Snub cubic antiprism

The snub cubic antiprism, omnisnub cubic antiprism, or sniccap, also known as the alternated great rhombicuboctahedral prism, is a convex isogonal polychoron that consists of 2 snub cubes, 6 square gyroprisms, 8 triangular gyroprisms, 12 rhombic disphenoids, and 48 irregular tetrahedra. 4 tetrahedra and one each of the other 4 types of cells join at each vertex. It can be obtained through the process of alternating the great rhombicuboctahedral prism. However, it cannot be made uniform, as it generally has 6 edge lengths, which can be minimzed to no fewer than 3 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$$\frac{\sqrt5}{2}$$ ≈ 1:1.11803.

Vertex coordinates
Vertex coordinates for a snub cubic antiprism, created from the vertices of a great rhombicuboctahedral prism of edge length 1, are given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes in the first three coordinates, of:


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

A variant with uniform snub cubes as bases is given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes excluding the last coordinate of:


 * $$\left(c_1,\,c_2,\,c_3,\,c_4\right),$$
 * $$\left(c_2,\,c_1,\,c_3,\,-c_4\right),$$

where


 * $$c_1=\sqrt{\frac{1}{12}\left(4-\sqrt[3]{17+3\sqrt{33}}-\sqrt[3]{17-3\sqrt{33}}\right)},$$
 * $$c_2=\sqrt{\frac{1}{12}\left(2+\sqrt[3]{17+3\sqrt{33}}+\sqrt[3]{17-3\sqrt{33}}\right)},$$
 * $$c_3=\sqrt{\frac{1}{12}\left(4+\sqrt[3]{199+3\sqrt{33}}+\sqrt[3]{199-3\sqrt{33}}\right)}.$$
 * $$c_4=\sqrt{\frac{1}{12}\left(-2+\sqrt[3]{19+3\sqrt{33}}+\sqrt[3]{19-3\sqrt{33}}\right)}.$$

An alternate set of coordinates, assuming that the edge length differences are minimized, centered at the origin, are given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes excluding the last coordinate of:


 * $$\left(c_1,\,c_2,\,c_3,\,c_4\right),$$
 * $$\left(c_2,\,c_1,\,c_3,\,-c_4\right),$$

where

which has rhombic disphenoids (via the absolute value method), or
 * $$c_1=\text{root}(9472x^8-5504x^6+1136x^4-98x^2+3, 6) ≈ 0.3135135258027234561506493,$$
 * $$c_2=\text{root}(14208x^8-19840x^6+7432x^4-679x^2+8, 7) ≈ 0.6542869462841313854118897,$$
 * $$c_3=\text{root}(14208x^8-20800x^6+3640x^4-159x^2+2, 8) ≈ 1.1264771628934748152619404,$$
 * $$c_4=\text{root}(18944x^8-7936x^6+1120x^4-60x^2+1, 7) ≈ 0.3894987408692678350179804,$$

where the ratio of the largest edge length to the smallest edge length is lowest (via the ratio method).
 * $$\left(\frac{\sqrt{10}}{10},\,\frac{\sqrt{10}}{5},\,\frac{\sqrt{15+10\sqrt2}}{5},\,\frac{\sqrt{15}}{10}\right),$$
 * $$\left(\frac{\sqrt{10}}{5},\,\frac{\sqrt{10}}{10},\,\frac{\sqrt{15+10\sqrt2}}{5},\,-\frac{\sqrt{15}}{10}\right),$$