Segmentotope (EntityClass, 11)

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(Redirected from Segmentochoron)

A segmentotope is an orbiform polytope whose vertices lie on two parallel hyperplanes and whose edge lengths are all equal. The set of all convex segmentotopes up to and including 4D has been enumerated by Dr. Richard Klitzing. There are 177 four-dimensional segmentotopes, 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 segmentotope is denoted by the notation A || B, where A and B are lower-dimensional polytopes. For segmentochora, A and B are usually polyhedra, although one of them can be lower-dimensional, as is the case with the wedges and pyramids.

Some segmentotopes 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)CellsValues

K 4.7

digon || square_pyramid

  • Height of triangular prism pyramid: E*sqrt(5/12)
  • Dichoral angle between tetrahedron and triangular prism: atan(sqrt(5/3)) ≈ 52.24°
  • Dichoral angle between square pyramid and triangular prism: atan(sqrt(5)) ≈ 65.91°

K 4.26

square || square_pyramid

  • Dichoral angle between square pyramid and cube: 45° (exact)
  • Dichoral angle between adjacent square pyramids: 180° - acos(1/3) ≈ 109.47°

K 4.45

triangular_cupola || triangular_cupola

  • Distance between hexagonal prism and antipodal triangular prism: E*sqrt(2/3). (Same as the height of a triangular cupola.)

K 4.51

hexagon || trigonal_cupola

  • Distance from hexagonal prism to triangle: E*sqrt(5/12)
  • Dichoral angle between trigonal cupola and hexagonal prism: atan(sqrt(5/3)) ≈ 52.23°
  • Dichoral angle between triangular prism and hexagonal prism: acos(2/3) ≈ 48.19°
  • Dichoral angle between square pyramid and hexagonal prism: atan(sqrt(5)) ≈ 65.91°

K 4.69

square_cupola || square_cupola

  • Distance between octagonal prism and antipodal cube: E*sqrt(2)/2. (Same as height of square cupola.)

K 4.105

octagon || square_cupola

  • Distance from octagonal prism to square: E*sqrt(2)/2
  • Dichoral angle between square cupola and octagonal prism: 45° (exact)
  • Dichoral angle between square pyramid and octagonal prism: 45° (exact)
  • Dichoral angle between triangular prism and octagonal prism: acos(sqrt(2/3)) ≈ 35.26°

K 4.117

pentagonal_cupola || pentagonal_cupola

  • Distance between pentagonal prism and decagonal prism: E*(sqrt(2*sqrt(2*(3*sqrt(5)+7)) - (12*sqrt(5)+20)/5)/2). (Same as height of pentagonal cupola.)

K 4.141

pentagon || pentagonal_pyramid

  • Distance between pentagonal prism and antipodal point: (E/2)*sqrt((5-2*sqrt(5))/5)
  • Dichoral angle between pentagonal pyramid and pentagonal prism: 18° (exact)
  • Dichoral angle between square pyramid and pentagonal prism: atan(sqrt(5)-2) ≈ 13.28°

K 4.165

decagon || pentagonal_cupola

  • Distance between decagonal prism and pentagon: (E/2)*sqrt((5-2*sqrt(5))/5)
  • Dichoral angle between pentagonal cupola and decagonal prism: 18° (exact)
  • Dichoral angle between square pyramid and decagonal prism: atan(sqrt(9-4*sqrt(5))) ≈ 13.28°
  • Dichoral angle between triangular prism and decagonal prism: asin(sqrt((5-2*sqrt(5))/15)) ≈ 10.81°

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 10 bicupolic rings (excluding the prisms of the tetrahedron and square pyramid, already counted above)
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 tridiminished-icosahedral 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 of segmentochora

# Name Symbol Cells Comments
1pyrochoronx3o3o3o5x tetrahedron
2aerochoronx3o3o4o16x tetrahedron
3octahedral pyramidxo3oo4oo8x tetrahedron + 1x octahedronhalf of aerochoron
4square dipyramidxo4oo2ox4x tetrahedron + 2x square pyramidquarter of aerochoron
53-pyrotomochorono3x3o3o5x tetrahedron + 5x octahedrontetrahedral semicupola
6K4.6xox3oxo&#x3x tetrahedron + 2x octahedron + 3x square pyramid + 1x trigonal prism
7triangular prism pyramidxo3oo2xo2x tetrahedron + 3x square pyramid + 1x trigonal prism
8digonal gyrobicupolic ringxxo2xox&#x1x tetrahedron + 4x square pyramid + 2x trigonal prismbidiminished 3-pyrotomochoron
9tetrahedral prismx3o3o2x2x tetrahedron + 4x trigonal prism
10duotrianglinderx3o2x3o6x trigonal prism
11octahedral prismx3o4o2x2x octahedron + 8x trigonal prism
12square pyramid prismxx2oo4xo2x square pyramid + 4x trigonal prism + 1x cube
13gyrated octahedral prism??4x square pyramid + (4+4)x trigonal prism
14K4.14xox4oxo&#x4x tetrahedron + 4x square pyramid + 2x square antiprism + 1x cube
15staurohedral antiprismxo4oo3ox(8+12)x tetrahedron + 1x octahedron + 6x square pyramid + 1x cube
16K4.16??(4+4+4)x tetrahedron + (1+1+4)x square pyramid + 1x square antiprism + 1x cube
17square antiprism pyramidoxo4oox&#x8x tetrahedron + 2x square pyramid + 1x square antiprism
18triangular diprismx3o2x4o4x trigonal prism + 3x cube
19square antiprism prism??8x trigonal prism + 2x square antiprism + 2x cube
20geochoronx4o3o3o8x cube
21snubdis antiprismxo4os3os8x tetrahedron + 12x square pyramid + 6x trigonal prism + 1x cube + 1x icosahedronsnubdis antiprism
22K4.22xox5oxo&#x5x tetrahedron + 5x square pyramid + 2x pentagonal antiprism + 1x pentagonal prism
23tetrahedral pericupolaxx3oo3ox(1+4)x tetrahedron + (4+6)x trigonal prism + 1x cuboctahedron
24K4.24??2x tetrahedron + 6x trigonal prism + 2x trigonal cupola
25triangular orthobicupolic ringxx3xo2ox3x tetrahedron + (1+3)x trigonal prism + 2x trigonal cupola
26cubic pyramidxo4oo3oo6x square pyramid + 1x cube
27triangular gyrobicupolic ringxxo3xox&#x1x octahedron + 6x square pyramid + 2x trigonal cupola
28K4.28??(4+8)x square pyramid + 2x cube + 1x cuboctahedron
29octahedral semicupolaxo3ox4oo(1+8)x octahedron + 6x square pyramid + 1x cuboctahedron
30K4.30??(2+3)x octahedron + 6x square pyramid + 2x trigonal cupola
31K4.31??4x octahedron + (1+1+4+4)x square pyramid + 1x cube + 1x cuboctahedron
32K4.32??1x octahedron + (2+2+4)x square pyramid + 1x cube + 2x trigonal cupola
33K4.33??3x tetrahedron + 1x octahedron + 3x square pyramid + 1x trigonal prism + 3x pentagonal pyramid + 1x tridiminished icosahedrontrigonal tridiminished-icosahedral wedge
345,3-duoprismx5o2x3o5x trigonal prism + 3x pentagonal prism
35cubic semicupolaxo4ox3oo8x tetrahedron + 6x square antiprism + 1x cube + 1x cuboctahedron
36icosahedral prismx3o5o2x20x trigonal prism + 2x icosahedron
37gyroelongated pentagonal pyramid prism??(5+5+5)x trigonal prism + 1x pentagonal prism + 2x gyroelongated pentagonal pyramid
38pentagonal pyramid prismxx2oo5xo5x trigonal prism + 2x pentagonal pyramid + 1x pentagonal prism
39pentagonal antiprism prism??10x trigonal prism + 2x pentagonal antiprism + 2x pentagonal prism
40metabidiminished icosahedron prism??(2+2+2+4)x trigonal prism + 2x pentagonal prism + 2x metabidiminished icosahedron
41tridiminished icosahedron prismxxx2xfo3oox(1+1+3)x trigonal prism + 3x pentagonal prism + 2x tridiminished icosahedron
42pentagonal diprismx5o2x4o5x cube + 4x pentagonal prism
43cuboctahedron prismo4x3o2x8x trigonal prism + 6x cube + 2x cuboctahedron
44trigonal orthobicupola prismxxx3oxo2xxx(2+6)x trigonal prism + 6x cube + 2x trigonal orthobicupola
45trigonal cupola prismxx3xo2xx(1+3)x trigonal prism + 3x cube + 2x trigonal cupola + 1x hexagonal prismhalf of cuboctahedron prism
46K4.46xox6oxo&#x6x tetrahedron + 6x square pyramid + 2x hexagonal antiprism + 1x hexagonal prism
476,3-duoprismx6o2x3o6x trigonal prism + 3x hexagonal prism
48tetrahedral semipericupolaxx3xo3ox4x octahedron + 6x trigonal prism + 1x cuboctahedron + 4x trigonal cupola + 1x truncated tetrahedron
49K4.49??1x octahedron + (3+3)x square pyramid + (3+3)x trigonal prism + 1x trigonal orthobicupola + (1+3)x trigonal cupola + 1x truncated tetrahedron
50K4.50??1x octahedron + 3x square pyramid + 3x trigonal prism + (1+3)x trigonal cupola + 1x hexagonal prism + 1x truncated tetrahedron
51triangular magnabicupolic ringxx3xo2xo3x square pyramid + 3x trigonal prism + 2x trigonal cupola + 1x hexagonal prism
52tetrahedral semimesocupolaxo3xx3oo1x octahedron + 4x trigonal prism + 4x trigonal cupola + 1x truncated tetrahedron
53hexagonal antiprism prism??12x trigonal prism + 2x hexagonal antiprism + 2x hexagonal prism
54hexagonal diprismx6o2x4o6x cube + 4x hexagonal prism
55tetrahedral semiantiprismxo3xx3ox6x tetrahedron + 8x trigonal cupola + 2x truncated tetrahedron
56tetrahedral quasicupolaxx3xo3oo(1+4)x tetrahedron + 4x trigonal cupola + 1x truncated tetrahedron
57truncated tetrahedron prismx3x3o2x4x trigonal prism + 4x hexagonal prism + 2x truncated tetrahedron
58K4.58xox8oxo&#x8x tetrahedron + 8x square pyramid + 2x octagonal antiprism + 1x octagonal prism
598,3-duoprismx8o2x3o8x trigonal prism + 3x octagonal prism
60snub cube prisms4s3s2x(8+24)x trigonal prism + 6x cube + 2x snub cube
61staurohedral mesocupolaxo4ox3xo8x octahedron + 12x square pyramid + 6x square antiprism + 1x cuboctahedron + 1x rhombicuboctahedron
62K4.62??4x octahedron + (4+4+4)x square pyramid + (1+4)x square antiprism + 1x cuboctahedron + 1x elongated square cupola + 1 square cupola
63K4.63??(4+8)x square pyramid + 4x square antiprism + 1x cuboctahedron + 2x square cupola + 1x octagonal prism
64square gyrobicupolic ringxxo4oxx&#x8x square pyramid + 1x square antiprism + 2x square cupola
65octagonal antiprism prism??16x trigonal prism + 2x octagonal antiprism + 2x octagonal prism
66rhombicuboctahedral prismx4o3x2x8x trigonal prism + (6+12)x cube + 2x rhombicuboctahedron
67elongated square gyrobicupola prism??8x trigonal prism + (2+8+8)x cube + 2x elongated square gyrobicupola
68elongated square cupola prism??4x trigonal prism + (1+4+4+4)x cube + 2x elongated square cupola + 1x octagonal prism
69square cupola prismxx4xo2xx4x trigonal prism + (1+4)x cube + 2x square cupola + 1x octagonal prism
70octagonal diprismx8o2x4o8x cube + 4x octagonal prism
71cubic pericupolaxx4oo3ox8x tetrahedron + 12x trigonal prism + (1+6)x cube + 1x rhombicuboctahedron
72K4.72??4x tetrahedron + (4+4)x trigonal prism + (1+1+4)x cube + 1x elongated square cupola + 1 square cupola
73square orthobicupolic ringxx4xo2ox4x tetrahedron + 2x square pyramid + 4x trigonal prism + 1x cube
74dodecahedron prismx5o3o2x12x pentagonal prism + 2x dodecahedron
75octahedral semipericupolaxx3xo4ox12x trigonal prism + 6x square antiprism + 8x trigonal cupola + 1x rhombicuboctahedron + 1x truncated octahedron
76tetrahedral canticupolaxx3xx3xo6x trigonal prism + 4x trigonal cupola + 4x hexagonal prism + 1 truncated tetrahedron + 1 truncated octahedron
77dodecahedral semicupolaxo5ox3oo20x tetrahedron + 12x pentagonal antiprism + 1x dodecahedron + 1x icosidodecahedron
78rhodohedral antiprismxo5oo3ox(20+30)x tetrahedron + 1x icosahedron + 12x pentagonal pyramid + 1x dodecahedron
79K4.79??(5+5+5+5+5+5+10)x tetrahedron + 1x gyroelongated pentagonal pyramid + (1+5+5)x pentagonal pyramid + 1x pentagonal antiprism + 1x dodecahedron
80pentagonal antiprism pyramid??10x tetrahedron + 2x pentagonal pyramid + 1x pentagonal antiprism
81K4.81??(10+10+10)x tetrahedron + 10x pentagonal pyramid + (1+2)x pentagonal antiprism + 1x dodecahedron
82K4.82??(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
83K4.83??(1+1+3+3+3+3+6)x tetrahedron + (3+3+3)x pentagonal pyramid + 3x pentagonal antiprism + 1x tridiminished icosahedron + 1x dodecahedron
84icosahedron pyramidxo3oo5oo20x tetrahedron + 1x icosahedron
85gyroelongated pentagonal pyramid pyramid??(5+5+5)x tetrahedron + 1x gyroelongated pentagonal pyramid + 1x pentagonal pyramid
86pentagonal dipyramidxo5oo2ox5x tetrahedron + 2x pentagonal pyramid
87metabidiminished icosahedron pyramid??(2+2+2+4)x tetrahedron + 2x pentagonal pyramid + 1x metabidiminished icosahedron
88tridiminished icosahedron pyramid??(1+1+3)x tetrahedron + 3x pentagonal pyramid + 1x tridiminished icosahedron
89truncated octahedron prismx3x4o2x6x cube + 8x hexagonal prism + 2x truncated octahedron
90icosidodecahedron prism??20x trigonal prism + 12x pentagonal prism + 2x icosidodecahedron
91orthobirotunda prism??(10+10)x trigonal prism + (2+10)x pentagonal prism + 2x orthobirotunda
92rotunda prism??(5+5)x trigonal prism + (1+5)x pentagonal prism + 2x rotunda + 1x decagonal prism
93K4.93xox10oxo&#x10x tetrahedron + 10x square pyramid + 2x decagonal antiprism + 1x decagonal prism
9410,3-duoprismx10o2x3o10x trigonal prism + 3x decagonal prism
95octahedral semimesocupolaxo3xx4oo6x cube + 1x cuboctahedron + 8x trigonal cupola + 1x truncated octahedron
96decagonal antiprism prism??20x trigonal prism + 2x decagonal antiprism + 2x decagonal prism
97decagonal diprismx10o2x4o10x cube + 4x decagonal prism
98staurohedral semiantiprismxo4xx3ox12x tetrahedron + 8x trigonal cupola + 6x square cupola + 1x truncated octahedron + 1x truncated cube
99truncated cube prismx4x3o2x8x trigonal prism + 6x octagonal prism + 2x truncated cube
100cubic semipericupolaxx4xo3ox8x octahedron + 12x trigonal prism + 1x rhombicuboctahedron + 6x square cupola + 1x truncated cube
101K4.101??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
102K4.102??(8+8)x square pyramid + (4+8)x trigonal prism + 1x rhombicuboctahedron + (2+4)x square cupola + 1x truncated cube
103K4.103??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
104K4.104??(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
105square magnabicupolic ringxx4xo2xo4x square pyramid + 4x trigonal prism + 2x square cupola + 1x octagonal prism
106K4.106??8x square pyramid + 4x trigonal prism + 4x square cupola + (1+2)x octagonal prism + 1x truncated cube
107octahedral pericupolaxx3oo4ox1x octahedron + 6x square pyramid + (8+12)x trigonal prism + 1x rhombicuboctahedron
108K4.108??(1+1+4)x square pyramid + (4+4+4)x trigonal prism + 1x elongated square cupola + 1x square cupola
109K4.109??8x square pyramid + 2x trigonal prism + 2x square cupola
110snub dodecahedron prisms5s3s2x(20+60)x trigonal prism + 12x pentagonal prism + 2x snub dodecahedron
111rhombicosidodecahedron prismx5o3x2x20x trigonal prism + 30x cube + 12x pentagonal prism + 2x rhombicosidodecahedron
112gyrated 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
113parabigyrated rhombicosidodecahedron prism??(10+10)x trigonal prism + (10+10+10)x cube + (2+10)x pentagonal prism + 2x parabigyrated rhombicosidodecahedron
114metabigyrated 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
115trigyrated 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
116diminished 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
117pentagonal cupola prismxx5xo2xx5x trigonal prism + 5x cube + 1x pentagonal prism + 1x decagonal prism + 2x pentagonal cupola
118gyrated 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
119gyrated 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
120bigyrated 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
121parabidiminished rhombicosidodecahedron prism??10x trigonal prism + (10+10)x cube + 10x pentagonal prism + 2x decagonal prism + 2x parabidiminished rhombicosidodecahedron
122metabidiminished 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
123gyrated 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
124tridiminished 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
125truncated cuboctahedron prismx4x3x2x12x cube + 8x hexagonal prism + 6x octagonal prism + 2x truncated cuboctahedron
126icosahedral semipericupolaxx3xo5ox30x trigonal prism + 12x pentagonal antiprism + 20x trigonal cupola + 1x rhombicosidodecahedron + 1x truncated icosahedron
127truncated icosahedron prismx3x5o2x12x pentagonal prism + 20x hexagonal prism + 2x truncated icosahedron
128cubic canticupolaxx4xx3ox12x trigonal prism + 8x trigonal cupola + 6x octagonal prism + 1x truncated cube + 1x truncated cuboctahedron
129cubic semimesocupolaxo4xx3oo8x trigonal prism + 1x cuboctahedron + 6x square cupola + 1x truncated cube
130truncated dodecahedron prismx5x3o2x20x trigonal prism + 12x decagonal prism + 2x truncated dodecahedron
131rhodohedral mesocupolaxo5ox3xo20x octahedron + 30x square pyramid + 12x pentagonal antiprism + 1x icosidodecahedron + 1x rhombicosidodecahedron
132K4.132??(5+5+5)x octahedron + (5+5+5+5+10)x square pyramid + (1+5+5)x pentagonal antiprism + 1x icosidodecahedron + 1x diminished rhombicosidodecahedron + 1x pentagonal cupola
133pentagonal gyrobicupolic ringxxo5oxx&#x10x square pyramid + 1x pentagonal antiprism + 2x pentagonal cupola
134K4.134??10x octahedon + (10+10+10)x square pyramid + 10x pentagonal antiprism + 1x icosidodecahedron + 2x pentagoal cupola + 1x parabidiminished rhombicosidodecahedron
135K4.135??(2+2+2+4)x octahedron + (1+1+2+2+4+4+4+4+4+4)x square pyramid + (2+2+2+4)x pentagonal antiprism + 1x icosidodecahedron + 2x pentagonal cupola + 1x metabidiminished rhombicosidodecahedron
136K4.136??(1+1+3)x octahedron + (3+3+3+3+6+6+6)x square pyramid + (3+3+3)x pentagonal antiprism + 1x icosidodecahedron + 3x pentagonal cupola + 1x tridiminished rhombicosidodecahedron
137K4.137xo3ox5oo20x octahedron + 1x icosahedron + 12x pentagonal pyramid + 1x icosidodecahedron
138K4.138??(5+5+5)x octahedron + 5x square pyramid + 1x gyroelongated pentagonal pyramid + (1+5+5)x pentagonal pyramid + 1x pentagonal prism + 1x icosidodecahedron
139K4.139??5x octahedron + 5x square pyramid + (1+1+5)x pentagonal pyramid + 1x pentagonal rotnda +1x pentagonal cupola
140K4.140??(5+5)x octahedron + 5x square pyramid + 1x gyroelongated pentagonal pyramid + (1+5)x pentagonal pyramid + 1x pentagonal rotunda + 1x pentagonal cupola
141pentagonal prism pyramidxox5ooo&#x5x square pyramid + 2x pentagonal pyramid + 1x pentagonal prism
142K4.142??10x octahedron + 10x square pyramid + 10x pentagonal pyramid + 1x pentagonal antiprism + 2x pentagonal prism + 1x icosidodecahedron
143K4.143??(2+2+2+4)x octahedron +(2+4+4)x square pyramid + (2+2+2+4)x pentagonal pyramid + 1x metabidiminished icosahedron + 1x pentagonal prism + 1x icosidodecahedron
144K4.144??5x octahedron + (5+5)x square pyramid + 5x pentagonal pyramid +1x pentagonal antiprism + 1x pentagonal prism + 1x pentagonal rotunda + 1x pentagonal cupola
145K4.145??(1+1+2+2)x octahedron + (2+2+2+2)x square pyramid + (1+2+2)x pentagonal pyramid + 1x metabidiminished icosahedron + 1x pentagonal prism + 1x pentagonal rotunda + 1x pentagonal cupola
146K4.146??(5+5)x square pyramid + 5x pentagonal pyramid + 1x pentagonal prism + 1x rotunda +1x pentagonal cupola
147K4.147??(1+1+3)x octahedron + (3+6+6)x square pyramid + (3+3+3)x pentagonal pyramid + 1x tridiminished icosahedron + 3x pentagonal prism + 1x icosidodecahedron
148K4.148??(1+1)x octahedron + (1+2+2+2+2+2)x square pyramid + (1+1+2)x pentagonal pyramid + 1x tridiminished icosahedron + 2x pentagonal prism + 1x pentagonal rotunda + 1x pentagonal cupola

(To be completed)

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