Segmentotope (EntityClass, 11)
From Hi.gher. Space
A segmentochoron is an orbiform polychoron whose vertices lie on two parallel hyperplanes and whose edge lengths are all equal. The set of all convex segmentochora has been enumerated by Dr. Richard Klitzing. There are 177 of them, 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 segmentochoron is denoted by the notation A  B, where A and B are lowerdimensional polytopes. A and B are usually polyhedra, although one of them can be lowerdimensional, as is the case with the wedges and pyramids.
Some segmentochora 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 

 
K 4.26 



K 4.45 
triangular_cupola  triangular_cupola 
 
K 4.51 
 
K 4.69 
square_cupola  square_cupola 
 
K 4.105 
 
K 4.117 
pentagonal_cupola  pentagonal_cupola 
 
K 4.141 
 
K 4.165 

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:
 3 regular polychora
 1, 2, 20
 the 3pyrotomochoron, a uniform polychoron
 5
 21 members of infinite families
 6, 10, 14, 18, 19, 22, 34, 39, 42, 46, 47, 53, 54, 58, 59, 65, 70, 93, 94, 96, 97
 4 infinite families (not individual polychora)
 174, 175, 176, 177
 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
 12 pyramids of CRF polyhedra
 3, 4, 7, 17, 26, 80, 84, 85, 86, 87, 88, 141
 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
 30 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
 all 10 bicupolic rings
 8, 25, 27, 51, 64, 73, 105, 133, 154, 165
 3 fragments of cupolic forms (excluding the bicupolic rings)
 109, 139, 146
 the gyrated octahedral prism
 13
 the trigonal tridiminishedicosahedral wedge
 33
 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
#  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  3pyrotomochoron  o3x3o3o  5x tetrahedron + 5x octahedron  tetrahedral semicupola 
6  K4.6  ??  3x tetrahedron + 2x octahedron + 3x square pyramid + 1x trigonal prism  
7  triangular prism pyramid  xo3oo2xo  2x tetrahedron + 3x square pyramid + 1x trigonal prism  
8  K4.8  ??  1x tetrahedron + 4x square pyramid + 2x trigonal prism  bidiminished 3pyrotomochoron 
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  K4.14  ??  4x tetrahedron + 4x square pyramid + 2x square antiprism + 1x cube  
15  stauroantiprism  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  ??  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  K4.21  xo4os3os  8x tetrahedron + 12x square pyramid + 6x trigonal prism + 1x cube + 1x icosahedron  snubdis antiprism 
22  K4.22  ??  5x tetrahedron + 5x square pyramid + 2x pentagonal antiprism + 1x pentagonal prism  
23  pyropericupola  xx3oo3ox  (1+4)x tetrahedron + (4+6)x trigonal prism + 1x cuboctahedron  
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  ??  1x octahedron + 6x square pyramid + 2x trigonal cupola  
28  K4.28  ??  (4+8)x square pyramid + 2x cube + 1x cuboctahedron  
29  aerosemicupola  xo3ox4oo  (1+8)x octahedron + 6x square pyramid + 1x cuboctahedron  
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 cuboctahedron  
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 tridiminishedicosahedral wedge 
34  5,3duoprism  x5o2x3o  5x trigonal prism + 3x pentagonal prism  
35  geosemicupola  xo4ox3oo  8x tetrahedron + 6x square antiprism + 1x cube + 1x cuboctahedron  
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  cuboctahedron prism  o4x3o2x  8x trigonal prism + 6x cube + 2x cuboctahedron  
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 cuboctahedron prism 
46  K4.46  ??  6x tetrahedron + 6x square pyramid + 2x hexagonal antiprism + 1x hexagonal prism  
47  6,3duoprism  x6o2x3o  6x trigonal prism + 3x hexagonal prism  
48  pyrosemipericupola  xx3xo3ox  4x octahedron + 6x trigonal prism + 1x cuboctahedron + 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  pyrosemimesocupola  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  pyrosemiantiprism  xo3xx3ox  6x tetrahedron + 8x trigonal cupola + 2x truncated tetrahedron  
56  pyroquasicupola  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  ??  8x tetrahedron + 8x square pyramid + 2x octagonal antiprism + 1x octagonal prism  
59  8,3duoprism  x8o2x3o  8x trigonal prism + 3x octagonal prism  
60  snub cube prism  s4s3s2x  (8+24)x trigonal prism + 6x cube + 2x snub cube  
61  stauromesocupola  xo4ox3xo  8x octahedron + 12x square pyramid + 6x square antiprism + 1x cuboctahedron + 1x rhombicuboctahedron  
62  K4.62  ??  4x octahedron + (4+4+4)x square pyramid + (1+4)x square antiprism + 1x cuboctahedron + 1x elongated square cupola + 1 square cupola  
63  K4.63  ??  (4+8)x square pyramid + 4x square antiprism + 1x cuboctahedron + 2x square cupola + 1x octagonal prism  
64  square gyrobicupolic ring  ??  8x square pyramid + 1x square antiprism + 2x square cupola  
65  octagonal antiprism prism  ??  16x trigonal prism + 2x octagonal antiprism + 2x octagonal prism  
66  rhombicuboctahedral prism  x4o3x2x  8x trigonal prism + (6+12)x cube + 2x rhombicuboctahedron  
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  geopericupola  xx4oo3ox  8x tetrahedron + 12x trigonal prism + (1+6)x cube + 1x rhombicuboctahedron  
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  aerosemipericupola  xx3xo4ox  12x trigonal prism + 6x square antiprism + 8x trigonal cupola + 1x rhombicuboctahedron + 1x truncated octahedron  
76  pyrocanticupola  xx3xx3xo  6x trigonal prism + 4x trigonal cupola + 4x hexagonal prism + 1 truncated tetrahedron + 1 truncated octahedron  
77  cosmosemicupola  xo5ox3oo  20x tetrahedron + 12x pentagonal antiprism + 1x dodecahedron + 1x icosidodecahedron  
78  rhodoantiprism  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  icosidodecahedron prism  ??  20x trigonal prism + 12x pentagonal prism + 2x icosidodecahedron  
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  ??  10x tetrahedron + 10x square pyramid + 2x decagonal antiprism + 1x decagonal prism  
94  10,3duoprism  x10o2x3o  10x trigonal prism + 3x decagonal prism  
95  aerosemimesocupola  xo3xx4oo  6x cube + 1x cuboctahedron + 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  staurosemiantiprism  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  geosemipericupola  xx4xo3ox  8x octahedron + 12x trigonal prism + 1x rhombicuboctahedron + 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 rhombicuboctahedron + (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  aeropericupola  xx3oo4ox  1x octahedron + 6x square pyramid + (8+12)x trigonal prism + 1x rhombicuboctahedron  
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  rhombicosidodecahedron prism  x5o3x2x  20x trigonal prism + 30x cube + 12x pentagonal prism + 2x rhombicosidodecahedron  
112  gyrated rhombicosidodecahedron prism  ??  (5+5+5+5)x trigonal prism + (5+5+5+5+10)x cube + (1+1+5+5)x pentagonal prism + 2x gyrated rhombicosidodecahedron  
113  parabigyrated rhombicosidodecahedron prism  ??  (10+10)x trigonal prism + (10+10+10)x cube + (2+10)x pentagonal prism + 2x parabigyrated rhombicosidodecahedron  
114  metabigyrated rhombicosidodecahedron  ??  (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 rhombicosidodecahedron  
115  trigyrated rhombicosidodecahedron  ??  (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 rhombicosidodecahedron  
116  diminished rhombicosidodecahedron prism  ??  (5+5+5)x trigonal prism + (5+5+5+10)x cube + (1+5+5)x pentagonal prism + 1x decagonal prism + 2x diminished rhombicosidodecahedron  
117  pentagonal cupola prism  xx5xo2xx  5x trigonal prism + 5x cube + 1x pentagonal prism + 1x decagonal prism + 2x pentagonal cupola  
118  gyrated paradiminished rhombicosidodecahedron 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 rhombicosidodecahedron  
119  gyrated metadiminished rhombicosidodecahedron 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 rhombicosidodecahedron  
120  bigyrated diminished rhombicosidodecahedron 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 rhombicosidodecahedron  
121  parabidiminished rhombicosidodecahedron prism  ??  10x trigonal prism + (10+10)x cube + 10x pentagonal prism + 2x decagonal prism + 2x parabidiminished rhombicosidodecahedron  
122  metabidiminished rhombicosidodecahedron 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 rhombicosidodecahedron  
123  gyrated bidiminished rhombicosidodecahedron 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 rhombicosidodecahedron  
124  tridiminished rhombicosidodecahedron prism  ??  (1+1+3)x trigonal prism + (3+3+3+6)x cube + (3+3+3)x pentagonal prism + 3x decagonal prism + 2x tridiminished rhombicosidodecahedron  
125  truncated cuboctahedron prism  x4x3x2x  12x cube + 8x hexagonal prism + 6x octagonal prism + 2x truncated cuboctahedron  
126  hydrosemipericupola  xx3xo5ox  30x trigonal prism + 12x pentagonal antiprism + 20x trigonal cupola + 1x rhombicosidodecahedron + 1x truncated icosahedron  
127  truncated icosahedron prism  x3x5o2x  12x pentagonal prism + 20x hexagonal prism + 2x truncated icosahedron  
128  geocanticupola  xx4xx3ox  12x trigonal prism + 8x trigonal cupola + 6x octagonal prism + 1x truncated cube + 1x truncated cuboctahedron  
129  geosemimesocupola  xo4xx3oo  8x trigonal prism + 1x cuboctahedron + 6x square cupola + 1x truncated cube  
130  truncated dodecahedron prism  x5x3o2x  20x trigonal prism + 12x decagonal prism + 2x truncated dodecahedron  
131  rhodomesocupola  xo5ox3xo  20x octahedron + 30x square pyramid + 12x pentagonal antiprism + 1x icosidodecahedron + 1x rhombicosidodecahedron 
(To be completed)