User:Tomruen/uniform polyteron

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5-cube (Penteract)	5-orthoplex (Pentacross)	5-simplex (Hexateron)
Graphs of the three regular polyterons

A uniform 5-polytope is a uniform polytope that exists in 5-dimensional Euclidean space. Using a Wythoff construction, the set of uniform 5-polytopes are enumerated below, grouped with the generation symmetry, although there are overlaps as different generators can create the same forms.

Regulars and truncations The three regular 5-polytopes above create 2 families of uniform 5-polytopes. Using a naming scheme proposed by Norman Johnson, these are:

Contents 1 The hexateron family {3,3,3,3} 2 The penteract/pentacross family [4,3,3,3] 2.1 The penteract subfamily 2.2 Pentacross subfamily 3 Demipenteract family [31,2,1] 4 Nonwythoffian 5 Prismatic forms 6 References

[edit] The hexateron family {3,3,3,3}

There are 19 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. (2⁵-1 - 12 symmetry cases)

#	Coxeter-Dynkin andSchläfli symbols	Name	Facet counts by location: [3,3,3,3]					Element counts
			4	3	2	1	0
			[3,3,3] (6)	[3,3]×[ ] (15)	[3]×[3] (20)	[ ]×[3,3] (15)	[3,3,3] (6)	Facets	Cells	Faces	Edges	Vertices
1	t0{3,3,3,3}	Hexateron Hix	{3,3,3}	-	-	-	-	6	15	20	15	6
2	t1{3,3,3,3}	Rectified hexateron Rix	t1{3,3,3}	-	-	-	{3,3,3}	12	45	80	60	15
3	t2{3,3,3,3}	Birectified hexateron Dot	t2{3,3,3}	-	-	-	t1{3,3,3}	12	60	120	90	20
4	t0,1{3,3,3,3}	Truncated hexateron Tix	t0,1{3,3,3}	-	-	-	{3,3,3}	12	45	80	75	30
5	t1,2{3,3,3,3}	Bitruncated hexateron Bittix	t1,2{3,3,3}	-	-	-	t0,1{3,3,3}	12	60	140	150	60
6	t0,2{3,3,3,3}	Cantellated hexateron Sarx	t0,2{3,3,3}	-	-	{}×{3,3}	t1{3,3,3}	27	135	290	240	60
7	t1,3{3,3,3,3}	Bicantellated hexateron Sibrid	t1,3{3,3,3}	-	{3}×{3}	-	t0,2{3,3,3}	32	180	420	360	90
8	t0,3{3,3,3,3}	Runcinated hexateron Spix	t0,3{3,3,3}	-	{3}×{3}	{}×t1{3,3}	t1{3,3,3}	47	255	420	270	60
9	t0,4{3,3,3,3}	Stericated hexateron Scad	{3,3,3}	{}×{3,3}	{3}×{3}	{}×{3,3}	{3,3,3}	62	180	210	120	30
10	t0,1,2{3,3,3,3}	Cantitruncated hexateron Garx	t0,1,2{3,3,3}	-	-	{}×{3,3}	t0,1{3,3,3}	27	135	290	300	120
11	t1,2,3{3,3,3,3}	Bicantitruncated hexateron Gibrid	t0,1,2{3,3,3}	-	{3}×{3}	-	t0,1,2{3,3,3}	32	180	450	420	180
12	t0,1,3{3,3,3,3}	Runcitruncated hexateron Pattix	t0,1,3{3,3,3}	-	{6}×{3}	{}×t1{3,3}	t0,2{3,3,3}	47	315	720	630	180
13	t0,2,3{3,3,3,3}	Runcicantellated hexateron Pirx	t0,1,3{3,3,3}	-	{3}×{3}	{}×t0,1{3,3}	t1,2{3,3,3}	47	255	570	540	180
14	t0,1,4{3,3,3,3}	Steritruncated hexateron Cappix	t0,1{3,3,3}	{}×t0,1{3,3}	{3}×{6}	{}×{3,3}	t0,3{3,3,3}	62	330	570	420	120
15	t0,2,4{3,3,3,3}	Stericantellated hexateron Card	t0,2{3,3,3}	{}×t0,2{3,3}	{3}×{3}	{}×t0,2{3,3}	t0,2{3,3,3}	62	420	900	720	180
16	t0,1,2,3{3,3,3,3}	Runcicantitruncated hexateron Gippix	t0,1,2,3{3,3,3}	-	{3}×{6}	{}×t0,1{3,3}	t0,2{3,3,3}	47	315	810	900	360
17	t0,1,2,4{3,3,3,3}	Stericantitruncated hexateron Cograx	t0,1,2{3,3,3}	{}×t0,1,2{3,3}	{3}×{6}	{}×t0,2{3,3}	t0,1,3{3,3,3}	62	480	1140	1080	360
18	t0,1,3,4{3,3,3,3}	Steriruncitruncated hexateron Captid	t0,1,3{3,3,3}	{}×t0,1{3,3}	{6}×{6}	{}×t0,1,3{3,3}	t0,1,3{3,3,3}	62	450	1110	1080	360
19	t0,1,2,3,4{3,3,3,3}	Omnitruncated hexateron Gocard	t0,1,2,3{3,3,3}	{}×t0,1,2{3,3}	{6}×{6}	{}×t0,1,2{3,3}	t0,1,2,3{3,3,3}	62	540	1560	1800	720

[edit] The penteract/pentacross family [4,3,3,3]

This family has 31 Wythoffian uniform polyhedra, from 2⁵-1 permutations of the Coxeter-Dynkin diagram with one or more rings.

For simplicity it divided into two subfamilies, each with 12 forms, and 7 "middle" forms which equally belong in both subfamilies.

[edit] The penteract subfamily

There are 20 forms here, 7 shared with the pentacross family. Four are shared with the demipenteract family.

#	Coxeter-Dynkin andSchläfli symbols	Name	Facet counts by location: [4,3,3,3]					Element counts
			4	3	2	1	0
			[4,3,3] (32)	[4,3]×[ ] (80)	[4]×[3] (80)	[ ]×[3,3] (40)	[3,3,3] (10)	Facets	Cells	Faces	Edges	Vertices
20	t0{4,3,3,3}	Penteract Pent	{4,3,3}	-	-	-	-	10	40	80	80	32
21	t1{4,3,3,3}	Rectified penteract Rin	t1{4,3,3}	-	-	-	{3,3,3}	42	200	400	320	80
22	t2{4,3,3,3}	Birectified penteract Nit	t1{4,3,3}	-	-	-	t1{3,3,3}	42	280	640	480	80
23	t0,1{4,3,3,3}	Truncated penteract Tan	t0,1{4,3,3}	-	-	-	{3,3,3}	42	200	400	400	160
24	t1,2{4,3,3,3}	Bitruncated penteract Bittin	t1,2{4,3,3}	-	-	-	t0,1{3,3,3}	42	280	720	800	320
25	t0,2{4,3,3,3}	Cantellated penteract Sirn	t0,2{4,3,3}	-	-	{}×{3,3}	t1{3,3,3}	122	680	1520	1280	320
26	t1,3{4,3,3,3}	Bicantellated penteract Sibrant	t0,2{4,3,3}	-	{4}×{3}	-	t0,2{3,3,3}	122	840	2160	1920	480
27	t0,3{4,3,3,3}	Runcinated penteract Span	t0,3{4,3,3}	-	{4}×{3}	{}×t1{3,3}	{3,3,3}	202	1240	2160	1440	320
28	t0,4{4,3,3,3}	Stericated penteract Scant	{4,3,3}	{4,3}×{}	{4}×{3}	{}×{3,3}	{3,3,3}	242	800	1040	640	160
29	t0,1,2{4,3,3,3}	Cantitruncated penteract Girn	t0,1,2{4,3,3}	-	-	{}×{3,3}	t0,1{3,3,3}	122	680	1520	1600	640
30	t1,2,3{4,3,3,3}	Bicantitruncated penteract Gibrant	t0,1,2{4,3,3}	-	{4}×{3}	-	t0,1,2{3,3,3}	122	840	2160	2400	960
31	t0,1,3{4,3,3,3}	Runcitruncated penteract Pattin	t0,1,3{4,3,3}	-	{8}×{3}	{}×t1{3,3}	t0,2{3,3,3}	202	1560	3760	3360	960
32	t0,2,3{4,3,3,3}	Runcicantellated penteract Prin	t0,1,3{4,3,3}	-	{4}×{3}	{}×t0,1{3,3}	t1,2{3,3,3}	202	1240	2960	2880	960
33	t0,1,4{4,3,3,3}	Steritruncated penteract Capt	t0,1{4,3,3}	t0,1{4,3}×{}	{8}×{3}	{}×{3,3}	t0,3{3,3,3}	242	1600	2960	2240	640
34	t0,2,4{4,3,3,3}	Stericantellated penteract Carnit	t0,2{4,3,3}	t0,2{4,3}×{}	{4}×{3}	{}×t0,2{3,3}	t0,2{3,3,3}	242	2080	4720	3840	960
35	t0,1,2,3{4,3,3,3}	Runcicantitruncated penteract Gippin	t0,1,2,3{4,3,3}	-	{8}×{3}	{}×t0,1{3,3}	t0,1,2{3,3,3}	202	1560	4240	4800	1920
36	t0,1,2,4{4,3,3,3}	Stericantitruncated penteract Cogrin	t0,1,2{4,3,3}	t0,1,2{4,3}×{}	{8}×{3}	{}×t0,2{3,3}	t0,1,3{3,3,3}	242	2400	6000	5760	1920
37	t0,1,3,4{4,3,3,3}	Steriruncitruncated penteract Captint	t0,1,3{4,3,3}	t0,1{4,3}×{}	{8}×{6}	{}×t0,1{3,3}	t0,1,3{3,3,3}	242	2160	5760	5760	1920
38	t0,1,2,3,4{4,3,3,3}	Omnitruncated penteract Gacnet	t0,1,2{4,3}×{}	t0,1,2{4,3}×{}	{8}×{6}	{}×t0,1,2{3,3}	t0,1,2,3{3,3,3}	242	2640	8160	9600	3840
[51]	h0{4,3,3,3}	Demipenteract Hin	(16) {3,3,3}	-	-	-	{3,3,4}	26	120	160	80	16

[edit] Pentacross subfamily

There are 19 forms, 12 new ones. 7 are shared from the penteract family, and 10 shared with the demipenteract family.

#	Coxeter-Dynkin andSchläfli symbols	Name	Facet counts by location: [4,3,3,3]					Element counts
			4	3	2	1	0
			[3,3,3] (10)	[3,3]×[ ] (40)	[3]×[4] (80)	[ ]×[3,4] (80)	[3,3,4] (32)	Facets	Cells	Faces	Edges	Vertices
39	t0{3,3,3,4}	Pentacross Tac	{3,3,3}	-	-	-	-	10	40	80	80	32
40	t1{3,3,3,4}	Rectified pentacross Rat	t1{3,3,3}	-	-	-	?	42	240	400	240	40
[22]	t2{3,3,3,4}	Birectified pentacross Nit	t1{3,3,3}	-	-	-	?	42	280	640	480	80
41	t0,1{3,3,3,4}	Truncated pentacross Tot	t0,1{3,3,3}	-	-	-	?	42	240	400	280	80
42	t1,2{3,3,3,4}	Bitruncated pentacross Bittit	t1,2{3,3,3}	?	?	?	?	42	280	720	720	240
43	t0,2{3,3,3,4}	Cantellated pentacross Sart	t0,2{3,3,3}	?	?	?	?	82	640	1520	1200	240
[26]	t1,3{3,3,3,4}	Bicantellated pentacross Sibrant	t1,3{3,3,3}	?	?	?	?	122	840	2160	1920	480
44	t0,3{3,3,3,4}	Runcinated pentacross Spat	t0,3{3,3,3}	?	?	?	?	162	1200	2160	1440	320
[28]	t0,4{3,3,3,4}	Stericated pentacross Scant	{3,3,3}	?	?	?	?	242	800	1040	640	160
45	t0,1,2{3,3,3,4}	Cantitruncated pentacross Gart	t0,1,2{3,3,3}	?	?	?	?	82	640	1520	1440	480
[30]	t1,2,3{3,3,3,4}	Bicantitruncated pentacross Gibrant	t1,2,3{3,3,3}	?	?	?	?	122	840	2160	2400	960
46	t0,1,3{3,3,3,4}	Runcitruncated pentacross Pattit	t0,1,3{3,3,3}	?	?	?	?	162	1440	3680	3360	960
47	t0,2,3{3,3,3,4}	Runcicantellated pentacross Pirt	t0,1,3{3,3,3}	?	?	?	?	162	1200	2660	2880	960
48	t0,1,4{3,3,3,4}	Steritruncated pentacross Cappin	t0,1{3,3,3}	?	?	?	?	242	1520	2880	2240	640
[34]	t0,2,4{3,3,3,4}	Stericantellated pentacross Carnit	t0,2{3,3,3}	?	?	?	?	242	2080	4720	3840	960
49	t0,1,2,3{3,3,3,4}	Runcicantitruncated pentacross Gippit	t0,1,2,3{3,3,3}	?	?	?	?	162	1440	4160	4800	1920
50	t0,1,2,4{3,3,3,4}	Stericantitruncated pentacross Cogart	t0,1,2{3,3,3}	?	?	?	?	242	2320	5920	5760	1920
[37]	t0,1,3,4{3,3,3,4}	Steriruncitruncated pentacross Captint	t0,1,3{3,3,3}	?	?	?	?	242	2160	5760	5760	1920
[38]	t0,1,2,3,4{3,3,3,4}	Omnitruncated pentacross Gacnet	t0,1,2,3{3,3,3}	?	?	?	?	242	2640	8160	9600	3840

[edit] Demipenteract family [3^1,2,1]

There are 23 forms. 16 are repeated from the [4,3,3,3] family and 7 are new ones.

#	Coxeter-Dynkin andSchläfli symbols	Name	Facet counts by location: [31,2,1]					Element counts
			4	3	2	1	0
			[3,3,3] (16)	[31,1,1] (10)	[3,3,3]×[ ] (40)	[ ]×[3]×[ ] (80)	[3,3,3] (16)	Facets	Cells	Faces	Edges	Vertices
51	{31,2,1}	Demipenteract Hin	{3,3,3}	t0{31,1,1}	-	-	-	26	120	160	80	16
[22]	t1{31,2,1}	(Birectified penteract) Nit	t1{3,3,3}	t1{31,1,1}	-	-	t1{3,3,3}	42	280	640	480	80
[40]	t2{31,2,1}	(Rectified pentacross) Rat	t1{3,3,3}	t0{31,1,1}	-	-	t1{3,3,3}	42	240	400	240	40
[39]	t3{31,2,1}	(Pentacross) Tac	{3,3,3}	-	-	-	{3,3,3}	32	80	80	40	10
52	t0,1{31,2,1}	Truncated demipenteract Thin	-	-	-	-	-	42	280	640	560	160
53	t0,2{31,2,1}	Cantellated demipenteract Sirhin	-	-	-	-	-	42	360	880	720	160
54	t0,3{31,2,1}	Runcinated demipenteract Siphin	-	-	-	-	-	82	480	720	400	80
[21]	t0,4{31,2,1}	(Rectified penteract) Rin	-	-	-	-	-	42	200	400	320	80
[42]	t1,2{31,2,1}	(Bitruncated pentacross) Bittit	-	-	-	-	-	42	280	720	720	240
[43]	t1,3{31,2,1}	(Cantellated pentacross) Sart	-	-	-	-	-	82	640	1520	1200	240
[41]	t2,3{31,2,1}	(Truncated pentacross) Tot	-	-	-	-	-	42	240	400	280	80
[24]	t0,1,4{31,2,1}	(Bitruncated penteract) Bittin	-	-	-	-	-	42	280	720	800	320
55	t0,1,2{31,2,1}	Cantitruncated demipenteract Girhin	-	-	-	-	-	42	360	1040	1200	480
56	t0,1,3{31,2,1}	Runcitruncated demipenteract Pithin	-	-	-	-	-	82	720	1840	1680	480
[26]	t0,2,4{31,2,1}	(Bicantellated penteract) Sibrant	-	-	-	-	-	122	840	2160	1920	480
[44]	t0,3,4{31,2,1}	(Runcinated pentacross) Spat	-	-	-	-	-	162	1200	2160	1440	320
57	t0,2,3{31,2,1}	Runcicantellated demipenteract Pirhin	-	-	-	-	-	82	560	1280	1120	320
[45]	t1,2,3{31,2,1}	(Cantitruncated pentacross) Gart	-	-	-	-	-	82	640	1520	1440	480
[30]	t0,1,2,4{31,2,1}	(Bicantitruncated pentacross) Gibrant	-	-	-	-	-	122	840	2160	2400	960
[46]	t0,1,3,4{31,2,1}	(Runcicantellated pentacross) Pirt	-	-	-	-	-	162	1440	3680	3360	960
58	t0,1,2,3{31,2,1}	Runcicantitruncated demipenteract Giphin	-	-	-	-	-	82	720	2080	2400	960
[47]	t0,2,3,4{31,2,1}	(Runcitruncated pentacross) Pattit	-	-	-	-	-	162	1200	2660	2880	960
[49]	t0,1,2,3,4{31,2,1}	(Runcicantitruncated pentacross) Gippit	-	-	-	-	-	162	1440	4160	4800	1920

[edit] Nonwythoffian

The great-antiprism prism is the only nonwythoffian uniform polyteron. It has 200 vertices, 1100 edges, 1940 faces (40 pentagons, 500 squares, 1400 triangles), 1360 cells (300 tetrahedrons, 20 pentagonal antiprisms, 700 triangular prisms, 20 pentagonal prisms), 322 hypercells (2 grand antiprisms , 20 pentagonal antiprism prisms , and 300 tetrahedral prisms ).

#	Name	Element counts
		Facets	Cells	Faces	Edges	Vertices
59	Great-antiprism prism Gappip	322	1360	1940	1100	200

[edit] Prismatic forms

There are 3 categorical uniform prismatic forms:

• [ ] × [p,q,r] – uniform polychoron prisms (Each uniform polychoron forms one uniform prism)
  • [ ] × [3,3,3] – 9 forms
  • [ ] × [3,3,4] – 15 forms (Three shared with [ ]×[3,4,3] family)
  • [ ] × [3,4,3] – 10 forms
  • [ ] × [3,3,5] – 15 forms
  • Grand antiprism prism
• [p] × [q,r] – Regular polygon × uniform polyhedron duoprisms
  • [p] × [3,3] – 5 forms for each (p≥3) (Three shared with [p]×[3,4] family)
  • [p] × [3,4] – 7 forms for each (p≥3)
  • [p] × [3,5] – 7 forms for each (p≥3)
• [ ] × [p] × [q] – Uniform duoprism prisms – 1 form for each p and q, (each ≥3).

[edit] References

• T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
• Norman Johnson Uniform Polytopes, Manuscript (1991)
• Richard Klitzing 5D quasiregulars, (multi)prisms, non-prismatic Wythoffian polyterons

• Polytope names
• Polytopes of Various Dimensions, Jonathan Bowers
• Multi-dimensional Glossary
• Glossary for hyperspace