentellated 8-simplex	(0,0,0,1,1,1,1,1,2)							5040	504
16		t1,5	Bistericated 8-simplex	(0,0,0,1,1,1,1,2,2)							12600	1260
17		t2,5	Triruncinated 8-simplex	(0,0,0,1,1,1,2,2,2)							15120	1680
18		t0,6	Hexicated 8-simplex	(0,0,1,1,1,1,1,1,2)							2268	252
19		t1,6	Bipentellated 8-simplex	(0,0,1,1,1,1,1,2,2)							7560	756
20		t0,7	Heptellated 8-simplex	(0,1,1,1,1,1,1,1,2)							504	72
21		t0,1,2	Cantitruncated 8-simplex	(0,0,0,0,0,0,1,2,3)							2016	504
22		t0,1,3	Runcitruncated 8-simplex	(0,0,0,0,0,1,1,2,3)							9828	1512
23		t0,2,3	Runcicantellated 8-simplex	(0,0,0,0,0,1,2,2,3)							6804	1512
24		t1,2,3	Bicantitruncated 8-simplex	(0,0,0,0,0,1,2,3,3)							6048	1512
25		t0,1,4	Steritruncated 8-simplex	(0,0,0,0,1,1,1,2,3)							20160	2520
26		t0,2,4	Stericantellated 8-simplex	(0,0,0,0,1,1,2,2,3)							26460	3780
27		t1,2,4	Biruncitruncated 8-simplex	(0,0,0,0,1,1,2,3,3)							22680	3780
28		t0,3,4	Steriruncinated 8-simplex	(0,0,0,0,1,2,2,2,3)							12600	2520
29		t1,3,4	Biruncicantellated 8-simplex	(0,0,0,0,1,2,2,3,3)							18900	3780
30		t2,3,4	Tricantitruncated 8-simplex	(0,0,0,0,1,2,3,3,3)							10080	2520
31		t0,1,5	Pentitruncated 8-simplex	(0,0,0,1,1,1,1,2,3)							21420	2520
32		t0,2,5	Penticantellated 8-simplex	(0,0,0,1,1,1,2,2,3)							42840	5040
33		t1,2,5	Bisteritruncated 8-simplex	(0,0,0,1,1,1,2,3,3)							35280	5040
34		t0,3,5	Pentiruncinated 8-simplex	(0,0,0,1,1,2,2,2,3)							37800	5040
35		t1,3,5	Bistericantellated 8-simplex	(0,0,0,1,1,2,2,3,3)							52920	7560
36		t2,3,5	Triruncitruncated 8-simplex	(0,0,0,1,1,2,3,3,3)							27720	5040
37		t0,4,5	Pentistericated 8-simplex	(0,0,0,1,2,2,2,2,3)							13860	2520
38		t1,4,5	Bisteriruncinated 8-simplex	(0,0,0,1,2,2,2,3,3)							30240	5040
39		t0,1,6	Hexitruncated 8-simplex	(0,0,1,1,1,1,1,2,3)							12096	1512
40		t0,2,6	Hexicantellated 8-simplex	(0,0,1,1,1,1,2,2,3)							34020	3780
41		t1,2,6	Bipentitruncated 8-simplex	(0,0,1,1,1,1,2,3,3)							26460	3780
42		t0,3,6	Hexiruncinated 8-simplex	(0,0,1,1,1,2,2,2,3)							45360	5040
43		t1,3,6	Bipenticantellated 8-simplex	(0,0,1,1,1,2,2,3,3)							60480	7560
44		t0,4,6	Hexistericated 8-simplex	(0,0,1,1,2,2,2,2,3)							30240	3780
45		t0,5,6	Hexipentellated 8-simplex	(0,0,1,2,2,2,2,2,3)							9072	1512
46		t0,1,7	Heptitruncated 8-simplex	(0,1,1,1,1,1,1,2,3)							3276	504
47		t0,2,7	Hepticantellated 8-simplex	(0,1,1,1,1,1,2,2,3)							12852	1512
48		t0,3,7	Heptiruncinated 8-simplex	(0,1,1,1,1,2,2,2,3)							23940	2520
49		t0,1,2,3	Runcicantitruncated 8-simplex	(0,0,0,0,0,1,2,3,4)							12096	3024
50		t0,1,2,4	Stericantitruncated 8-simplex	(0,0,0,0,1,1,2,3,4)							45360	7560
51		t0,1,3,4	Steriruncitruncated 8-simplex	(0,0,0,0,1,2,2,3,4)							34020	7560
52		t0,2,3,4	Steriruncicantellated 8-simplex	(0,0,0,0,1,2,3,3,4)							34020	7560
53		t1,2,3,4	Biruncicantitruncated 8-simplex	(0,0,0,0,1,2,3,4,4)							30240	7560
54		t0,1,2,5	Penticantitruncated 8-simplex	(0,0,0,1,1,1,2,3,4)							70560	10080
55		t0,1,3,5	Pentiruncitruncated 8-simplex	(0,0,0,1,1,2,2,3,4)							98280	15120
56		t0,2,3,5	Pentiruncicantellated 8-simplex	(0,0,0,1,1,2,3,3,4)							90720	15120
57		t1,2,3,5	Bistericantitruncated 8-simplex	(0,0,0,1,1,2,3,4,4)							83160	15120
58		t0,1,4,5	Pentisteritruncated 8-simplex	(0,0,0,1,2,2,2,3,4)							50400	10080
59		t0,2,4,5	Pentistericantellated 8-simplex	(0,0,0,1,2,2,3,3,4)							83160	15120
60		t1,2,4,5	Bisteriruncitruncated 8-simplex	(0,0,0,1,2,2,3,4,4)							68040	15120
61		t0,3,4,5	Pentisteriruncinated 8-simplex	(0,0,0,1,2,3,3,3,4)							50400	10080
62		t1,3,4,5	Bisteriruncicantellated 8-simplex	(0,0,0,1,2,3,3,4,4)							75600	15120
63		t2,3,4,5	Triruncicantitruncated 8-simplex	(0,0,0,1,2,3,4,4,4)							40320	10080
64		t0,1,2,6	Hexicantitruncated 8-simplex	(0,0,1,1,1,1,2,3,4)							52920	7560
65		t0,1,3,6	Hexiruncitruncated 8-simplex	(0,0,1,1,1,2,2,3,4)							113400	15120
66		t0,2,3,6	Hexiruncicantellated 8-simplex	(0,0,1,1,1,2,3,3,4)							98280	15120
67		t1,2,3,6	Bipenticantitruncated 8-simplex	(0,0,1,1,1,2,3,4,4)							90720	15120
68		t0,1,4,6	Hexisteritruncated 8-simplex	(0,0,1,1,2,2,2,3,4)							105840	15120
69		t0,2,4,6	Hexistericantellated 8-simplex	(0,0,1,1,2,2,3,3,4)							158760	22680
70		t1,2,4,6	Bipentiruncitruncated 8-simplex	(0,0,1,1,2,2,3,4,4)							136080	22680
71		t0,3,4,6	Hexisteriruncinated 8-simplex	(0,0,1,1,2,3,3,3,4)							90720	15120
72		t1,3,4,6	Bipentiruncicantellated 8-simplex	(0,0,1,1,2,3,3,4,4)							136080	22680
73		t0,1,5,6	Hexipentitruncated 8-simplex	(0,0,1,2,2,2,2,3,4)							41580	7560
74		t0,2,5,6	Hexipenticantellated 8-simplex	(0,0,1,2,2,2,3,3,4)							98280	15120
75		t1,2,5,6	Bipentisteritruncated 8-simplex	(0,0,1,2,2,2,3,4,4)							75600	15120
76		t0,3,5,6	Hexipentiruncinated 8-simplex	(0,0,1,2,2,3,3,3,4)							98280	15120
77		t0,4,5,6	Hexipentistericated 8-simplex	(0,0,1,2,3,3,3,3,4)							41580	7560
78		t0,1,2,7	Hepticantitruncated 8-simplex	(0,1,1,1,1,1,2,3,4)							18144	3024
79		t0,1,3,7	Heptiruncitruncated 8-simplex	(0,1,1,1,1,2,2,3,4)							56700	7560
80		t0,2,3,7	Heptiruncicantellated 8-simplex	(0,1,1,1,1,2,3,3,4)							45360	7560
81		t0,1,4,7	Heptisteritruncated 8-simplex	(0,1,1,1,2,2,2,3,4)							80640	10080
82		t0,2,4,7	Heptistericantellated 8-simplex	(0,1,1,1,2,2,3,3,4)							113400	15120
83		t0,3,4,7	Heptisteriruncinated 8-simplex	(0,1,1,1,2,3,3,3,4)							60480	10080
84		t0,1,5,7	Heptipentitruncated 8-simplex	(0,1,1,2,2,2,2,3,4)							56700	7560
85		t0,2,5,7	Heptipenticantellated 8-simplex	(0,1,1,2,2,2,3,3,4)							120960	15120
86		t0,1,6,7	Heptihexitruncated 8-simplex	(0,1,2,2,2,2,2,3,4)							18144	3024
87		t0,1,2,3,4	Steriruncicantitruncated 8-simplex	(0,0,0,0,1,2,3,4,5)							60480	15120
88		t0,1,2,3,5	Pentiruncicantitruncated 8-simplex	(0,0,0,1,1,2,3,4,5)							166320	30240
89		t0,1,2,4,5	Pentistericantitruncated 8-simplex	(0,0,0,1,2,2,3,4,5)							136080	30240
90		t0,1,3,4,5	Pentisteriruncitruncated 8-simplex	(0,0,0,1,2,3,3,4,5)							136080	30240
91		t0,2,3,4,5	Pentisteriruncicantellated 8-simplex	(0,0,0,1,2,3,4,4,5)							136080	30240
92		t1,2,3,4,5	Bisteriruncicantitruncated 8-simplex	(0,0,0,1,2,3,4,5,5)							120960	30240
93		t0,1,2,3,6	Hexiruncicantitruncated 8-simplex	(0,0,1,1,1,2,3,4,5)							181440	30240
94		t0,1,2,4,6	Hexistericantitruncated 8-simplex	(0,0,1,1,2,2,3,4,5)							272160	45360
95		t0,1,3,4,6	Hexisteriruncitruncated 8-simplex	(0,0,1,1,2,3,3,4,5)							249480	45360
96		t0,2,3,4,6	Hexisteriruncicantellated 8-simplex	(0,0,1,1,2,3,4,4,5)							249480	45360
97		t1,2,3,4,6	Bipentiruncicantitruncated 8-simplex	(0,0,1,1,2,3,4,5,5)							226800	45360
98		t0,1,2,5,6	Hexipenticantitruncated 8-simplex	(0,0,1,2,2,2,3,4,5)							151200	30240
99		t0,1,3,5,6	Hexipentiruncitruncated 8-simplex	(0,0,1,2,2,3,3,4,5)							249480	45360
100		t0,2,3,5,6	Hexipentiruncicantellated 8-simplex	(0,0,1,2,2,3,4,4,5)							226800	45360
101		t1,2,3,5,6	Bipentistericantitruncated 8-simplex	(0,0,1,2,2,3,4,5,5)							204120	45360
102		t0,1,4,5,6	Hexipentisteritruncated 8-simplex	(0,0,1,2,3,3,3,4,5)							151200	30240
103		t0,2,4,5,6	Hexipentistericantellated 8-simplex	(0,0,1,2,3,3,4,4,5)							249480	45360
104		t0,3,4,5,6	Hexipentisteriruncinated 8-simplex	(0,0,1,2,3,4,4,4,5)							151200	30240
105		t0,1,2,3,7	Heptiruncicantitruncated 8-simplex	(0,1,1,1,1,2,3,4,5)							83160	15120
106		t0,1,2,4,7	Heptistericantitruncated 8-simplex	(0,1,1,1,2,2,3,4,5)							196560	30240
107		t0,1,3,4,7	Heptisteriruncitruncated 8-simplex	(0,1,1,1,2,3,3,4,5)							166320	30240
108		t0,2,3,4,7	Heptisteriruncicantellated 8-simplex	(0,1,1,1,2,3,4,4,5)							166320	30240
109		t0,1,2,5,7	Heptipenticantitruncated 8-simplex	(0,1,1,2,2,2,3,4,5)							196560	30240
110		t0,1,3,5,7	Heptipentiruncitruncated 8-simplex	(0,1,1,2,2,3,3,4,5)							294840	45360
111		t0,2,3,5,7	Heptipentiruncicantellated 8-simplex	(0,1,1,2,2,3,4,4,5)							272160	45360
112		t0,1,4,5,7	Heptipentisteritruncated 8-simplex	(0,1,1,2,3,3,3,4,5)							166320	30240
113		t0,1,2,6,7	Heptihexicantitruncated 8-simplex	(0,1,2,2,2,2,3,4,5)							83160	15120
114		t0,1,3,6,7	Heptihexiruncitruncated 8-simplex	(0,1,2,2,2,3,3,4,5)							196560	30240
115		t0,1,2,3,4,5	Pentisteriruncicantitruncated 8-simplex	(0,0,0,1,2,3,4,5,6)							241920	60480
116		t0,1,2,3,4,6	Hexisteriruncicantitruncated 8-simplex	(0,0,1,1,2,3,4,5,6)							453600	90720
117		t0,1,2,3,5,6	Hexipentiruncicantitruncated 8-simplex	(0,0,1,2,2,3,4,5,6)							408240	90720
118		t0,1,2,4,5,6	Hexipentistericantitruncated 8-simplex	(0,0,1,2,3,3,4,5,6)							408240	90720
119		t0,1,3,4,5,6	Hexipentisteriruncitruncated 8-simplex	(0,0,1,2,3,4,4,5,6)							408240	90720
120		t0,2,3,4,5,6	Hexipentisteriruncicantellated 8-simplex	(0,0,1,2,3,4,5,5,6)							408240	90720
121		t1,2,3,4,5,6	Bipentisteriruncicantitruncated 8-simplex	(0,0,1,2,3,4,5,6,6)							362880	90720
122		t0,1,2,3,4,7	Heptisteriruncicantitruncated 8-simplex	(0,1,1,1,2,3,4,5,6)							302400	60480
123		t0,1,2,3,5,7	Heptipentiruncicantitruncated 8-simplex	(0,1,1,2,2,3,4,5,6)							498960	90720
124		t0,1,2,4,5,7	Heptipentistericantitruncated 8-simplex	(0,1,1,2,3,3,4,5,6)							453600	90720
125		t0,1,3,4,5,7	Heptipentisteriruncitruncated 8-simplex	(0,1,1,2,3,4,4,5,6)							453600	90720
126		t0,2,3,4,5,7	Heptipentisteriruncicantellated 8-simplex	(0,1,1,2,3,4,5,5,6)							453600	90720
127		t0,1,2,3,6,7	Heptihexiruncicantitruncated 8-simplex	(0,1,2,2,2,3,4,5,6)							302400	60480
128		t0,1,2,4,6,7	Heptihexistericantitruncated 8-simplex	(0,1,2,2,3,3,4,5,6)							498960	90720
129		t0,1,3,4,6,7	Heptihexisteriruncitruncated 8-simplex	(0,1,2,2,3,4,4,5,6)							453600	90720
130		t0,1,2,5,6,7	Heptihexipenticantitruncated 8-simplex	(0,1,2,3,3,3,4,5,6)							302400	60480
131		t0,1,2,3,4,5,6	Hexipentisteriruncicantitruncated 8-simplex	(0,0,1,2,3,4,5,6,7)							725760	181440
132		t0,1,2,3,4,5,7	Heptipentisteriruncicantitruncated 8-simplex	(0,1,1,2,3,4,5,6,7)							816480	181440
133		t0,1,2,3,4,6,7	Heptihexisteriruncicantitruncated 8-simplex	(0,1,2,2,3,4,5,6,7)							816480	181440
134		t0,1,2,3,5,6,7	Heptihexipentiruncicantitruncated 8-simplex	(0,1,2,3,3,4,5,6,7)							816480	181440
135		t0,1,2,3,4,5,6,7	Omnitruncated 8-simplex	(0,1,2,3,4,5,6,7,8)							1451520	362880

Б ₈ Семья

Семейство B ₈ имеет симметрию порядка 10321920 (8 факториал x 2 ⁸ ). Существует 255 форм, основанных на всех перестановках диаграмм Кокстера-Динкина с одним или несколькими кольцами.

См. также список многогранников B8 для симметричных плоских графов Кокстера этих многогранников.

showB 8 однородных многогранников
#	Coxeter-Dynkin diagram	Schläfli symbol	Name	Element counts
				7	6	5	4	3	2	1	0
1		t0{36,4}	8-orthoplex Diacosipentacontahexazetton (ek)	256	1024	1792	1792	1120	448	112	16
2		t1{36,4}	Rectified 8-orthoplex Rectified diacosipentacontahexazetton (rek)	272	3072	8960	12544	10080	4928	1344	112
3		t2{36,4}	Birectified 8-orthoplex Birectified diacosipentacontahexazetton (bark)	272	3184	16128	34048	36960	22400	6720	448
4		t3{36,4}	Trirectified 8-orthoplex Trirectified diacosipentacontahexazetton (tark)	272	3184	16576	48384	71680	53760	17920	1120
5		t3{4,36}	Trirectified 8-cube Trirectified octeract (tro)	272	3184	16576	47712	80640	71680	26880	1792
6		t2{4,36}	Birectified 8-cube Birectified octeract (bro)	272	3184	14784	36960	55552	50176	21504	1792
7		t1{4,36}	Rectified 8-cube Rectified octeract (recto)	272	2160	7616	15456	19712	16128	7168	1024
8		t0{4,36}	8-cube Octeract (octo)	16	112	448	1120	1792	1792	1024	256
9		t0,1{36,4}	Truncated 8-orthoplex Truncated diacosipentacontahexazetton (tek)							1456	224
10		t0,2{36,4}	Cantellated 8-orthoplex Small rhombated diacosipentacontahexazetton (srek)							14784	1344
11		t1,2{36,4}	Bitruncated 8-orthoplex Bitruncated diacosipentacontahexazetton (batek)							8064	1344
12		t0,3{36,4}	Runcinated 8-orthoplex Small prismated diacosipentacontahexazetton (spek)							60480	4480
13		t1,3{36,4}	Bicantellated 8-orthoplex Small birhombated diacosipentacontahexazetton (sabork)							67200	6720
14		t2,3{36,4}	Tritruncated 8-orthoplex Tritruncated diacosipentacontahexazetton (tatek)							24640	4480
15		t0,4{36,4}	Stericated 8-orthoplex Small cellated diacosipentacontahexazetton (scak)							125440	8960
16		t1,4{36,4}	Biruncinated 8-orthoplex Small biprismated diacosipentacontahexazetton (sabpek)							215040	17920
17		t2,4{36,4}	Tricantellated 8-orthoplex Small trirhombated diacosipentacontahexazetton (satrek)							161280	17920
18		t3,4{4,36}	Quadritruncated 8-cube Octeractidiacosipentacontahexazetton (oke)							44800	8960
19		t0,5{36,4}	Pentellated 8-orthoplex Small terated diacosipentacontahexazetton (setek)							134400	10752
20		t1,5{36,4}	Bistericated 8-orthoplex Small bicellated diacosipentacontahexazetton (sibcak)							322560	26880
21		t2,5{4,36}	Triruncinated 8-cube Small triprismato-octeractidiacosipentacontahexazetton (sitpoke)							376320	35840
22		t2,4{4,36}	Tricantellated 8-cube Small trirhombated octeract (satro)							215040	26880
23		t2,3{4,36}	Tritruncated 8-cube Tritruncated octeract (tato)							48384	10752
24		t0,6{36,4}	Hexicated 8-orthoplex Small petated diacosipentacontahexazetton (supek)							64512	7168
25		t1,6{4,36}	Bipentellated 8-cube Small biteri-octeractidiacosipentacontahexazetton (sabtoke)							215040	21504
26		t1,5{4,36}	Bistericated 8-cube Small bicellated octeract (sobco)							358400	35840
27		t1,4{4,36}	Biruncinated 8-cube Small biprismated octeract (sabepo)							322560	35840
28		t1,3{4,36}	Bicantellated 8-cube Small birhombated octeract (subro)							150528	21504
29		t1,2{4,36}	Bitruncated 8-cube Bitruncated octeract (bato)							28672	7168
30		t0,7{4,36}	Heptellated 8-cube Small exi-octeractidiacosipentacontahexazetton (saxoke)							14336	2048
31		t0,6{4,36}	Hexicated 8-cube Small petated octeract (supo)							64512	7168
32		t0,5{4,36}	Pentellated 8-cube Small terated octeract (soto)							143360	14336
33		t0,4{4,36}	Stericated 8-cube Small cellated octeract (soco)							179200	17920
34		t0,3{4,36}	Runcinated 8-cube Small prismated octeract (sopo)							129024	14336
35		t0,2{4,36}	Cantellated 8-cube Small rhombated octeract (soro)							50176	7168
36		t0,1{4,36}	Truncated 8-cube Truncated octeract (tocto)							8192	2048
37		t0,1,2{36,4}	Cantitruncated 8-orthoplex Great rhombated diacosipentacontahexazetton							16128	2688
38		t0,1,3{36,4}	Runcitruncated 8-orthoplex Prismatotruncated diacosipentacontahexazetton							127680	13440
39		t0,2,3{36,4}	Runcicantellated 8-orthoplex Prismatorhombated diacosipentacontahexazetton							80640	13440
40		t1,2,3{36,4}	Bicantitruncated 8-orthoplex Great birhombated diacosipentacontahexazetton							73920	13440
41		t0,1,4{36,4}	Steritruncated 8-orthoplex Cellitruncated diacosipentacontahexazetton							394240	35840
42		t0,2,4{36,4}	Stericantellated 8-orthoplex Cellirhombated diacosipentacontahexazetton							483840	53760
43		t1,2,4{36,4}	Biruncitruncated 8-orthoplex Biprismatotruncated diacosipentacontahexazetton							430080	53760
44		t0,3,4{36,4}	Steriruncinated 8-orthoplex Celliprismated diacosipentacontahexazetton							215040	35840
45		t1,3,4{36,4}	Biruncicantellated 8-orthoplex Biprismatorhombated diacosipentacontahexazetton							322560	53760
46		t2,3,4{36,4}	Tricantitruncated 8-orthoplex Great trirhombated diacosipentacontahexazetton							179200	35840
47		t0,1,5{36,4}	Pentitruncated 8-orthoplex Teritruncated diacosipentacontahexazetton							564480	53760
48		t0,2,5{36,4}	Penticantellated 8-orthoplex Terirhombated diacosipentacontahexazetton							1075200	107520
49		t1,2,5{36,4}	Bisteritruncated 8-orthoplex Bicellitruncated diacosipentacontahexazetton							913920	107520
50		t0,3,5{36,4}	Pentiruncinated 8-orthoplex Teriprismated diacosipentacontahexazetton							913920	107520
51		t1,3,5{36,4}	Bistericantellated 8-orthoplex Bicellirhombated diacosipentacontahexazetton							1290240	161280
52		t2,3,5{36,4}	Triruncitruncated 8-orthoplex Triprismatotruncated diacosipentacontahexazetton							698880	107520
53		t0,4,5{36,4}	Pentistericated 8-orthoplex Tericellated diacosipentacontahexazetton							322560	53760
54		t1,4,5{36,4}	Bisteriruncinated 8-orthoplex Bicelliprismated diacosipentacontahexazetton							698880	107520
55		t2,3,5{4,36}	Triruncitruncated 8-cube Triprismatotruncated octeract							645120	107520
56		t2,3,4{4,36}	Tricantitruncated 8-cube Great trirhombated octeract							241920	53760
57		t0,1,6{36,4}	Hexitruncated 8-orthoplex Petitruncated diacosipentacontahexazetton							344064	43008
58		t0,2,6{36,4}	Hexicantellated 8-orthoplex Petirhombated diacosipentacontahexazetton							967680	107520
59		t1,2,6{36,4}	Bipentitruncated 8-orthoplex Biteritruncated diacosipentacontahexazetton							752640	107520
60		t0,3,6{36,4}	Hexiruncinated 8-orthoplex Petiprismated diacosipentacontahexazetton							1290240	143360
61		t1,3,6{36,4}	Bipenticantellated 8-orthoplex Biterirhombated diacosipentacontahexazetton							1720320	215040
62		t1,4,5{4,36}	Bisteriruncinated 8-cube Bicelliprismated octeract							860160	143360
63		t0,4,6{36,4}	Hexistericated 8-orthoplex Peticellated diacosipentacontahexazetton							860160	107520
64		t1,3,6{4,36}	Bipenticantellated 8-cube Biterirhombated octeract							1720320	215040
65		t1,3,5{4,36}	Bistericantellated 8-cube Bicellirhombated octeract							1505280	215040
66		t1,3,4{4,36}	Biruncicantellated 8-cube Biprismatorhombated octeract							537600	107520
67		t0,5,6{36,4}	Hexipentellated 8-orthoplex Petiterated diacosipentacontahexazetton							258048	43008
68		t1,2,6{4,36}	Bipentitruncated 8-cube Biteritruncated octeract							752640	107520
69		t1,2,5{4,36}	Bisteritruncated 8-cube Bicellitruncated octeract							1003520	143360
70		t1,2,4{4,36}	Biruncitruncated 8-cube Biprismatotruncated octeract							645120	107520
71		t1,2,3{4,36}	Bicantitruncated 8-cube Great birhombated octeract							172032	43008
72		t0,1,7{36,4}	Heptitruncated 8-orthoplex Exitruncated diacosipentacontahexazetton							93184	14336
73		t0,2,7{36,4}	Hepticantellated 8-orthoplex Exirhombated diacosipentacontahexazetton							365568	43008
74		t0,5,6{4,36}	Hexipentellated 8-cube Petiterated octeract							258048	43008
75		t0,3,7{36,4}	Heptiruncinated 8-orthoplex Exiprismated diacosipentacontahexazetton							680960	71680
76		t0,4,6{4,36}	Hexistericated 8-cube Peticellated octeract							860160	107520
77		t0,4,5{4,36}	Pentistericated 8-cube Tericellated octeract							394240	71680
78		t0,3,7{4,36}	Heptiruncinated 8-cube Exiprismated octeract							680960	71680
79		t0,3,6{4,36}	Hexiruncinated 8-cube Petiprismated octeract							1290240	143360
80		t0,3,5{4,36}	Pentiruncinated 8-cube Teriprismated octeract							1075200	143360
81		t0,3,4{4,36}	Steriruncinated 8-cube Celliprismated octeract							358400	71680
82		t0,2,7{4,36}	Hepticantellated 8-cube Exirhombated octeract							365568	43008
83		t0,2,6{4,36}	Hexicantellated 8-cube Petirhombated octeract							967680	107520
84		t0,2,5{4,36}	Penticantellated 8-cube Terirhombated octeract							1218560	143360
85		t0,2,4{4,36}	Stericantellated 8-cube Cellirhombated octeract							752640	107520
86		t0,2,3{4,36}	Runcicantellated 8-cube Prismatorhombated octeract							193536	43008
87		t0,1,7{4,36}	Heptitruncated 8-cube Exitruncated octeract							93184	14336
88		t0,1,6{4,36}	Hexitruncated 8-cube Petitruncated octeract							344064	43008
89		t0,1,5{4,36}	Pentitruncated 8-cube Teritruncated octeract							609280	71680
90		t0,1,4{4,36}	Steritruncated 8-cube Cellitruncated octeract							573440	71680
91		t0,1,3{4,36}	Runcitruncated 8-cube Prismatotruncated octeract							279552	43008
92		t0,1,2{4,36}	Cantitruncated 8-cube Great rhombated octeract							57344	14336
93		t0,1,2,3{36,4}	Runcicantitruncated 8-orthoplex Great prismated diacosipentacontahexazetton							147840	26880
94		t0,1,2,4{36,4}	Stericantitruncated 8-orthoplex Celligreatorhombated diacosipentacontahexazetton							860160	107520
95		t0,1,3,4{36,4}	Steriruncitruncated 8-orthoplex Celliprismatotruncated diacosipentacontahexazetton							591360	107520
96		t0,2,3,4{36,4}	Steriruncicantellated 8-orthoplex Celliprismatorhombated diacosipentacontahexazetton							591360	107520
97		t1,2,3,4{36,4}	Biruncicantitruncated 8-orthoplex Great biprismated diacosipentacontahexazetton							537600	107520
98		t0,1,2,5{36,4}	Penticantitruncated 8-orthoplex Terigreatorhombated diacosipentacontahexazetton							1827840	215040
99		t0,1,3,5{36,4}	Pentiruncitruncated 8-orthoplex Teriprismatotruncated diacosipentacontahexazetton							2419200	322560
100		t0,2,3,5{36,4}	Pentiruncicantellated 8-orthoplex Teriprismatorhombated diacosipentacontahexazetton							2257920	322560
101		t1,2,3,5{36,4}	Bistericantitruncated 8-orthoplex Bicelligreatorhombated diacosipentacontahexazetton							2096640	322560
102		t0,1,4,5{36,4}	Pentisteritruncated 8-orthoplex Tericellitruncated diacosipentacontahexazetton							1182720	215040
103		t0,2,4,5{36,4}	Pentistericantellated 8-orthoplex Tericellirhombated diacosipentacontahexazetton							1935360	322560
104		t1,2,4,5{36,4}	Bisteriruncitruncated 8-orthoplex Bicelliprismatotruncated diacosipentacontahexazetton							1612800	322560
105		t0,3,4,5{36,4}	Pentisteriruncinated 8-orthoplex Tericelliprismated diacosipentacontahexazetton							1182720	215040
106		t1,3,4,5{36,4}	Bisteriruncicantellated 8-orthoplex Bicelliprismatorhombated diacosipentacontahexazetton							1774080	322560
107		t2,3,4,5{4,36}	Triruncicantitruncated 8-cube Great triprismato-octeractidiacosipentacontahexazetton							967680	215040
108		t0,1,2,6{36,4}	Hexicantitruncated 8-orthoplex Petigreatorhombated diacosipentacontahexazetton							1505280	215040
109		t0,1,3,6{36,4}	Hexiruncitruncated 8-orthoplex Petiprismatotruncated diacosipentacontahexazetton							3225600	430080
110		t0,2,3,6{36,4}	Hexiruncicantellated 8-orthoplex Petiprismatorhombated diacosipentacontahexazetton							2795520	430080
111		t1,2,3,6{36,4}	Bipenticantitruncated 8-orthoplex Biterigreatorhombated diacosipentacontahexazetton							2580480	430080
112		t0,1,4,6{36,4}	Hexisteritruncated 8-orthoplex Peticellitruncated diacosipentacontahexazetton							3010560	430080
113		t0,2,4,6{36,4}	Hexistericantellated 8-orthoplex Peticellirhombated diacosipentacontahexazetton							4515840	645120
114		t1,2,4,6{36,4}	Bipentiruncitruncated 8-orthoplex Biteriprismatotruncated diacosipentacontahexazetton							3870720	645120
115		t0,3,4,6{36,4}	Hexisteriruncinated 8-orthoplex Peticelliprismated diacosipentacontahexazetton							2580480	430080
116		t1,3,4,6{4,36}	Bipentiruncicantellated 8-cube Biteriprismatorhombi-octeractidiacosipentacontahexazetton							3870720	645120
117		t1,3,4,5{4,36}	Bisteriruncicantellated 8-cube Bicelliprismatorhombated octeract							2150400	430080
118		t0,1,5,6{36,4}	Hexipentitruncated 8-orthoplex Petiteritruncated diacosipentacontahexazetton							1182720	215040
119		t0,2,5,6{36,4}	Hexipenticantellated 8-orthoplex Petiterirhombated diacosipentacontahexazetton							2795520	430080
120		t1,2,5,6{4,36}	Bipentisteritruncated 8-cube Bitericellitrunki-octeractidiacosipentacontahexazetton							2150400	430080
121		t0,3,5,6{36,4}	Hexipentiruncinated 8-orthoplex Petiteriprismated diacosipentacontahexazetton							2795520	430080
122		t1,2,4,6{4,36}	Bipentiruncitruncated 8-cube Biteriprismatotruncated octeract							3870720	645120
123		t1,2,4,5{4,36}	Bisteriruncitruncated 8-cube Bicelliprismatotruncated octeract							1935360	430080
124		t0,4,5,6{36,4}	Hexipentistericated 8-orthoplex Petitericellated diacosipentacontahexazetton							1182720	215040
125		t1,2,3,6{4,36}	Bipenticantitruncated 8-cube Biterigreatorhombated octeract							2580480	430080
126		t1,2,3,5{4,36}	Bistericantitruncated 8-cube Bicelligreatorhombated octeract							2365440	430080
127		t1,2,3,4{4,36}	Biruncicantitruncated 8-cube Great biprismated octeract							860160	215040
128		t0,1,2,7{36,4}	Hepticantitruncated 8-orthoplex Exigreatorhombated diacosipentacontahexazetton							516096	86016
129		t0,1,3,7{36,4}	Heptiruncitruncated 8-orthoplex Exiprismatotruncated diacosipentacontahexazetton							1612800	215040
130		t0,2,3,7{36,4}	Heptiruncicantellated 8-orthoplex Exiprismatorhombated diacosipentacontahexazetton							1290240	215040
131		t0,4,5,6{4,36}	Hexipentistericated 8-cube Petitericellated octeract							1182720	215040
132		t0,1,4,7{36,4}	Heptisteritruncated 8-orthoplex Exicellitruncated diacosipentacontahexazetton							2293760	286720
133		t0,2,4,7{36,4}	Heptistericantellated 8-orthoplex Exicellirhombated diacosipentacontahexazetton							3225600	430080
134		t0,3,5,6{4,36}	Hexipentiruncinated 8-cube Petiteriprismated octeract							2795520	430080
135		t0,3,4,7{4,36}	Heptisteriruncinated 8-cube Exicelliprismato-octeractidiacosipentacontahexazetton							1720320	286720
136		t0,3,4,6{4,36}	Hexisteriruncinated 8-cube Peticelliprismated octeract							2580480	430080
137		t0,3,4,5{4,36}	Pentisteriruncinated 8-cube Tericelliprismated octeract							1433600	286720
138		t0,1,5,7{36,4}	Heptipentitruncated 8-orthoplex Exiteritruncated diacosipentacontahexazetton							1612800	215040
139		t0,2,5,7{4,36}	Heptipenticantellated 8-cube Exiterirhombi-octeractidiacosipentacontahexazetton							3440640	430080
140		t0,2,5,6{4,36}	Hexipenticantellated 8-cube Petiterirhombated octeract							2795520	430080
141		t0,2,4,7{4,36}	Heptistericantellated 8-cube Exicellirhombated octeract							3225600	430080
142		t0,2,4,6{4,36}	Hexistericantellated 8-cube Peticellirhombated octeract							4515840	645120
143		t0,2,4,5{4,36}	Pentistericantellated 8-cube Tericellirhombated octeract							2365440	430080
144		t0,2,3,7{4,36}	Heptiruncicantellated 8-cube Exiprismatorhombated octeract							1290240	215040
145		t0,2,3,6{4,36}	Hexiruncicantellated 8-cube Petiprismatorhombated octeract							2795520	430080
146		t0,2,3,5{4,36}	Pentiruncicantellated 8-cube Teriprismatorhombated octeract							2580480	430080
147		t0,2,3,4{4,36}	Steriruncicantellated 8-cube Celliprismatorhombated octeract							967680	215040
148		t0,1,6,7{4,36}	Heptihexitruncated 8-cube Exipetitrunki-octeractidiacosipentacontahexazetton							516096	86016
149		t0,1,5,7{4,36}	Heptipentitruncated 8-cube Exiteritruncated octeract							1612800	215040
150		t0,1,5,6{4,36}	Hexipentitruncated 8-cube Petiteritruncated octeract							1182720	215040
151		t0,1,4,7{4,36}	Heptisteritruncated 8-cube Exicellitruncated octeract							2293760	286720
152		t0,1,4,6{4,36}	Hexisteritruncated 8-cube Peticellitruncated octeract							3010560	430080
153		t0,1,4,5{4,36}	Pentisteritruncated 8-cube Tericellitruncated octeract							1433600	286720
154		t0,1,3,7{4,36}	Heptiruncitruncated 8-cube Exiprismatotruncated octeract							1612800	215040
155		t0,1,3,6{4,36}	Hexiruncitruncated 8-cube Petiprismatotruncated octeract							3225600	430080
156		t0,1,3,5{4,36}	Pentiruncitruncated 8-cube Teriprismatotruncated octeract							2795520	430080
157		t0,1,3,4{4,36}	Steriruncitruncated 8-cube Celliprismatotruncated octeract							967680	215040
158		t0,1,2,7{4,36}	Hepticantitruncated 8-cube Exigreatorhombated octeract							516096	86016
159		t0,1,2,6{4,36}	Hexicantitruncated 8-cube Petigreatorhombated octeract							1505280	215040
160		t0,1,2,5{4,36}	Penticantitruncated 8-cube Terigreatorhombated octeract							2007040	286720
161		t0,1,2,4{4,36}	Stericantitruncated 8-cube Celligreatorhombated octeract							1290240	215040
162		t0,1,2,3{4,36}	Runcicantitruncated 8-cube Great prismated octeract							344064	86016
163		t0,1,2,3,4{36,4}	Steriruncicantitruncated 8-orthoplex Great cellated diacosipentacontahexazetton							1075200	215040
164		t0,1,2,3,5{36,4}	Pentiruncicantitruncated 8-orthoplex Terigreatoprismated diacosipentacontahexazetton							4193280	645120
165		t0,1,2,4,5{36,4}	Pentistericantitruncated 8-orthoplex Tericelligreatorhombated diacosipentacontahexazetton							3225600	645120
166		t0,1,3,4,5{36,4}	Pentisteriruncitruncated 8-orthoplex Tericelliprismatotruncated diacosipentacontahexazetton							3225600	645120
167		t0,2,3,4,5{36,4}	Pentisteriruncicantellated 8-orthoplex Tericelliprismatorhombated diacosipentacontahexazetton							3225600	645120
168		t1,2,3,4,5{36,4}	Bisteriruncicantitruncated 8-orthoplex Great bicellated diacosipentacontahexazetton							2903040	645120
169		t0,1,2,3,6{36,4}	Hexiruncicantitruncated 8-orthoplex Petigreatoprismated diacosipentacontahexazetton							5160960	860160
170		t0,1,2,4,6{36,4}	Hexistericantitruncated 8-orthoplex Peticelligreatorhombated diacosipentacontahexazetton							7741440	1290240
171		t0,1,3,4,6{36,4}	Hexisteriruncitruncated 8-orthoplex Peticelliprismatotruncated diacosipentacontahexazetton							7096320	1290240
172		t0,2,3,4,6{36,4}	Hexisteriruncicantellated 8-orthoplex Peticelliprismatorhombated diacosipentacontahexazetton							7096320	1290240
173		t1,2,3,4,6{36,4}	Bipentiruncicantitruncated 8-orthoplex Biterigreatoprismated diacosipentacontahexazetton							6451200	1290240
174		t0,1,2,5,6{36,4}	Hexipenticantitruncated 8-orthoplex Petiterigreatorhombated diacosipentacontahexazetton							4300800	860160
175		t0,1,3,5,6{36,4}	Hexipentiruncitruncated 8-orthoplex Petiteriprismatotruncated diacosipentacontahexazetton							7096320	1290240
176		t0,2,3,5,6{36,4}	Hexipentiruncicantellated 8-orthoplex Petiteriprismatorhombated diacosipentacontahexazetton							6451200	1290240
177		t1,2,3,5,6{36,4}	Bipentistericantitruncated 8-orthoplex Bitericelligreatorhombated diacosipentacontahexazetton							5806080	1290240
178		t0,1,4,5,6{36,4}	Hexipentisteritruncated 8-orthoplex Petitericellitruncated diacosipentacontahexazetton							4300800	860160
179		t0,2,4,5,6{36,4}	Hexipentistericantellated 8-orthoplex Petitericellirhombated diacosipentacontahexazetton							7096320	1290240
180		t1,2,3,5,6{4,36}	Bipentistericantitruncated 8-cube Bitericelligreatorhombated octeract							5806080	1290240
181		t0,3,4,5,6{36,4}	Hexipentisteriruncinated 8-orthoplex Petitericelliprismated diacosipentacontahexazetton							4300800	860160
182		t1,2,3,4,6{4,36}	Bipentiruncicantitruncated 8-cube Biterigreatoprismated octeract							6451200	1290240
183		t1,2,3,4,5{4,36}	Bisteriruncicantitruncated 8-cube Great bicellated octeract							3440640	860160
184		t0,1,2,3,7{36,4}	Heptiruncicantitruncated 8-orthoplex Exigreatoprismated diacosipentacontahexazetton							2365440	430080
185		t0,1,2,4,7{36,4}	Heptistericantitruncated 8-orthoplex Exicelligreatorhombated diacosipentacontahexazetton							5591040	860160
186		t0,1,3,4,7{36,4}	Heptisteriruncitruncated 8-orthoplex Exicelliprismatotruncated diacosipentacontahexazetton							4730880	860160
187		t0,2,3,4,7{36,4}	Heptisteriruncicantellated 8-orthoplex Exicelliprismatorhombated diacosipentacontahexazetton							4730880	860160
188		t0,3,4,5,6{4,36}	Hexipentisteriruncinated 8-cube Petitericelliprismated octeract							4300800	860160
189		t0,1,2,5,7{36,4}	Heptipenticantitruncated 8-orthoplex Exiterigreatorhombated diacosipentacontahexazetton							5591040	860160
190		t0,1,3,5,7{36,4}	Heptipentiruncitruncated 8-orthoplex Exiteriprismatotruncated diacosipentacontahexazetton							8386560	1290240
191		t0,2,3,5,7{36,4}	Heptipentiruncicantellated 8-orthoplex Exiteriprismatorhombated diacosipentacontahexazetton							7741440	1290240
192		t0,2,4,5,6{4,36}	Hexipentistericantellated 8-cube Petitericellirhombated octeract							7096320	1290240
193		t0,1,4,5,7{36,4}	Heptipentisteritruncated 8-orthoplex Exitericellitruncated diacosipentacontahexazetton							4730880	860160
194		t0,2,3,5,7{4,36}	Heptipentiruncicantellated 8-cube Exiteriprismatorhombated octeract							7741440	1290240
195		t0,2,3,5,6{4,36}	Hexipentiruncicantellated 8-cube Petiteriprismatorhombated octeract							6451200	1290240
196		t0,2,3,4,7{4,36}	Heptisteriruncicantellated 8-cube Exicelliprismatorhombated octeract							4730880	860160
197		t0,2,3,4,6{4,36}	Hexisteriruncicantellated 8-cube Peticelliprismatorhombated octeract							7096320	1290240
198		t0,2,3,4,5{4,36}	Pentisteriruncicantellated 8-cube Tericelliprismatorhombated octeract							3870720	860160
199		t0,1,2,6,7{36,4}	Heptihexicantitruncated 8-orthoplex Exipetigreatorhombated diacosipentacontahexazetton							2365440	430080
200		t0,1,3,6,7{36,4}	Heptihexiruncitruncated 8-orthoplex Exipetiprismatotruncated diacosipentacontahexazetton							5591040	860160
201		t0,1,4,5,7{4,36}	Heptipentisteritruncated 8-cube Exitericellitruncated octeract							4730880	860160
202		t0,1,4,5,6{4,36}	Hexipentisteritruncated 8-cube Petitericellitruncated octeract							4300800	860160
203		t0,1,3,6,7{4,36}	Heptihexiruncitruncated 8-cube Exipetiprismatotruncated octeract							5591040	860160
204		t0,1,3,5,7{4,36}	Heptipentiruncitruncated 8-cube Exiteriprismatotruncated octeract							8386560	1290240
205		t0,1,3,5,6{4,36}	Hexipentiruncitruncated 8-cube Petiteriprismatotruncated octeract							7096320	1290240
206		t0,1,3,4,7{4,36}	Heptisteriruncitruncated 8-cube Exicelliprismatotruncated octeract							4730880	860160
207		t0,1,3,4,6{4,36}	Hexisteriruncitruncated 8-cube Peticelliprismatotruncated octeract							7096320	1290240
208		t0,1,3,4,5{4,36}	Pentisteriruncitruncated 8-cube Tericelliprismatotruncated octeract							3870720	860160
209		t0,1,2,6,7{4,36}	Heptihexicantitruncated 8-cube Exipetigreatorhombated octeract							2365440	430080
210		t0,1,2,5,7{4,36}	Heptipenticantitruncated 8-cube Exiterigreatorhombated octeract							5591040	860160
211		t0,1,2,5,6{4,36}	Hexipenticantitruncated 8-cube Petiterigreatorhombated octeract							4300800	860160
212		t0,1,2,4,7{4,36}	Heptistericantitruncated 8-cube Exicelligreatorhombated octeract							5591040	860160
213		t0,1,2,4,6{4,36}	Hexistericantitruncated 8-cube Peticelligreatorhombated octeract							7741440	1290240
214		t0,1,2,4,5{4,36}	Pentistericantitruncated 8-cube Tericelligreatorhombated octeract							3870720	860160
215		t0,1,2,3,7{4,36}	Heptiruncicantitruncated 8-cube Exigreatoprismated octeract							2365440	430080
216		t0,1,2,3,6{4,36}	Hexiruncicantitruncated 8-cube Petigreatoprismated octeract							5160960	860160
217		t0,1,2,3,5{4,36}	Pentiruncicantitruncated 8-cube Terigreatoprismated octeract							4730880	860160
218		t0,1,2,3,4{4,36}	Steriruncicantitruncated 8-cube Great cellated octeract							1720320	430080
219		t0,1,2,3,4,5{36,4}	Pentisteriruncicantitruncated 8-orthoplex Great terated diacosipentacontahexazetton							5806080	1290240
220		t0,1,2,3,4,6{36,4}	Hexisteriruncicantitruncated 8-orthoplex Petigreatocellated diacosipentacontahexazetton							12902400	2580480
221		t0,1,2,3,5,6{36,4}	Hexipentiruncicantitruncated 8-orthoplex Petiterigreatoprismated diacosipentacontahexazetton							11612160	2580480
222		t0,1,2,4,5,6{36,4}	Hexipentistericantitruncated 8-orthoplex Petitericelligreatorhombated diacosipentacontahexazetton							11612160	2580480
223		t0,1,3,4,5,6{36,4}	Hexipentisteriruncitruncated 8-orthoplex Petitericelliprismatotruncated diacosipentacontahexazetton							11612160	2580480
224		t0,2,3,4,5,6{36,4}	Hexipentisteriruncicantellated 8-orthoplex Petitericelliprismatorhombated diacosipentacontahexazetton							11612160	2580480
225		t1,2,3,4,5,6{4,36}	Bipentisteriruncicantitruncated 8-cube Great biteri-octeractidiacosipentacontahexazetton							10321920	2580480
226		t0,1,2,3,4,7{36,4}	Heptisteriruncicantitruncated 8-orthoplex Exigreatocellated diacosipentacontahexazetton							8601600	1720320
227		t0,1,2,3,5,7{36,4}	Heptipentiruncicantitruncated 8-orthoplex Exiterigreatoprismated diacosipentacontahexazetton							14192640	2580480
228		t0,1,2,4,5,7{36,4}	Heptipentistericantitruncated 8-orthoplex Exitericelligreatorhombated diacosipentacontahexazetton							12902400	2580480
229		t0,1,3,4,5,7{36,4}	Heptipentisteriruncitruncated 8-orthoplex Exitericelliprismatotruncated diacosipentacontahexazetton							12902400	2580480
230		t0,2,3,4,5,7{4,36}	Heptipentisteriruncicantellated 8-cube Exitericelliprismatorhombi-octeractidiacosipentacontahexazetton							12902400	2580480
231		t0,2,3,4,5,6{4,36}	Hexipentisteriruncicantellated 8-cube Petitericelliprismatorhombated octeract							11612160	2580480
232		t0,1,2,3,6,7{36,4}	Heptihexiruncicantitruncated 8-orthoplex Exipetigreatoprismated diacosipentacontahexazetton							8601600	1720320
233		t0,1,2,4,6,7{36,4}	Heptihexistericantitruncated 8-orthoplex Exipeticelligreatorhombated diacosipentacontahexazetton							14192640	2580480
234		t0,1,3,4,6,7{4,36}	Heptihexisteriruncitruncated 8-cube Exipeticelliprismatotrunki-octeractidiacosipentacontahexazetton							12902400	2580480
235		t0,1,3,4,5,7{4,36}	Heptipentisteriruncitruncated 8-cube Exitericelliprismatotruncated octeract							12902400	2580480
236		t0,1,3,4,5,6{4,36}	Hexipentisteriruncitruncated 8-cube Petitericelliprismatotruncated octeract							11612160	2580480
237		t0,1,2,5,6,7{4,36}	Heptihexipenticantitruncated 8-cube Exipetiterigreatorhombi-octeractidiacosipentacontahexazetton							8601600	1720320
238		t0,1,2,4,6,7{4,36}	Heptihexistericantitruncated 8-cube Exipeticelligreatorhombated octeract							14192640	2580480
239		t0,1,2,4,5,7{4,36}	Heptipentistericantitruncated 8-cube Exitericelligreatorhombated octeract							12902400	2580480
240		t0,1,2,4,5,6{4,36}	Hexipentistericantitruncated 8-cube Petitericelligreatorhombated octeract							11612160	2580480
241		t0,1,2,3,6,7{4,36}	Heptihexiruncicantitruncated 8-cube Exipetigreatoprismated octeract							8601600	1720320
242		t0,1,2,3,5,7{4,36}	Heptipentiruncicantitruncated 8-cube Exiterigreatoprismated octeract							14192640	2580480
243		t0,1,2,3,5,6{4,36}	Hexipentiruncicantitruncated 8-cube Petiterigreatoprismated octeract							11612160	2580480
244		t0,1,2,3,4,7{4,36}	Heptisteriruncicantitruncated 8-cube Exigreatocellated octeract							8601600	1720320
245		t0,1,2,3,4,6{4,36}	Hexisteriruncicantitruncated 8-cube Petigreatocellated octeract							12902400	2580480
246		t0,1,2,3,4,5{4,36}	Pentisteriruncicantitruncated 8-cube Great terated octeract							6881280	1720320
247		t0,1,2,3,4,5,6{36,4}	Hexipentisteriruncicantitruncated 8-orthoplex Great petated diacosipentacontahexazetton							20643840	5160960
248		t0,1,2,3,4,5,7{36,4}	Heptipentisteriruncicantitruncated 8-orthoplex Exigreatoterated diacosipentacontahexazetton							23224320	5160960
249		t0,1,2,3,4,6,7{36,4}	Heptihexisteriruncicantitruncated 8-orthoplex Exipetigreatocellated diacosipentacontahexazetton							23224320	5160960
250		t0,1,2,3,5,6,7{36,4}	Heptihexipentiruncicantitruncated 8-orthoplex Exipetiterigreatoprismated diacosipentacontahexazetton							23224320	5160960
251		t0,1,2,3,5,6,7{4,36}	Heptihexipentiruncicantitruncated 8-cube Exipetiterigreatoprismated octeract							23224320	5160960
252		t0,1,2,3,4,6,7{4,36}	Heptihexisteriruncicantitruncated 8-cube Exipetigreatocellated octeract							23224320	5160960
253		t0,1,2,3,4,5,7{4,36}	Heptipentisteriruncicantitruncated 8-cube Exigreatoterated octeract							23224320	5160960
254		t0,1,2,3,4,5,6{4,36}	Hexipentisteriruncicantitruncated 8-cube Great petated octeract							20643840	5160960
255		t0,1,2,3,4,5,6,7{4,36}	Omnitruncated 8-cube Great exi-octeractidiacosipentacontahexazetton							41287680	10321920

Д ₈ Семья

Семейство D ₈ имеет симметрию порядка 5 160 960 (8 факториал x 2 ⁷ ).

В этом семействе имеется 191 однородный многогранник Витоффа, представляющий собой перестановку 3x64-1 D ₈ диаграммы Кокстера-Динкина с одним или несколькими кольцами. 127 (2x64-1) повторяются из семейства B ₈ , а 64 уникальны для этого семейства, все они перечислены ниже.

См. список многогранников D8 для плоских графов Кокстера этих многогранников.

showD 8 однородных многогранников
#	Coxeter-Dynkin diagram	Name	Base point (Alternately signed)	Element counts								Circumrad
				7	6	5	4	3	2	1	0
1	=	8-demicube h{4,3,3,3,3,3,3}	(1,1,1,1,1,1,1,1)	144	1136	4032	8288	10752	7168	1792	128	1.0000000
2	=	cantic 8-cube h2{4,3,3,3,3,3,3}	(1,1,3,3,3,3,3,3)							23296	3584	2.6457512
3	=	runcic 8-cube h3{4,3,3,3,3,3,3}	(1,1,1,3,3,3,3,3)							64512	7168	2.4494896
4	=	steric 8-cube h4{4,3,3,3,3,3,3}	(1,1,1,1,3,3,3,3)							98560	8960	2.2360678
5	=	pentic 8-cube h5{4,3,3,3,3,3,3}	(1,1,1,1,1,3,3,3)							89600	7168	1.9999999
6	=	hexic 8-cube h6{4,3,3,3,3,3,3}	(1,1,1,1,1,1,3,3)							48384	3584	1.7320508
7	=	heptic 8-cube h7{4,3,3,3,3,3,3}	(1,1,1,1,1,1,1,3)							14336	1024	1.4142135
8	=	runcicantic 8-cube h2,3{4,3,3,3,3,3,3}	(1,1,3,5,5,5,5,5)							86016	21504	4.1231055
9	=	stericantic 8-cube h2,4{4,3,3,3,3,3,3}	(1,1,3,3,5,5,5,5)							349440	53760	3.8729835
10	=	steriruncic 8-cube h3,4{4,3,3,3,3,3,3}	(1,1,1,3,5,5,5,5)							179200	35840	3.7416575
11	=	penticantic 8-cube h2,5{4,3,3,3,3,3,3}	(1,1,3,3,3,5,5,5)							573440	71680	3.6055512
12	=	pentiruncic 8-cube h3,5{4,3,3,3,3,3,3}	(1,1,1,3,3,5,5,5)							537600	71680	3.4641016
13	=	pentisteric 8-cube h4,5{4,3,3,3,3,3,3}	(1,1,1,1,3,5,5,5)							232960	35840	3.3166249
14	=	hexicantic 8-cube h2,6{4,3,3,3,3,3,3}	(1,1,3,3,3,3,5,5)							456960	53760	3.3166249
15	=	hexicruncic 8-cube h3,6{4,3,3,3,3,3,3}	(1,1,1,3,3,3,5,5)							645120	71680	3.1622777
16	=	hexisteric 8-cube h4,6{4,3,3,3,3,3,3}	(1,1,1,1,3,3,5,5)							483840	53760	3
17	=	hexipentic 8-cube h5,6{4,3,3,3,3,3,3}	(1,1,1,1,1,3,5,5)							182784	21504	2.8284271
18	=	hepticantic 8-cube h2,7{4,3,3,3,3,3,3}	(1,1,3,3,3,3,3,5)							172032	21504	3
19	=	heptiruncic 8-cube h3,7{4,3,3,3,3,3,3}	(1,1,1,3,3,3,3,5)							340480	35840	2.8284271
20	=	heptsteric 8-cube h4,7{4,3,3,3,3,3,3}	(1,1,1,1,3,3,3,5)							376320	35840	2.6457512
21	=	heptipentic 8-cube h5,7{4,3,3,3,3,3,3}	(1,1,1,1,1,3,3,5)							236544	21504	2.4494898
22	=	heptihexic 8-cube h6,7{4,3,3,3,3,3,3}	(1,1,1,1,1,1,3,5)							78848	7168	2.236068
23	=	steriruncicantic 8-cube h2,3,4{4,36}	(1,1,3,5,7,7,7,7)							430080	107520	5.3851647
24	=	pentiruncicantic 8-cube h2,3,5{4,36}	(1,1,3,5,5,7,7,7)							1182720	215040	5.0990195
25	=	pentistericantic 8-cube h2,4,5{4,36}	(1,1,3,3,5,7,7,7)							1075200	215040	4.8989797
26	=	pentisterirunic 8-cube h3,4,5{4,36}	(1,1,1,3,5,7,7,7)							716800	143360	4.7958317
27	=	hexiruncicantic 8-cube h2,3,6{4,36}	(1,1,3,5,5,5,7,7)							1290240	215040	4.7958317
28	=	hexistericantic 8-cube h2,4,6{4,36}	(1,1,3,3,5,5,7,7)							2096640	322560	4.5825758
29	=	hexisterirunic 8-cube h3,4,6{4,36}	(1,1,1,3,5,5,7,7)							1290240	215040	4.472136
30	=	hexipenticantic 8-cube h2,5,6{4,36}	(1,1,3,3,3,5,7,7)							1290240	215040	4.3588991
31	=	hexipentirunic 8-cube h3,5,6{4,36}	(1,1,1,3,3,5,7,7)							1397760	215040	4.2426405
32	=	hexipentisteric 8-cube h4,5,6{4,36}	(1,1,1,1,3,5,7,7)							698880	107520	4.1231055
33	=	heptiruncicantic 8-cube h2,3,7{4,36}	(1,1,3,5,5,5,5,7)							591360	107520	4.472136
34	=	heptistericantic 8-cube h2,4,7{4,36}	(1,1,3,3,5,5,5,7)							1505280	215040	4.2426405
35	=	heptisterruncic 8-cube h3,4,7{4,36}	(1,1,1,3,5,5,5,7)							860160	143360	4.1231055
36	=	heptipenticantic 8-cube h2,5,7{4,36}	(1,1,3,3,3,5,5,7)							1612800	215040	4
37	=	heptipentiruncic 8-cube h3,5,7{4,36}	(1,1,1,3,3,5,5,7)							1612800	215040	3.8729835
38	=	heptipentisteric 8-cube h4,5,7{4,36}	(1,1,1,1,3,5,5,7)							752640	107520	3.7416575
39	=	heptihexicantic 8-cube h2,6,7{4,36}	(1,1,3,3,3,3,5,7)							752640	107520	3.7416575
40	=	heptihexiruncic 8-cube h3,6,7{4,36}	(1,1,1,3,3,3,5,7)							1146880	143360	3.6055512
41	=	heptihexisteric 8-cube h4,6,7{4,36}	(1,1,1,1,3,3,5,7)							913920	107520	3.4641016
42	=	heptihexipentic 8-cube h5,6,7{4,36}	(1,1,1,1,1,3,5,7)							365568	43008	3.3166249
43	=	pentisteriruncicantic 8-cube h2,3,4,5{4,36}	(1,1,3,5,7,9,9,9)							1720320	430080	6.4031243
44	=	hexisteriruncicantic 8-cube h2,3,4,6{4,36}	(1,1,3,5,7,7,9,9)							3225600	645120	6.0827627
45	=	hexipentiruncicantic 8-cube h2,3,5,6{4,36}	(1,1,3,5,5,7,9,9)							2903040	645120	5.8309517
46	=	hexipentistericantic 8-cube h2,4,5,6{4,36}	(1,1,3,3,5,7,9,9)							3225600	645120	5.6568542
47	=	hexipentisteriruncic 8-cube h3,4,5,6{4,36}	(1,1,1,3,5,7,9,9)							2150400	430080	5.5677648
48	=	heptsteriruncicantic 8-cube h2,3,4,7{4,36}	(1,1,3,5,7,7,7,9)							2150400	430080	5.7445626
49	=	heptipentiruncicantic 8-cube h2,3,5,7{4,36}	(1,1,3,5,5,7,7,9)							3548160	645120	5.4772258
50	=	heptipentistericantic 8-cube h2,4,5,7{4,36}	(1,1,3,3,5,7,7,9)							3548160	645120	5.291503
51	=	heptipentisteriruncic 8-cube h3,4,5,7{4,36}	(1,1,1,3,5,7,7,9)							2365440	430080	5.1961527
52	=	heptihexiruncicantic 8-cube h2,3,6,7{4,36}	(1,1,3,5,5,5,7,9)							2150400	430080	5.1961527
53	=	heptihexistericantic 8-cube h2,4,6,7{4,36}	(1,1,3,3,5,5,7,9)							3870720	645120	5
54	=	heptihexisteriruncic 8-cube h3,4,6,7{4,36}	(1,1,1,3,5,5,7,9)							2365440	430080	4.8989797
55	=	heptihexipenticantic 8-cube h2,5,6,7{4,36}	(1,1,3,3,3,5,7,9)							2580480	430080	4.7958317
56	=	heptihexipentiruncic 8-cube h3,5,6,7{4,36}	(1,1,1,3,3,5,7,9)							2795520	430080	4.6904159
57	=	heptihexipentisteric 8-cube h4,5,6,7{4,36}	(1,1,1,1,3,5,7,9)							1397760	215040	4.5825758
58	=	hexipentisteriruncicantic 8-cube h2,3,4,5,6{4,36}	(1,1,3,5,7,9,11,11)							5160960	1290240	7.1414285
59	=	heptipentisteriruncicantic 8-cube h2,3,4,5,7{4,36}	(1,1,3,5,7,9,9,11)							5806080	1290240	6.78233
60	=	heptihexisteriruncicantic 8-cube h2,3,4,6,7{4,36}	(1,1,3,5,7,7,9,11)							5806080	1290240	6.480741
61	=	heptihexipentiruncicantic 8-cube h2,3,5,6,7{4,36}	(1,1,3,5,5,7,9,11)							5806080	1290240	6.244998
62	=	heptihexipentistericantic 8-cube h2,4,5,6,7{4,36}	(1,1,3,3,5,7,9,11)							6451200	1290240	6.0827627
63	=	heptihexipentisteriruncic 8-cube h3,4,5,6,7{4,36}	(1,1,1,3,5,7,9,11)							4300800	860160	6.0000000
64	=	heptihexipentisteriruncicantic 8-cube h2,3,4,5,6,7{4,36}	(1,1,3,5,7,9,11,13)							2580480	10321920	7.5498347

Е ₈ Семья

Семейство E8 _{имеет порядок симметрии 696729600} .

Существует 255 форм, основанных на всех перестановках диаграмм Кокстера-Динкина с одним или несколькими кольцами. Ниже показаны восемь форм, ниже приведены 4 однокольцевые, 3 усеченные (2 кольца) и финальная омниусеченность. Названия аббревиатур в стиле Бауэрса даны для перекрестных ссылок.

См. также список многогранников E8 для плоских графов Кокстера этого семейства.

showE 8 однородных многогранников
#	Coxeter-Dynkin diagram	Names	Element counts
			7-faces	6-faces	5-faces	4-faces	Cells	Faces	Edges	Vertices
1		421 (fy)	19440	207360	483840	483840	241920	60480	6720	240
2		Truncated 421 (tiffy)							188160	13440
3		Rectified 421 (riffy)	19680	375840	1935360	3386880	2661120	1028160	181440	6720
4		Birectified 421 (borfy)	19680	382560	2600640	7741440	9918720	5806080	1451520	60480
5		Trirectified 421 (torfy)	19680	382560	2661120	9313920	16934400	14515200	4838400	241920
6		Rectified 142 (buffy)	19680	382560	2661120	9072000	16934400	16934400	7257600	483840
7		Rectified 241 (robay)	19680	313440	1693440	4717440	7257600	5322240	1451520	69120
8		241 (bay)	17520	144960	544320	1209600	1209600	483840	69120	2160
9		Truncated 241								138240
10		142 (bif)	2400	106080	725760	2298240	3628800	2419200	483840	17280
11		Truncated 142								967680
12		Omnitruncated 421								696729600

Регулярные и однородные соты

Соответствия диаграмм Коксетера-Дынкина между семействами и более высокая симметрия внутри диаграмм. Узлы одного цвета в каждом ряду представляют собой одинаковые зеркала. Черные узлы в переписке не активны.

Существует пять фундаментальных аффинных групп Кокстера , которые генерируют регулярные и равномерные мозаики в 7-мерном пространстве:

#	Группа Коксетера		Диаграмма Кокстера	Формы
1	${\tilde {A}}_{7}$	[3 [8] ]		29
2	${\tilde {C}}_{7}$	[4,3 5 ,4]		135
3	${\tilde {B}}_{7}$	[4,3 4 ,3 1,1 ]		191 (64 новых)
4	${\tilde {D}}_{7}$	[3 1,1 ,3 3 ,3 1,1 ]		77 (10 новых)
5	${\tilde {E}}_{7}$	[3 3,3,1 ]		143

Регулярные и однородные тесселяции включают в себя:

• ${\tilde {A}}_{7}$ 29 форм с уникальными кольцами, в том числе:
  • 7-симплексные соты : {3 ^[8] }
• ${\tilde {C}}_{7}$ 135 форм с уникальными кольцами, в том числе:
  • Обычные соты из 7 кубов : {4,3 ⁴ ,4} = {4,3 ⁴ ,3 ^1,1 }, =
• ${\tilde {B}}_{7}$ 191 форма с уникальным кольцом, 127 общие с ${\tilde {C}}_{7}$и 64 новых, в том числе:
  • 7-кубовые соты : h{4,3 ⁴ ,4} = {3 ^1,1 ,3 ⁴ ,4}, =
• ${\tilde {D}}_{7}$, [3 ^1,1 ,3 ³ ,3 ^1,1 ]: 77 уникальных перестановок колец, 10 из них новые, первый Коксетер назвал четверть 7-кубических сот .
  • , , , , , , , , ,
• ${\tilde {E}}_{7}$ 143 формы с уникальными кольцами, в том числе:
  • 1 ₃₃ соты : {3,3 ^3,3 },
  • 3 ₃₁ соты : {3,3,3,3 ^3,1 },

Правильные и однородные гиперболические соты

Не существует компактных гиперболических групп Кокстера ранга 8, групп, которые могут порождать соты со всеми конечными гранями, и конечной вершинной фигуры . Однако существует 4 паракомпактные гиперболические группы Кокстера ранга 8, каждая из которых порождает однородные соты в 7-пространстве как перестановки колец диаграмм Кокстера.

${\bar {P}}_{7}$ = [3,3 [7] ]:

${\bar {Q}}_{7}$ = [3 1,1 ,3 2 ,3 2,1 ]:

${\bar {S}}_{7}$ = [4,3 3 ,3 2,1 ]:

${\bar {T}}_{7}$ = [3 3,2,2 ]:

Ссылки

• Т. Госсет : О правильных и полуправильных фигурах в пространстве n измерений , Вестник математики , Макмиллан, 1900 г.
• А. Буль Стотт : Геометрический вывод полуправильных многогранников из правильных многогранников и пространственного заполнения , Трактаты о единице ширины Королевской академии наук Амстердам, Первый раздел 11,1, Амстердам, 1910 г.
• ХСМ Коксетер :
  • HSM Коксетер, М. С. Лонге-Хиггинс и Дж. К. П. Миллер: однородные многогранники , Философские труды Лондонского королевского общества, Лондон, 1954 г.
  • HSM Coxeter, Правильные многогранники , 3-е издание, Дувр, Нью-Йорк, 1973 г.
• Калейдоскопы: Избранные сочинения HSM Коксетера , под редакцией Ф. Артура Шерка, Питера Макмаллена, Энтони К. Томпсона, Азии Ивик Вайс, Wiley-Interscience Publication, 1995, ISBN   978-0-471-01003-6 Wiley::Калейдоскопы: Избранные сочинения HSM Коксетера
  • (Документ 22) HSM Коксетер, Правильные и полуправильные многогранники I , [Math. Зейт. 46 (1940) 380–407, МР 2,10]
  • (Документ 23) HSM Коксетер, Правильные и полуправильные многогранники II , [Math. Зейт. 188 (1985) 559-591]
  • (Документ 24) HSM Коксетер, Правильные и полуправильные многогранники III , [Math. Зейт. 200 (1988) 3-45]
• Н. В. Джонсон : Теория однородных многогранников и сот , доктор философии. Диссертация, Университет Торонто, 1966 г.
• Клитцинг, Ричард. «8D однородные многогранники (polyzetta)» .

Внешние ссылки

• Имена многогранников
• Многогранники различных размерностей
• Многомерный глоссарий