Segmentotope (EntityClass, 11)

From Higher Dimensions Database

(Difference between revisions)
(Properties: expand)
(Pentagon||pentagonal pyramid: K number)
Line 52: Line 52:
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.)
-
===Pentagon||pentagonal pyramid===
+
===Pentagon||pentagonal pyramid (K 4.141)===
Other names: point||pentagonal prism
Other names: point||pentagonal prism

Revision as of 18:56, 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.

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.)

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.)

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°

Pages in this category (5)