Snub decachoron

The snub decachoron, or snad, also commonly called the omninsub pentachoron or omnisnub 5-cell, is a convex isogonal polychoron that consists of 10 snub tetrahedra, 20 triangular antiprisms, and 60 phyllic disphenoids. 2 snub tetrahedra, 2 triangular antiprisms, and 4 disphenoids join at each vertex. It can be obtained through the process of alternating the great prismatodecachoron. However, it cannot be made uniform.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\sqrt{\frac{3+\sqrt2}{3}}$$ ≈ 1:1.21301.

This polychoron generally has a doubled symmetry. A variant without this extended symmetry also exists, known as the snub pentachoron.

Vertex coordinates
A snub decachoron formed directly from alternating a great prismatodecachoron of edge length 1 has coordinates in 5 dimensions given by all even permutations of:


 * $$\left(2\sqrt2,\,\tfrac{3\sqrt2}{2},\,\sqrt2,\,\tfrac{\sqrt2}{2},\,0\right).$$

an optimized snub decachoron using the absolute value method, where the phyllic disphenoids become rhombic disphenoids, is given by all even permutations of:


 * $$\left(0,\,\frac12,\,\frac{3+\sqrt6}{6},\,\frac{3+2\sqrt6}{6},\,\frac{3+\sqrt6}{3}\right).$$

Finally, a variant optimized by the ratio method is given by all even permutations of:


 * $$\left(0,\,\frac12,\,\frac{2+\sqrt2}{4},\,\frac{1+\sqrt2}{2},\,\frac{2+\sqrt2}{2}\right).$$