# Segmentotope (EntityClass, 11)

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)CellsValues

K 4.7

• 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

• Dichoral angle between square pyramid and cube: 45° (exact)

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

• 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 square pyramid and hexagonal prism: atan(sqrt(5/2)) ≈ 57.69°
• Dichoral angle between triangular prism 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

• 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: asin(sqrt(2/3)) ≈ 54.74°
• Dichoral angle between triangular prism and octagonal prism: 45° (exact)

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

• 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

• 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
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 Symbol Cells Comments
1pyrochoronx3o3o3o5x tetrahedron
2aerochoronx3o3o4o16x tetrahedron
3octahedral pyramidxo3oo4oo8x tetrahedron + 1x octahedronhalf of aerochoron
4square dipyramidxo2xo2ox4x tetrahedron + 2x square pyramidquarter of aerochoron
53-pyrotomochorono3x3o3o5x tetrahedron + 5x octahedrontetrahedral semicupola
6K4.6??3x tetrahedron + 2x octahedron + 3x square pyramid + 1x trigonal prism
7triangular prism pyramidxo3oo2xo2x tetrahedron + 3x square pyramid + 1x trigonal prism
8K4.8??1x 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
14K4.14??4x tetrahedron + 4x square pyramid + 2x square antiprism + 1x cube
15stauroantiprismxo4oo3ox(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 pyramid??8x 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
21K4.21??8x tetrahedron + 12x square pyramid + 6x trigonal prism + 1x cube + 1x icosahedronsnubdis antiprism

(To be completed)

## Pages in this category (5)

 Square dipyramid