Snub decachoron

The snub decachoron, or snad, also commonly called the omnisnub pentachoron or omnisnub 5-cell, is a convex isogonal polychoron that consists of 10 snub tetrahedra, 20 triangular gyroprisms, 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, 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:$$\sqrt{\frac{3+\sqrt2}{3}}$$ ≈ 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:


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