Pyritosnub cube

Pyritosnub cube
Rank	3
Type	Isogonal
Notation
Bowers style acronym	Pysnic
Coxeter diagram	x4s3s ()
Elements
Faces	8 triangles, 12 isosceles trapezoids, 6 rectangles
Edges	12+12+24
Vertices	24
Vertex figure	Irregular tetragon
Measures (edge lengths a (long edge of rectangle), b (short edge of rectangle), c (triangle-trapezoid))
Central density	1
Related polytopes
Army	Pysnic
Regiment	Pysnic
Dual	Tetragonal icositetrahedron
Abstract & topological properties
Euler characteristic	2
Surface	Sphere
Orientable	Yes
Genus	0
Properties
Symmetry	B3/2, order 24
Convex	Yes
Nature	Tame

The pyritosnub cube, or pysnic, is a convex isogonal polyhedron that is a variant of the small rhombicuboctahedron with pyritohedral symmetry. It has 8 equilateral triangles, 6 rectangles, and 12 isosceles trapezoids for faces.

It can generally be formed by alternating one set of 24 edges of a general great rhombicuboctahedron, such that the octagons become long rectangles.

This polyhedron generally has 3 types of edges, as the 24 edges of the small rhombicuboctahedron's squares split into 2 groups of 12, turning the squares into rectangles.

The variant derived from the uniform great rhombicuboctahedron has rectangles with edge lengths 1 and $1+{\sqrt {2}}$ and triangles of side ${\sqrt {3}}$.

Another case of this polyhedron, with 6 golden rectangles, can be obtained by removing the 6 vertices of an inscribed octahedron from a uniform icosidodecahedron.

The Conway-Thurston symbol for this is 3x * x2x, the first 'x' represents the triangle, and the other two x's are for the rectangle.

Vertex coordinates

A pyritosnub cube with edges of length a (long edge of rectangle), b (short edge of rectangle), and c (triangle-trapezoid) has vertex coordinates given by all cyclic permutations of:

• $\left(\pm {\frac {a}{2}},\,\pm {\frac {b}{2}},\,\pm {\frac {a+b+{\sqrt {8c^{2}-3(a-b)^{2}}}}{4}}\right).$

External links

• Klitzing, Richard. "pysnic".