# Segmentotope (EntityClass, 11)

### From Hi.gher. Space

(→Properties: add K 4.165) |
(link to Klitzing's paper) |
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- | A ''segmentochoron'' is a polychoron whose vertices lie on two parallel hyperplanes. The set of all convex segmentochora having regular polygon ridges has been enumerated by Dr. Richard Klitzing. There are 177 of them, some of which includes polychora from other categories (such as cube||cube, which is the same as the tesseract). | + | A ''segmentochoron'' is a polychoron whose vertices lie on two parallel hyperplanes. The set of all convex segmentochora having regular polygon ridges has been enumerated by Dr. Richard Klitzing. There are 177 of them, some of which includes polychora from other categories (such as cube||cube, which is the same as the tesseract). The full list can be obtained from [http://www.orchidpalms.com/polyhedra/segmentochora/artConvSeg_7.pdf Klitzing's paper] (PDF format). |

==Nomenclature== | ==Nomenclature== |

## Revision as of 19:31, 9 January 2012

A *segmentochoron* is a polychoron whose vertices lie on two parallel hyperplanes. The set of all convex segmentochora having regular polygon ridges has been enumerated by Dr. Richard Klitzing. There are 177 of them, some of which includes 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.

### Line||square pyramid (K 4.7)

Other names: triangular prism pyramid (K 4.7.2), point||trigonal prism

Height of triangular prism pyramid: E*sqrt(5/12)

Cells: 2 tetrahedra, 3 square pyramids, trigonal prism.

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°

### Square||square pyramid (K 4.26)

Other names: cubical pyramid, point||cube, square prism pyramid

Cells: 6 square pyramids, 1 cube

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

### Triangular cupola||triangular cupola (K 4.45)

Other names: triangular prism||hexagonal prism

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

### hexagon||trigonal cupola (K 4.51)

Other names: triangle||hexagonal prism, trigonal ortho-bicupolic ring

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°

### Square cupola||square cupola (K 4.69)

Other names: cube||octagonal prism

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

### Octagon||square cupola (K 4.105)

Other names: square || octagonal prism, square ortho-bicupolic ring

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)

### Pentagonal cupola||pentagonal cupola (K 4.117)

Other names: pentagonal prism||decagonal prism

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.)

### Pentagon||pentagonal pyramid (K 4.141)

Other names: point||pentagonal prism

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°

### Decagon||pentagonal cupola (K 4.165)

Other names: pentagon||decagonal prism, pentagonal ortho-bicupolic ring

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°