Pyritohedral icosahedral antiprism

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Pyritohedral icosahedral antiprism
Rank4
TypeIsogonal
Notation
Bowers style acronymPikap
Coxeter diagram
Elements
Cells24 sphenoids, 6 tetragonal disphenoids, 8 triangular gyroprisms, 2 pyritohedral icosahedra
Faces48 scalene triangles, 24+24 isosceles triangles, 16 triangles
Edges12+12+24+48
Vertices24
Vertex figureTriangular-pentagonal antiwedge
Measures (as derived from unit-edge truncated octahedral prism)
Edge lengthsDiagonals of original squares (12+12+24):
 Edges of equilateral triangles (48):
Circumradius
Central density1
Related polytopes
ArmyPikap
RegimentPikap
DualPyritohedral antitegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(B3/2×2×A1)/2, order 48
ConvexYes
NatureTame

The pyritohedral icosahedral antiprism, pyritohedral icosahedral alterprism or pikap, also known as the alternated truncated octahedral prism or omnisnub tetrahedral antiprism, is a convex isogonal polychoron that consists of 2 pyritohedral icosahedra, 8 triangular gyroprisms, 6 tetragonal disphenoids, and 24 sphenoids. 1 pyritohedral icosahedron, 2 triangular antiprisms, 1 tetragonal disphenoid, and 4 sphenoids join at each vertex. It can be obtained through the process of alternating the truncated octahedral prism. However, it cannot be made uniform, as it generally has 4 edge lengths, which can be minimized to no fewer than 2 different sizes..

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1: ≈ 1:1.09325. In this variant, none of the cells are regular, though the sphenoids and tetragonal disphenoids become identical to each other.

This polychoron has a kind of pyritohedral antiprismatic symmetry. A variant with chiral tetrahedral antiprismatic symmetry, called the snub tetrahedral antiprism, also exists.

Vertex coordinates[edit | edit source]

Vertex coordinates for a pyritohedral icosahedral antiprism, created from the vertices of a truncated octahedral prism of edge length 1, are given by all even permutations in the first 3 coordinates of:

A variant where the icosahedra and the tetragonal disphenoids are regular of edge length 1, centered at the origin, is given by all even permutations in the first 3 coordinates of:

A variant using regular octahedra of edge length 1, centered at the origin, is given by all even permutations in the first 3 coordinates of:

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by all even permutations in the first 3 coordinates of:


Isogonal derivatives[edit | edit source]

Substitution by vertices of these following elements will produce these convex isogonal polychora:

External links[edit | edit source]