Pyritohedral icosahedral alterprism
|Pyritohedral icosahedral alterprism|
|Bowers style acronym||Pikap|
|Cells||24 sphenoids, 6 tetragonal disphenoids, 8 triangular gyroprisms, 2 pyritohedral icosahedra|
|Faces||48 scalene triangles, 24+24 isosceles triangles, 16 triangles|
|Vertex figure||Triangular-pentagonal antiwedge|
|Measures (as derived from unit-edge truncated octahedral prism)|
|Edge lengths||Diagonals of original squares (12+12+24):|
|Edges of equilateral triangles (48):|
|Abstract & topological properties|
|Symmetry||(B3/2×2×A1)/2, order 48|
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: ≈ 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[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:
- 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[edit | edit source]
- Klitzing, Richard. "pikap".
- Wikipedia Contributors. "Omnisnub tetrahedral antiprism".