Segmentotope (EntityClass, 11)
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==Properties== | ==Properties== | ||
| - | Below are some useful properties of selected segmentochora. Measurements are given in terms of E, the edge length. | + | 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. |
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NOTE: this is intended to list common properties of certain segmentotopes that might be useful in the search for CRF polychora | NOTE: this is intended to list common properties of certain segmentotopes that might be useful in the search for CRF polychora | ||
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| - | ===Triangular cupola||triangular cupola=== | + | ===Triangular cupola||triangular cupola (K 4.45)=== |
Other names: triangular prism||hexagonal prism | Other names: triangular prism||hexagonal prism | ||
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Distance between hexagonal prism and antipodal triangular prism: E*sqrt(2/3). (Same as the height of a triangular cupola.) | Distance between hexagonal prism and antipodal triangular prism: E*sqrt(2/3). (Same as the height of a triangular cupola.) | ||
| - | ===Square cupola||square cupola=== | + | ===Square cupola||square cupola (K 4.69)=== |
Other names: cube||octagonal prism | Other names: cube||octagonal prism | ||
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Distance between octagonal prism and antipodal cube: E*sqrt(2)/2. (Same as height of square cupola.) | Distance between octagonal prism and antipodal cube: E*sqrt(2)/2. (Same as height of square cupola.) | ||
| - | ===Pentagonal cupola||pentagonal cupola=== | + | ===Pentagonal cupola||pentagonal cupola (K 4.117)=== |
Other names: pentagonal prism||decagonal prism | 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.) | 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.) | ||
Revision as of 18:34, 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).
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.
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.)
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.)
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.)
