Segmentotope (EntityClass, 11)

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K 4.7
K 4.7
</td><td>
</td><td>
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'''[[digon]] || [[square_pyramid]]'''
+
'''[[digon]] || [[square pyramid]]'''
*[[triangular prism pyramid]] (K 4.7.2)
*[[triangular prism pyramid]] (K 4.7.2)
-
*[[point]] || [[trigonal_prism]]
+
*[[point]] || [[trigonal prism]]
</td><td>
</td><td>
*2 [[tetrahedra]]
*2 [[tetrahedra]]
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K 4.26
K 4.26
</td><td>
</td><td>
-
'''[[square]] || [[square_pyramid]]'''
+
'''[[square]] || [[square pyramid]]'''
*[[cubical pyramid]]
*[[cubical pyramid]]
*[[point]] || [[cube]]
*[[point]] || [[cube]]
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K 4.45
K 4.45
</td><td>
</td><td>
-
'''[[triangular_cupola]] || triangular_cupola'''
+
'''[[triangular cupola]] || triangular cupola'''
-
*[[triangular_prism]] || [[hexagonal_prism]]
+
*[[triangular prism]] || [[hexagonal prism]]
</td><td>
</td><td>
</td><td>
</td><td>
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K 4.51
K 4.51
</td><td>
</td><td>
-
'''[[hexagon]] || [[trigonal_cupola]]'''
+
'''[[hexagon]] || [[trigonal cupola]]'''
-
*[[triangle]] || [[hexagonal_prism]]
+
*[[triangle]] || [[hexagonal prism]]
*[[trigonal orthobicupolic ring]]
*[[trigonal orthobicupolic ring]]
</td><td>
</td><td>
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K 4.69
K 4.69
</td><td>
</td><td>
-
'''[[square_cupola]] || square_cupola'''
+
'''[[square cupola]] || square cupola'''
-
*'''[[cube]] || [[octagonal_prism]]'''
+
*'''[[cube]] || [[octagonal prism]]'''
</td><td>
</td><td>
</td><td>
</td><td>
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K 4.105
K 4.105
</td><td>
</td><td>
-
'''[[octagon]] || [[square_cupola]]'''
+
'''[[octagon]] || [[square cupola]]'''
-
*[[square]] || [[octagonal_prism]]
+
*[[square]] || [[octagonal prism]]
*[[square orthobicupolic ring]]
*[[square orthobicupolic ring]]
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</td><td>
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K 4.117
K 4.117
</td><td>
</td><td>
-
'''[[pentagonal_cupola]] || pentagonal_cupola'''
+
'''[[pentagonal cupola]] || pentagonal cupola'''
-
*[[pentagonal_prism]] || [[decagonal_prism]]
+
*[[pentagonal prism]] || [[decagonal prism]]
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</td><td>
</td><td>
</td><td>
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K 4.141
K 4.141
</td><td>
</td><td>
-
'''[[pentagon]] || [[pentagonal_pyramid]]'''
+
'''[[pentagon]] || [[pentagonal pyramid]]'''
-
*[[point]] || [[pentagonal_prism]]
+
*[[point]] || [[pentagonal prism]]
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</td><td>
</td><td>
</td><td>
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K 4.165
K 4.165
</td><td>
</td><td>
-
'''[[decagon]] || [[pentagonal_cupola]]'''
+
'''[[decagon]] || [[pentagonal cupola]]'''
-
*[[pentagon]] || [[decagonal_prism]]
+
*[[pentagon]] || [[decagonal prism]]
*[[pentagonal orthobicupolic ring]]
*[[pentagonal orthobicupolic ring]]
</td><td>
</td><td>

Revision as of 07:06, 10 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, 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

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)

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 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

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: 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

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°
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