Segmentotope (EntityClass, 11)
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;3 fragments of cupolic forms (excluding the bicupolic rings)  ;3 fragments of cupolic forms (excluding the bicupolic rings)  
:109, 139, 146  :109, 139, 146  
  ;1 nonuniform scaliform (  +  ;1 nonuniform scaliform (truncated tetrahedral cupoliprism) 
:55  :55  
;the gyrated octahedral prism  ;the gyrated octahedral prism 
Revision as of 20:27, 16 June 2013
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
 29 cupolic forms
 15, 21, 23, 29, 35, 48, 52, 56, 61, 71, 75, 76, 77, 78, 95, 98, 100, 107, 126, 128, 129, 131, 137, 149, 151, 152, 158, 159, 173
 all 9 bicupolic rings
 25, 27, 51, 64, 73, 105, 133, 154, 165
 3 fragments of cupolic forms (excluding the bicupolic rings)
 109, 139, 146
 1 nonuniform scaliform (truncated tetrahedral cupoliprism)
 55
 the gyrated octahedral prism
 13
 the trigonal tridiminishedicosahedral wedge
 33
 the diminished 3gon antiprismatic ring/bidimished 3pyrotomochoron
 8
 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  xo2xo2ox  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  ??  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 
(To be completed)