Segmentotope (EntityClass, 11)
From Hi.gher. Space
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(→Full list: add K4.56-78) |
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|55||[[pyrosemiantiprism]]||xo3xx3ox||6x tetrahedron + 8x trigonal cupola + 2x truncated tetrahedron|| | |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||[[8,3-duoprism]]||x8o2x3o||8x trigonal prism + 3x octagonal prism|| | ||
+ | |- | ||
+ | |60||[[snub cube prism]]||s4s3s2x||(8+24)x trigonal prism + 6x cube + 2x [[snub cube]]|| | ||
+ | |- | ||
+ | |61||[[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|| | ||
|} | |} | ||
(To be completed) | (To be completed) |
Revision as of 22:45, 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 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
- 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 9 bicupolic rings
- 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 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 | 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 | 3-pyrotomochoron | 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 3-pyrotomochoron |
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 tridiminished-icosahedral wedge |
34 | 5,3-duoprism | 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 | 6,3-duoprism | 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 | 8,3-duoprism | x8o2x3o | 8x trigonal prism + 3x octagonal prism | |
60 | snub cube prism | s4s3s2x | (8+24)x trigonal prism + 6x cube + 2x snub cube | |
61 | 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 |
(To be completed)