Pyritohedral icosahedral alterprism

Pyritohedral icosahedral alterprism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Notation
Bowers style acronym	Pikap
Coxeter diagram
Elements
Cells	24 sphenoids, 6 tetragonal disphenoids, 8 triangular gyroprisms, 2 pyritohedral icosahedra
Faces	48 scalene triangles, 24+24 isosceles triangles, 16 triangles
Edges	12+12+24+48
Vertices	24
Vertex figure	Triangular-pentagonal antiwedge
Measures (as derived from unit-edge truncated octahedral prism)
Edge lengths	Diagonals of original squares (12+12+24): $\sqrt2 \approx 1.41421$
	Edges of equilateral triangles (48): $\sqrt3 \approx 1.73205$
Circumradius	$\frac{\sqrt{11}}{2} \approx 1.65831$
Central density	1
Related polytopes
Army	Pikap
Regiment	Pikap
Dual	Pyritohedral antitegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(B3/2×2×A1)/2, order 48
Convex	Yes
Nature	Tame

The pyritohedral icosahedral antiprism 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:$\frac{\sqrt{15+\sqrt{17}}}{4}$ ≈ 1:1.09325.

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:

• $\left(±\sqrt2,\,±\frac{\sqrt2}{2},\,0,\,\frac12\right),$
• $\left(±\frac{\sqrt2}{2},\,±\sqrt2,\,0,\,-\frac12\right).$

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:

• $\left(±\frac{1+\sqrt5}{4},\,±\frac12,\,0,\,\frac{\sqrt2}{4}\right),$
• $\left(±\frac12,\,±\frac{1+\sqrt5}{4},\,0,\,-\frac{\sqrt2}{4}\right).$

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:

• $\left(±\frac{\sqrt6}{3},\,±\frac{\sqrt6}{6},\,0,\,\frac{\sqrt6}{6}\right),$
• $\left(±\frac{\sqrt6}{6},\,±\frac{\sqrt6}{3},\,0,\,-\frac{\sqrt6}{6}\right).$

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:

• $\left(±\frac{3+\sqrt{17}}{8},\,±\frac12,\,0,\,\frac{\sqrt{7+\sqrt{17}}}{8}\right),$
• $\left(±\frac12,\,±\frac{3+\sqrt{17}}{8},\,0,\,-\frac{\sqrt{7+\sqrt{17}}}{8}\right).$

Isogonal derivatives

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

• Sphenoid (24): Pyritohedral icosahedral antiprism
• Triangle (16): Tesseract
• Isosceles triangle (24): Pyritohedral icosahedral antiprism
• Scalene triangle (48): Pyritosnub alterprism
• Edge (12): Octahedral prism
• Edge (24): Pyritohedral icosahedral antiprism
• Edge (48): Pyritosnub alterprism

External links

• Klitzing, Richard. "pikap".

• Wikipedia Contributors. "Omnisnub tetrahedral antiprism".