# Pyritohedral icosahedral antiprism

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): ${\displaystyle {\sqrt {2}}\approx 1.41421}$
Edges of equilateral triangles (48): ${\displaystyle {\sqrt {3}}\approx 1.73205}$
Circumradius${\displaystyle {\frac {\sqrt {11}}{2}}\approx 1.65831}$
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:${\displaystyle {\frac {\sqrt {15+{\sqrt {17}}}}{4}}}$ ≈ 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

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:

• ${\displaystyle \left(\pm {\sqrt {2}},\,\pm {\frac {\sqrt {2}}{2}},\,0,\,{\frac {1}{2}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {2}}{2}},\,\pm {\sqrt {2}},\,0,\,-{\frac {1}{2}}\right).}$

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:

• ${\displaystyle \left(\pm {\frac {1+{\sqrt {5}}}{4}},\,\pm {\frac {1}{2}},\,0,\,{\frac {\sqrt {2}}{4}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {1+{\sqrt {5}}}{4}},\,0,\,-{\frac {\sqrt {2}}{4}}\right).}$

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:

• ${\displaystyle \left(\pm {\frac {\sqrt {6}}{3}},\,\pm {\frac {\sqrt {6}}{6}},\,0,\,{\frac {\sqrt {6}}{6}}\right),}$
• ${\displaystyle \left(\pm {\frac {\sqrt {6}}{6}},\,\pm {\frac {\sqrt {6}}{3}},\,0,\,-{\frac {\sqrt {6}}{6}}\right).}$

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:

• ${\displaystyle \left(\pm {\frac {3+{\sqrt {17}}}{8}},\,\pm {\frac {1}{2}},\,0,\,{\frac {\sqrt {7+{\sqrt {17}}}}{8}}\right),}$
• ${\displaystyle \left(\pm {\frac {1}{2}},\,\pm {\frac {3+{\sqrt {17}}}{8}},\,0,\,-{\frac {\sqrt {7+{\sqrt {17}}}}{8}}\right).}$

## Isogonal derivatives

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