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 lower-dimensional polytopes. A and B are usually polyhedra, although one of them can be lower-dimensional, 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 3-pyrotomochoron, 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 non-uniform scaliform (Truncated tetrahedral cupoliprism)
- 55
- the gyrated octahedral prism
- 13
- the trigonal tridiminished-icosahedral wedge
- 33
- the diminished 3-gon antiprismatic ring/bidimished 3-pyrotomochoron
- 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 | Cells | Comments |
|---|---|---|---|
| 1 | pyrochoron | 5x tetrahedron | |
| 2 | aerochoron | 16x tetrahedron | |
| 3 | octahedral pyramid | 8x tetrahedron + 1x octahedron | half of aerochoron |
| 4 | square bipyramid | 4x tetrahedron + 2x square pyramid | quarter of aerochoron |
| 5 | 3-pyrotomochoron | 5x tetrahedron + 5x octahedron | tetrahedral semicupola |
| 6 | K4.6 | 3x tetrahedron + 2x octahedron + 3x square pyramid + 1x trigonal prism | |
| 7 | triangular prism pyramid | 2x tetrahedron + 3x square pyramid + 1x trigonal prism | |
| 8 | K4.8 | 1x tetrahedron + 4x square pyramid + 2x trigonal prism | bidiminished 3-pyrotomochoron |
| 9 | tetrahedral prism | 2x tetrahedron + 4x trigonal prism |
(To be completed)
