Segmentotope (EntityClass, 11)

From Hi.gher. Space

A segmentotope is an orbiform polytope whose vertices lie on two parallel hyperplanes and whose edge lengths are all equal. The set of all convex segmentotopes up to and including 4D has been enumerated by Dr. Richard Klitzing. There are 177 four-dimensional segmentotopes, including some polychora from other categories (such as cube || cube, which is the same as the tesseract). The full list can be obtained from Klitzing's paper (PDF format).

Nomenclature

A segmentotope is denoted by the notation A || B, where A and B are lower-dimensional polytopes. For segmentochora, A and B are usually polyhedra, although one of them can be lower-dimensional, as is the case with the wedges and pyramids.

Some segmentotopes may have multiple designations, for example, (triangular_prism || hexagonal_prism) is the same as (triangular_cupola || triangular_cupola). Where multiple names are possible, the name listed by Klitzing takes precedence.

Properties

Below are some useful properties of selected segmentochora. Klitzing's numbering is written as "K 4.n", as given in his PhD dissertation. Measurements are given in terms of E, the edge length.

#	Name(s)	Cells	Values
K 4.7	digon || square_pyramid triangular prism pyramid (K 4.7.2) point || trigonal_prism	2 tetrahedra 3 square pyramids 1 trigonal prism	Height of triangular prism pyramid: E*sqrt(5/12) Dichoral angle between tetrahedron and triangular prism: atan(sqrt(5/3)) ≈ 52.24° Dichoral angle between square pyramid and triangular prism: atan(sqrt(5)) ≈ 65.91°
K 4.26	square || square_pyramid cubical pyramid point || cube square prism pyramid	6 square pyramids 1 cube	Dichoral angle between square pyramid and cube: 45° (exact) Dichoral angle between adjacent square pyramids: 180° - acos(1/3) ≈ 109.47°
K 4.45	triangular_cupola || triangular_cupola triangular_prism || hexagonal_prism		Distance between hexagonal prism and antipodal triangular prism: E*sqrt(2/3). (Same as the height of a triangular cupola.)
K 4.51	hexagon || trigonal_cupola triangle || hexagonal_prism trigonal magnabicupolic ring		Distance from hexagonal prism to triangle: E*sqrt(5/12) Dichoral angle between trigonal cupola and hexagonal prism: atan(sqrt(5/3)) ≈ 52.23° Dichoral angle between triangular prism and hexagonal prism: acos(2/3) ≈ 48.19° Dichoral angle between square pyramid and hexagonal prism: atan(sqrt(5)) ≈ 65.91°
K 4.69	square_cupola || square_cupola cube || octagonal_prism		Distance between octagonal prism and antipodal cube: E*sqrt(2)/2. (Same as height of square cupola.)
K 4.105	octagon || square_cupola square || octagonal_prism square magnabicupolic ring		Distance from octagonal prism to square: E*sqrt(2)/2 Dichoral angle between square cupola and octagonal prism: 45° (exact) Dichoral angle between square pyramid and octagonal prism: 45° (exact) Dichoral angle between triangular prism and octagonal prism: acos(sqrt(2/3)) ≈ 35.26°
K 4.117	pentagonal_cupola || pentagonal_cupola pentagonal_prism || decagonal_prism		Distance between pentagonal prism and decagonal prism: E*(sqrt(2*sqrt(2*(3*sqrt(5)+7)) - (12*sqrt(5)+20)/5)/2). (Same as height of pentagonal cupola.)
K 4.141	pentagon || pentagonal_pyramid pentagonal prism pyramid point || pentagonal_prism		Distance between pentagonal prism and antipodal point: (E/2)*sqrt((5-2*sqrt(5))/5) Dichoral angle between pentagonal pyramid and pentagonal prism: 18° (exact) Dichoral angle between square pyramid and pentagonal prism: atan(sqrt(5)-2) ≈ 13.28°
K 4.165	decagon || pentagonal_cupola pentagon || decagonal_prism pentagonal magnabicupolic ring		Distance between decagonal prism and pentagon: (E/2)*sqrt((5-2*sqrt(5))/5) Dichoral angle between pentagonal cupola and decagonal prism: 18° (exact) Dichoral angle between square pyramid and decagonal prism: atan(sqrt(9-4*sqrt(5))) ≈ 13.28° Dichoral angle between triangular prism and decagonal prism: asin(sqrt((5-2*sqrt(5))/15)) ≈ 10.81°

Classification

Keiji has begun a project to classify all of Klitzing's segmentochora, so that more interesting ones stand out to help future research.

The current classification is as follows, where each number is a K 4.X index:

Ur - 3 regular polychora

1, 2, 20

Ux - the pyrorectichoron, a uniform polychoron

5

Up - all 17 prisms of the uniform polyhedra (not 18, because the cube prism is the K 4.20, already counted above)

9, 11, 36, 43, 57, 60, 66, 74, 89, 90, 99, 110, 111, 125, 127, 130, 150

Inf - 4 infinite families (not individual polychora) and 21 members of those infinite families

family	{3}	{4}	{5}	{6}	{8}	{10}
174	6	14	22	46	58	93
175	10	(18)	34	47	59	94
176	(11)	19	39	53	65	96
177	18	(20)	42	54	70	97

Py - 12 pyramids of CRF polyhedra

3, 4, 7, 17, 26, 80, 84, 85, 86, 87, 88, 141

Po - 25 prisms of orbiform Johnson solids

12, 37, 38, 40, 41, 44, 45, 67, 68, 69, 91, 92, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124

Cu - 30 monostratic cupolic forms

15, 21, 23, 29, 35, 48, 52, 55, 56, 61, 71, 75, 76, 77, 78, 95, 98, 100, 107, 126, 128, 129, 131, 137, 149, 151, 152, 158, 159, 173

Bicu - all 10 bicupolic rings (excluding the prisms of the tetrahedron and square pyramid, already counted above)

8, 25, 27, 51, 64, 73, 105, 133, 154, 165

Xcu - 3 fragments of cupolic forms (excluding the bicupolic rings)

109, 139, 146

Xg - the gyrated octahedral prism

13

Xt - the trigonal tridiminished-icosahedral wedge

33

Gy - 49 gyrations and diminishes of the cupolic forms

16, 24, 28, 30, 31, 32, 49, 50, 62, 63, 72, 79, 81, 82, 83, 101, 102, 103, 104, 106, 108, 132, 134, 135, 136, 138, 140, 142, 143, 144, 145, 147, 148, 153, 155, 156, 157, 160, 161, 162, 163, 164, 166, 167, 168, 169, 170, 171, 172

Full list of segmentochora

• See also: Segmentochoron construction table

#	Name	Symbol	Cells	Comments
1	pyrochoron	x3o3o3o	5x tetrahedron
2	aerochoron	x3o3o4o	16x tetrahedron
3	octahedral pyramid	xo3oo4oo	8x tetrahedron + 1x octahedron	half of aerochoron
4	square dipyramid	xo4oo2ox	4x tetrahedron + 2x square pyramid	quarter of aerochoron
5	pyrorectichoron	o3x3o3o	5x tetrahedron + 5x octahedron	tetrahedral semicupola
6	trigonal biantiprismatic ring	xox3oxo&#x	3x tetrahedron + 2x octahedron + 3x square pyramid + 1x trigonal prism	(mono)diminished pyrorectichoron
7	triangular prism pyramid	xo3oo2xo	2x tetrahedron + 3x square pyramid + 1x trigonal prism
8	digonal gyrobicupolic ring	xxo2xox&#x	1x tetrahedron + 4x square pyramid + 2x trigonal prism	bidiminished pyrorectichoron
9	tetrahedral prism	x3o3o2x	2x tetrahedron + 4x trigonal prism
10	duotrianglinder	x3o2x3o	6x trigonal prism
11	octahedral prism	x3o4o2x	2x octahedron + 8x trigonal prism
12	square pyramid prism	xx2oo4xo	2x square pyramid + 4x trigonal prism + 1x cube
13	gyrated octahedral prism	??	4x square pyramid + (4+4)x trigonal prism
14	square biantiprismatic ring	xox4oxo&#x	4x tetrahedron + 4x square pyramid + 2x square antiprism + 1x cube
15	staurohedral antiprism	xo4oo3ox	(8+12)x tetrahedron + 1x octahedron + 6x square pyramid + 1x cube
16	K4.16	??	(4+4+4)x tetrahedron + (1+1+4)x square pyramid + 1x square antiprism + 1x cube
17	square antiprism pyramid	oxo4oox&#x	8x tetrahedron + 2x square pyramid + 1x square antiprism
18	triangular diprism	x3o2x4o	4x trigonal prism + 3x cube
19	square antiprism prism	??	8x trigonal prism + 2x square antiprism + 2x cube
20	geochoron	x4o3o3o	8x cube
21	snubdis antiprism	xo4os3os	8x tetrahedron + 12x square pyramid + 6x trigonal prism + 1x cube + 1x icosahedron	snubdis antiprism
22	K4.22	xox5oxo&#x	5x tetrahedron + 5x square pyramid + 2x pentagonal antiprism + 1x pentagonal prism
23	tetrahedral pericupola	xx3oo3ox	(1+4)x tetrahedron + (4+6)x trigonal prism + 1x stauromesohedron
24	K4.24	??	2x tetrahedron + 6x trigonal prism + 2x trigonal cupola
25	triangular orthobicupolic ring	xx3xo2ox	3x tetrahedron + (1+3)x trigonal prism + 2x trigonal cupola
26	cubic pyramid	xo4oo3oo	6x square pyramid + 1x cube
27	triangular gyrobicupolic ring	xxo3xox&#x	1x octahedron + 6x square pyramid + 2x trigonal cupola
28	K4.28	??	(4+8)x square pyramid + 2x cube + 1x stauromesohedron
29	octahedral semicupola	xo3ox4oo	(1+8)x octahedron + 6x square pyramid + 1x stauromesohedron
30	K4.30	??	(2+3)x octahedron + 6x square pyramid + 2x trigonal cupola
31	K4.31	??	4x octahedron + (1+1+4+4)x square pyramid + 1x cube + 1x stauromesohedron
32	K4.32	??	1x octahedron + (2+2+4)x square pyramid + 1x cube + 2x trigonal cupola
33	K4.33	??	3x tetrahedron + 1x octahedron + 3x square pyramid + 1x trigonal prism + 3x pentagonal pyramid + 1x tridiminished icosahedron	trigonal tridiminished-icosahedral wedge
34	5,3-duoprism	x5o2x3o	5x trigonal prism + 3x pentagonal prism
35	cubic semicupola	xo4ox3oo	8x tetrahedron + 6x square antiprism + 1x cube + 1x stauromesohedron
36	icosahedral prism	x3o5o2x	20x trigonal prism + 2x icosahedron
37	gyroelongated pentagonal pyramid prism	??	(5+5+5)x trigonal prism + 1x pentagonal prism + 2x gyroelongated pentagonal pyramid
38	pentagonal pyramid prism	xx2oo5xo	5x trigonal prism + 2x pentagonal pyramid + 1x pentagonal prism
39	pentagonal antiprism prism	??	10x trigonal prism + 2x pentagonal antiprism + 2x pentagonal prism
40	metabidiminished icosahedron prism	??	(2+2+2+4)x trigonal prism + 2x pentagonal prism + 2x metabidiminished icosahedron
41	tridiminished icosahedron prism	xxx2xfo3oox	(1+1+3)x trigonal prism + 3x pentagonal prism + 2x tridiminished icosahedron
42	pentagonal diprism	x5o2x4o	5x cube + 4x pentagonal prism
43	stauromesohedron prism	o4x3o2x	8x trigonal prism + 6x cube + 2x stauromesohedron
44	trigonal orthobicupola prism	xxx3oxo2xxx	(2+6)x trigonal prism + 6x cube + 2x trigonal orthobicupola
45	trigonal cupola prism	xx3xo2xx	(1+3)x trigonal prism + 3x cube + 2x trigonal cupola + 1x hexagonal prism	half of stauromesohedron prism
46	K4.46	xox6oxo&#x	6x tetrahedron + 6x square pyramid + 2x hexagonal antiprism + 1x hexagonal prism
47	6,3-duoprism	x6o2x3o	6x trigonal prism + 3x hexagonal prism
48	tetrahedral semipericupola	xx3xo3ox	4x octahedron + 6x trigonal prism + 1x stauromesohedron + 4x trigonal cupola + 1x truncated tetrahedron
49	K4.49	??	1x octahedron + (3+3)x square pyramid + (3+3)x trigonal prism + 1x trigonal orthobicupola + (1+3)x trigonal cupola + 1x truncated tetrahedron
50	K4.50	??	1x octahedron + 3x square pyramid + 3x trigonal prism + (1+3)x trigonal cupola + 1x hexagonal prism + 1x truncated tetrahedron
51	triangular magnabicupolic ring	xx3xo2xo	3x square pyramid + 3x trigonal prism + 2x trigonal cupola + 1x hexagonal prism
52	tetrahedral semimesocupola	xo3xx3oo	1x octahedron + 4x trigonal prism + 4x trigonal cupola + 1x truncated tetrahedron
53	hexagonal antiprism prism	??	12x trigonal prism + 2x hexagonal antiprism + 2x hexagonal prism
54	hexagonal diprism	x6o2x4o	6x cube + 4x hexagonal prism
55	tetrahedral semiantiprism	xo3xx3ox	6x tetrahedron + 8x trigonal cupola + 2x truncated tetrahedron
56	tetrahedral quasicupola	xx3xo3oo	(1+4)x tetrahedron + 4x trigonal cupola + 1x truncated tetrahedron
57	truncated tetrahedron prism	x3x3o2x	4x trigonal prism + 4x hexagonal prism + 2x truncated tetrahedron
58	K4.58	xox8oxo&#x	8x tetrahedron + 8x square pyramid + 2x octagonal antiprism + 1x octagonal prism
59	8,3-duoprism	x8o2x3o	8x trigonal prism + 3x octagonal prism
60	snub cube prism	s4s3s2x	(8+24)x trigonal prism + 6x cube + 2x snub cube
61	staurohedral mesocupola	xo4ox3xo	8x octahedron + 12x square pyramid + 6x square antiprism + 1x stauromesohedron + 1x stauroperihedron
62	K4.62	??	4x octahedron + (4+4+4)x square pyramid + (1+4)x square antiprism + 1x stauromesohedron + 1x elongated square cupola + 1 square cupola
63	K4.63	??	(4+8)x square pyramid + 4x square antiprism + 1x stauromesohedron + 2x square cupola + 1x octagonal prism
64	square gyrobicupolic ring	xxo4oxx&#x	8x square pyramid + 1x square antiprism + 2x square cupola
65	octagonal antiprism prism	??	16x trigonal prism + 2x octagonal antiprism + 2x octagonal prism
66	stauroperihedral prism	x4o3x2x	8x trigonal prism + (6+12)x cube + 2x stauroperihedron
67	elongated square gyrobicupola prism	??	8x trigonal prism + (2+8+8)x cube + 2x elongated square gyrobicupola
68	elongated square cupola prism	??	4x trigonal prism + (1+4+4+4)x cube + 2x elongated square cupola + 1x octagonal prism
69	square cupola prism	xx4xo2xx	4x trigonal prism + (1+4)x cube + 2x square cupola + 1x octagonal prism
70	octagonal diprism	x8o2x4o	8x cube + 4x octagonal prism
71	cubic pericupola	xx4oo3ox	8x tetrahedron + 12x trigonal prism + (1+6)x cube + 1x stauroperihedron
72	K4.72	??	4x tetrahedron + (4+4)x trigonal prism + (1+1+4)x cube + 1x elongated square cupola + 1 square cupola
73	square orthobicupolic ring	xx4xo2ox	4x tetrahedron + 2x square pyramid + 4x trigonal prism + 1x cube
74	dodecahedron prism	x5o3o2x	12x pentagonal prism + 2x dodecahedron
75	octahedral semipericupola	xx3xo4ox	12x trigonal prism + 6x square antiprism + 8x trigonal cupola + 1x stauroperihedron + 1x truncated octahedron
76	tetrahedral canticupola	xx3xx3xo	6x trigonal prism + 4x trigonal cupola + 4x hexagonal prism + 1 truncated tetrahedron + 1 truncated octahedron
77	dodecahedral semicupola	xo5ox3oo	20x tetrahedron + 12x pentagonal antiprism + 1x dodecahedron + 1x rhodomesohedron
78	rhodohedral antiprism	xo5oo3ox	(20+30)x tetrahedron + 1x icosahedron + 12x pentagonal pyramid + 1x dodecahedron
79	K4.79	??	(5+5+5+5+5+5+10)x tetrahedron + 1x gyroelongated pentagonal pyramid + (1+5+5)x pentagonal pyramid + 1x pentagonal antiprism + 1x dodecahedron
80	pentagonal antiprism pyramid	??	10x tetrahedron + 2x pentagonal pyramid + 1x pentagonal antiprism
81	K4.81	??	(10+10+10)x tetrahedron + 10x pentagonal pyramid + (1+2)x pentagonal antiprism + 1x dodecahedron
82	K4.82	??	(1+1+2+2+2+2+4+4+4+4+4)x tetrahedron + (2+2+2+4)x pentagonal pyramid + 2x pentagonal antiprism + 1x metabidiminished icosahedron + 1x dodecahedron
83	K4.83	??	(1+1+3+3+3+3+6)x tetrahedron + (3+3+3)x pentagonal pyramid + 3x pentagonal antiprism + 1x tridiminished icosahedron + 1x dodecahedron
84	icosahedron pyramid	xo3oo5oo	20x tetrahedron + 1x icosahedron
85	gyroelongated pentagonal pyramid pyramid	??	(5+5+5)x tetrahedron + 1x gyroelongated pentagonal pyramid + 1x pentagonal pyramid
86	pentagonal dipyramid	xo5oo2ox	5x tetrahedron + 2x pentagonal pyramid
87	metabidiminished icosahedron pyramid	??	(2+2+2+4)x tetrahedron + 2x pentagonal pyramid + 1x metabidiminished icosahedron
88	tridiminished icosahedron pyramid	??	(1+1+3)x tetrahedron + 3x pentagonal pyramid + 1x tridiminished icosahedron
89	truncated octahedron prism	x3x4o2x	6x cube + 8x hexagonal prism + 2x truncated octahedron
90	rhodomesohedron prism	??	20x trigonal prism + 12x pentagonal prism + 2x rhodomesohedron
91	orthobirotunda prism	??	(10+10)x trigonal prism + (2+10)x pentagonal prism + 2x orthobirotunda
92	rotunda prism	??	(5+5)x trigonal prism + (1+5)x pentagonal prism + 2x rotunda + 1x decagonal prism
93	K4.93	xox10oxo&#x	10x tetrahedron + 10x square pyramid + 2x decagonal antiprism + 1x decagonal prism
94	10,3-duoprism	x10o2x3o	10x trigonal prism + 3x decagonal prism
95	octahedral semimesocupola	xo3xx4oo	6x cube + 1x stauromesohedron + 8x trigonal cupola + 1x truncated octahedron
96	decagonal antiprism prism	??	20x trigonal prism + 2x decagonal antiprism + 2x decagonal prism
97	decagonal diprism	x10o2x4o	10x cube + 4x decagonal prism
98	staurohedral semiantiprism	xo4xx3ox	12x tetrahedron + 8x trigonal cupola + 6x square cupola + 1x truncated octahedron + 1x truncated cube
99	truncated cube prism	x4x3o2x	8x trigonal prism + 6x octagonal prism + 2x truncated cube
100	cubic semipericupola	xx4xo3ox	8x octahedron + 12x trigonal prism + 1x stauroperihedron + 6x square cupola + 1x truncated cube
101	K4.101	??	4x octahedron + (4+4)x square pyramid + (4+4+4)x trigonal prism + 1x elongated square gyrobicupola + (1+1+4)x square cupola + 1x truncated cube
102	K4.102	??	(8+8)x square pyramid + (4+8)x trigonal prism + 1x stauroperihedron + (2+4)x square cupola + 1x truncated cube
103	K4.103	??	4x octahedron + 4x square pyramid + (4+4)x triangular prism + 1x elongated square cupola + (1+4)x square cupola + 1x octagonal prism + 1x truncated cube
104	K4.104	??	(4+4+4)x square pyramid + (4+4)x trigonal prism + 1x elongated square cupola + (1+4)x square cupola + 1x octagonal prism + 1x truncated cube
105	square magnabicupolic ring	xx4xo2xo	4x square pyramid + 4x trigonal prism + 2x square cupola + 1x octagonal prism
106	K4.106	??	8x square pyramid + 4x trigonal prism + 4x square cupola + (1+2)x octagonal prism + 1x truncated cube
107	octahedral pericupola	xx3oo4ox	1x octahedron + 6x square pyramid + (8+12)x trigonal prism + 1x stauroperihedron
108	K4.108	??	(1+1+4)x square pyramid + (4+4+4)x trigonal prism + 1x elongated square cupola + 1x square cupola
109	K4.109	??	8x square pyramid + 2x trigonal prism + 2x square cupola
110	snub dodecahedron prism	s5s3s2x	(20+60)x trigonal prism + 12x pentagonal prism + 2x snub dodecahedron
111	rhodoperihedron prism	x5o3x2x	20x trigonal prism + 30x cube + 12x pentagonal prism + 2x rhodoperihedron
112	gyrated rhodoperihedron prism	??	(5+5+5+5)x trigonal prism + (5+5+5+5+10)x cube + (1+1+5+5)x pentagonal prism + 2x gyrated rhodoperihedron
113	parabigyrated rhodoperihedron prism	??	(10+10)x trigonal prism + (10+10+10)x cube + (2+10)x pentagonal prism + 2x parabigyrated rhodoperihedron
114	metabigyrated rhodoperihedron	??	(2+2+2+2+4+4+4)x trigonal prism + (1+1+2+2+4+4+4+4+4+4)x cube + (2+2+2+2+4)x pentagonal prism + 2x metabigyrated rhodoperihedron
115	trigyrated rhodoperihedron	??	(1+1+3+3+6+6)x trigonal prism + (3+3+3+3+6+6+6)x cube + (3+3+3+3)x pentagonal prism + 2x trigyrated rhodoperihedron
116	diminished rhodoperihedron prism	??	(5+5+5)x trigonal prism + (5+5+5+10)x cube + (1+5+5)x pentagonal prism + 1x decagonal prism + 2x diminished rhodoperihedron
117	pentagonal cupola prism	xx5xo2xx	5x trigonal prism + 5x cube + 1x pentagonal prism + 1x decagonal prism + 2x pentagonal cupola
118	gyrated paradiminished rhodoperihedron prism	??	(5+5+5)x trigonal prism + (5+5+5+10)x cube + (1+5+5)x pentagonal prism + 1x decagonal prism + 2x gyrated paradiminished rhodoperihedron
119	gyrated metadiminished rhodoperihedron prism	??	(1+1+1+2+2+2+2+2+2)x trigonal prism + (1+1+1+2+2+2+2+2+2+2+2+2+2+2)x cube + (1+1+1+2+2+2+2)x pentagonal prism + 1x decagonal prism + 2x gyrated metadiminished rhodoperihedron
120	bigyrated diminished rhodoperihedron prism	??	(1+1+1+2+2+2+2+2+2)x trigonal prism + (1+1+1+2+2+2+2+2+2+2+2+2+2+2)x cube + (1+1+1+2+2+2+2)x pentagonal prism + 1x decagonal prism + 2x bigyrated diminished rhodoperihedron
121	parabidiminished rhodoperihedron prism	??	10x trigonal prism + (10+10)x cube + 10x pentagonal prism + 2x decagonal prism + 2x parabidiminished rhodoperihedron
122	metabidiminished rhodoperihedron prism	??	(2+2+2+4)x trigonal prism + (1+1+2+4+4+4+4)x cube + (2+2+2+4)x pentagonal prism + 2x decagonal prism + 2x metabidiminished rhodoperihedron
123	gyrated bidiminished rhodoperihedron prism	??	(1+1+1+1+2+2+2)x trigonal prism + (1+1+1+1+2+2+2+2+2+2+2+2+2)x cube + (1+1+1+1+2+2+2)x pentagonal prism + 2x decagonal prism + 2x gyrated bidiminished rhodoperihedron
124	tridiminished rhodoperihedron prism	??	(1+1+3)x trigonal prism + (3+3+3+6)x cube + (3+3+3)x pentagonal prism + 3x decagonal prism + 2x tridiminished rhodoperihedron
125	stauropantohedron prism	x4x3x2x	12x cube + 8x hexagonal prism + 6x octagonal prism + 2x stauropantohedron
126	icosahedral semipericupola	xx3xo5ox	30x trigonal prism + 12x pentagonal antiprism + 20x trigonal cupola + 1x rhodoperihedron + 1x truncated icosahedron
127	truncated icosahedron prism	x3x5o2x	12x pentagonal prism + 20x hexagonal prism + 2x truncated icosahedron
128	cubic canticupola	xx4xx3ox	12x trigonal prism + 8x trigonal cupola + 6x octagonal prism + 1x truncated cube + 1x stauropantohedron
129	cubic semimesocupola	xo4xx3oo	8x trigonal prism + 1x stauromesohedron + 6x square cupola + 1x truncated cube
130	truncated dodecahedron prism	x5x3o2x	20x trigonal prism + 12x decagonal prism + 2x truncated dodecahedron
131	rhodohedral mesocupola	xo5ox3xo	20x octahedron + 30x square pyramid + 12x pentagonal antiprism + 1x rhodomesohedron + 1x rhodoperihedron
132	K4.132	??	(5+5+5)x octahedron + (5+5+5+5+10)x square pyramid + (1+5+5)x pentagonal antiprism + 1x rhodomesohedron + 1x diminished rhodoperihedron + 1x pentagonal cupola
133	pentagonal gyrobicupolic ring	xxo5oxx&#x	10x square pyramid + 1x pentagonal antiprism + 2x pentagonal cupola
134	K4.134	??	10x octahedon + (10+10+10)x square pyramid + 10x pentagonal antiprism + 1x rhodomesohedron + 2x pentagoal cupola + 1x parabidiminished rhodoperihedron
135	K4.135	??	(2+2+2+4)x octahedron + (1+1+2+2+4+4+4+4+4+4)x square pyramid + (2+2+2+4)x pentagonal antiprism + 1x rhodomesohedron + 2x pentagonal cupola + 1x metabidiminished rhodoperihedron
136	K4.136	??	(1+1+3)x octahedron + (3+3+3+3+6+6+6)x square pyramid + (3+3+3)x pentagonal antiprism + 1x rhodomesohedron + 3x pentagonal cupola + 1x tridiminished rhodoperihedron
137	K4.137	xo3ox5oo	20x octahedron + 1x icosahedron + 12x pentagonal pyramid + 1x rhodomesohedron
138	K4.138	??	(5+5+5)x octahedron + 5x square pyramid + 1x gyroelongated pentagonal pyramid + (1+5+5)x pentagonal pyramid + 1x pentagonal prism + 1x rhodomesohedron
139	K4.139	??	5x octahedron + 5x square pyramid + (1+1+5)x pentagonal pyramid + 1x pentagonal rotnda +1x pentagonal cupola
140	K4.140	??	(5+5)x octahedron + 5x square pyramid + 1x gyroelongated pentagonal pyramid + (1+5)x pentagonal pyramid + 1x pentagonal rotunda + 1x pentagonal cupola
141	pentagonal prism pyramid	xox5ooo&#x	5x square pyramid + 2x pentagonal pyramid + 1x pentagonal prism
142	K4.142	??	10x octahedron + 10x square pyramid + 10x pentagonal pyramid + 1x pentagonal antiprism + 2x pentagonal prism + 1x rhodomesohedron
143	K4.143	??	(2+2+2+4)x octahedron +(2+4+4)x square pyramid + (2+2+2+4)x pentagonal pyramid + 1x metabidiminished icosahedron + 1x pentagonal prism + 1x rhodomesohedron
144	K4.144	??	5x octahedron + (5+5)x square pyramid + 5x pentagonal pyramid +1x pentagonal antiprism + 1x pentagonal prism + 1x pentagonal rotunda + 1x pentagonal cupola
145	K4.145	??	(1+1+2+2)x octahedron + (2+2+2+2)x square pyramid + (1+2+2)x pentagonal pyramid + 1x metabidiminished icosahedron + 1x pentagonal prism + 1x pentagonal rotunda + 1x pentagonal cupola
146	K4.146	??	(5+5)x square pyramid + 5x pentagonal pyramid + 1x pentagonal prism + 1x rotunda +1x pentagonal cupola
147	K4.147	??	(1+1+3)x octahedron + (3+6+6)x square pyramid + (3+3+3)x pentagonal pyramid + 1x tridiminished icosahedron + 3x pentagonal prism + 1x rhodomesohedron
148	K4.148	??	(1+1)x octahedron + (1+2+2+2+2+2)x square pyramid + (1+1+2)x pentagonal pyramid + 1x tridiminished icosahedron + 2x pentagonal prism + 1x pentagonal rotunda + 1x pentagonal cupola
149	octahedral canticupola	xx3xx4ox	12x trigonal prism + 8x hexagonal prism + 6x square cupola + 1x truncated octahedron + 1x stauropantohedron
150	rhodopantohedral prism	x5x3x2x	30x cube + 20x hexagonal prism + 12x decagonal prism + 2x rhodopantohedron
151	rhodohedral semiantiprism	xo5xx3ox	30x tetrahedron + 20x trigonal cupola + 12x pentagonal cupola + 1x truncated icosahedron + 1x truncated dodecahedron
152	dodecahedral pericupola	xx5oo3ox	20x tetrahedron + 30x trigonal prism + 12x pentagonal prism + 1x dodecahedron + 1x rhodoperihedron
153	K4.153	??	(5+5+5)x tetrahedron + (5+5+5+10)x trigonal prism + (1+5+5+5)x pentagonal prism + 1x dodecahedron + 1x diminished rhodoperihedron + 1x pentagonal cupola
154	pentagonal orthobicupolic ring	xx5xo2ox	5x tetrahedron + 5x trigonal prism + 1x pentagonal prism + 2x pentagonal cupola
155	K4.155	??	10x tetrahedron + (10+10)x trigonal prism + 10x pentagonal prism + 1x dodecahedron + 2x pentagonal cupola + 1x parabidiminished rhodoperihedron
156	K4.156	??	(2+2+2+4)x tetrahedron + (1+1+2+4+4+4+4)x trigonal prism + (2+2+2+4)x pentagonal prism + 1x dodecahedron + 2x pentagonal cupola + 1x metabidiminished rhodoperihedron
157	K4.157	??	(1+1+3)x tetrahedron + (3+3+3+6)x trigonal prism + (3+3+3)x pentagonal prism + 1x dodecahedron + 3x pentagonal cupola + 1x tridiminished rhodoperihedron
158	icosahedral semimesocupola	ox3xx5oo	12x pentagonal prism + 20x trigonal cupola + 1x rhodomesohedron + 1x truncated icosahedron
159	dodecahedral semipericupola	xx5xo3ox	20x octahedron + 30x trigonal prism + 1x rhodoperihedron + 12x pentagonal cupola + 1x truncated dodecahedron
160	K4.160	??	(5+5+5)x octahedron + (5+5)x square pyramid + (5+5+5+5+10)x trigonal prism + 1x gyrated rhodoperihedron + (1+1+5+5)x pentagonal cupola + 1x truncated dodecahedron
161	K4.161	??	10x octahedron + (10+10)x square pyramid + (10+10+10)x trigonal prism + 1x parabigyrated rhodoperihedron + (2+10)x pentagonal cupola + 1x truncated dodecahedron
162	K4.162	??	(2+2+2+4)x octahedron + (2+2+4+4+4+4)x square pyramid + (1+1+2+2+4+4+4+4+4+4)x trigonal prism + 1x metabigyrated rhodoperihedron + (2+2+2+2+4)x pentagonal cupola + 1x truncated dodecahedron
163	K4.163	??	(1+1+3)x octahedron + (3+3+6+6+6+6)x square pyramid + (3+3+3+3+6+6+6)x trigonal prism + 1x trigyrated rhodoperihedron + (3+3+3+3)x pentagonal cupola + 1x truncated dodecahedron
164	K4.164	??	(5+5+5)x octahedron + 5x square pyramid + (5+5+5+10)x trigonal prism + 1x decagonal prism + 1x diminished rhodoperihedron + (1+5+5)x pentagonal cupola + 1x truncated dodecahedron
165	pentagonal magnabicupolic ring	xx5xo2ox	5x square pyramid + 5x trigonal prism + 1x decagonal prism + 2x pentagonal cupola
166	K4.166	??	(5+5)x octahedron + (5+5+5)x square pyramid + (5+5+5+10)x trigonal prism + (1+1+5)x pentagonal cupola + 1x gyrated paradiminished rhodoperihedron + 1x truncated dodecahedron
167	K4.167	??	(1+1+2+2+2+2)x octahedron + (1+1+1+2+2+2+2+2+2)x square pyramid + (1+1+1+2+2+2+2+2+2+2+2+2+2+2)x trigonal prism + (1+1+1+2+2+2+2)x pentagonal cupola + 1x gyrated metadiminished rhodoperihedron + 1x truncated dodecahedron
168	K4.168	??	(1+1+1+2)x octahedron + (1+2+2+2+2+2+2+2+2+2+2+2+2)x square pyramid + (1+1+1+2+2+2+2+2+2+2+2+2+2+2)x trigonal prism + (1+1+1+2+2+2+2)x pentagonal cupola + 1x bigyrated diminished rhodoperihedron + 1x truncated dodecahedron
169	K4.169	??	10x octahedron + 10x square pyramid + (10+10)x trigonal prism + 2x decagonal prism + 10x pentagonal cupola + 1x parabidiminished rhodoperihedron + 1x truncated dodecahedron
170	K4.170	??	(2+2+2+4)x octahedron + (2+4+4)x square pyramid + (1+1+2+4+4+4+4)x trigonal prism + 2x decagonal prism + (2+2+2+4)x pentagonal cupola + 1x metabidiminished rhodoperihedron + 1x truncated dodecahedron
171	K4.171	??	(1+1+1+2)x octahedron + (1+1+2+2+2+2+2+2+2+2+2)x square pyramid + (1+1+1+2+2+2+2+2+2+2+2+2+2+2)x trigonal prism + 2x decagonal prism + (1+1+1+1+2+2+2)x pentagonal cupola + 1x gyrated bidiminished rhodoperihedron + 1x truncated dodecahedron
172	K4.172	??	(1+1+3)x octahedron + (3+3+6)x square pyramid + (3+3+3+6)x trigonal prism + 3x decagonal prism + (3+3+3)x pentagonal cupola + 1x tridiminished rhodoperihedron + 1x truncated dodecahedron
173	dodecahedral canticupola	xx5xx3ox	30x trigonal prism + 20x trigonal cupola + 12x decagonal prism + 1x truncated dodecahedron + 1x rhodopantohedron
174	biantiprismatic rings	xoxnoxo&#x	nx tetrahedron + nx square pyramid + 2x n-antiprism + 1x n-prism
175	3,n-duoprisms	xno2x3o	nx trigonal prism + 3x n-gonal prism
176	n-antiprism prisms	??	2nx trigonal prism + 2x n-gonal antiprism + 2x n-gonal prism
177	4,n-duoprisms	xno2x4o	nx cube + 4x n-gonal prism

Pages in this category (5)

Bicupolic ring Gyrated octahedral prism

K4.109 Snubdis antiprism

Square dipyramid