Segmentotope (EntityClass, 11)
From Hi.gher. Space
m (ontology) 
(→Classification: divide into whether they have any pentagonal faces or not, cube  ike is special) 

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;3 regular polychora  ;3 regular polychora  
:1, 2, 20  :1, 2, 20  
  ;  +  ;the 3pyrotomochoron, a uniform polychoron 
:5  :5  
;21 members of infinite families  ;21 members of infinite families  
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;all 9 [[bicupolic ring]]s  ;all 9 [[bicupolic ring]]s  
:25, 27, 51, 64, 73, 105, 133, 154, 165  :25, 27, 51, 64, 73, 105, 133, 154, 165  
  ;  +  ;the special cube  icosahedron 
  :8, 13, 16  +  :21 
+  ;35 asyetunclassified polychora with no pentagonal faces  
+  :8, 13, 16, 23, 24, 28, 30, 31, 32, 48, 49, 50, 52, 55, 56, 62, 63, 71, 72, 75, 76, 95, 98, 100, 101, 102, 103, 104, 106, 107, 108, 109, 128, 129, 149  
+  ;42 asyetunclassified polychora with at least one pentagonal face  
+  :21, 33, 79, 81, 82, 83, 126, 131, 132, 134, 135, 136, 138, 139, 140, 142, 143, 144, 145, 146, 147, 148, 151, 152, 153, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 166, 167, 168, 169, 170, 171, 172, 173 
Revision as of 14:31, 27 August 2012
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
 7 prismatoid forms (cupolae and antiprisms of regular polyhedra)
 15, 29, 35, 61, 77, 78, 137
 all 9 bicupolic rings
 25, 27, 51, 64, 73, 105, 133, 154, 165
 the special cube  icosahedron
 21
 35 asyetunclassified polychora with no pentagonal faces
 8, 13, 16, 23, 24, 28, 30, 31, 32, 48, 49, 50, 52, 55, 56, 62, 63, 71, 72, 75, 76, 95, 98, 100, 101, 102, 103, 104, 106, 107, 108, 109, 128, 129, 149
 42 asyetunclassified polychora with at least one pentagonal face
 21, 33, 79, 81, 82, 83, 126, 131, 132, 134, 135, 136, 138, 139, 140, 142, 143, 144, 145, 146, 147, 148, 151, 152, 153, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 166, 167, 168, 169, 170, 171, 172, 173