Snub tesseract

The snub tesseract, omnisnub tesseract, omnisnub hexadecachoron, or snet is a convex isogonal polychoron that consists of 8 snub cubes, 16 snub tetrahedra, 24 square antiprisms, 32 triangular antiprisms, and 192 irregular tetrahedra. Each vertex joins 4 tetrahedra and one each of the other 4 cell types. It can be obtained through the process of alternating the great disprismatotesseractihexadecachoron. However, it cannot be made uniform, as it generally has 6 edge lengths, which can be minimized to no more than 3 different sizes.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:a ≈ 1:1.25002, where a is the largest real root of 5329x12-23652x10+41382x8-37052x6+18052x4-4560x2+468.

Vertex coordinates
Vertex coordinates for a snub tesseract, created from the vertices of a great disprismatotesseractihexadecachoron 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, of:


 * $$\left(\frac{1+3\sqrt2}{2},\,\frac{1+2\sqrt2}{2},\,\frac{1+\sqrt2}{2},\,\frac12\right).$$

A snub tesseract with uniform snub cubes and pyritohedral icosahedra of edge length 1 is given by all even permutations with an even number of sign changes, plus all odd permutations with an odd amount of sign changes, of: where
 * $$\left(c_1,\,c_2,\,c_3,\,c_4\right),$$


 * $$c_1=\sqrt{\frac{1}{12}\left(4-\sqrt[3]{17+3\sqrt{33}}-\sqrt[3]{17-3\sqrt{33}}\right)},$$
 * $$c_2=\sqrt{\frac{1}{12}\left(2+\sqrt[3]{17+3\sqrt{33}}+\sqrt[3]{17-3\sqrt{33}}\right)},$$
 * $$c_3=\sqrt{\frac{1}{12}\left(4+\sqrt[3]{199+3\sqrt{33}}+\sqrt[3]{199-3\sqrt{33}}\right)},$$
 * $$c_4=\sqrt{\frac{1}{12}\left(18+\sqrt[3]{189+33\sqrt{33}}+\sqrt[3]{189-33\sqrt{33}}\right)}.$$

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 of: where
 * $$\left(c_1,\,c_2,\,c_3,\,c_4\right),$$

which has rhombic disphenoids (via the absolute value method), or
 * $$c_1≈0.3441758380340759102270172,$$
 * $$c_2≈0.7362880830694947058117290,$$
 * $$c_3≈1.1604625551454112407307730,$$
 * $$c_4≈1.6733816374840225151413383,$$


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

where

where the ratio of the largest edge length to the smallest edge length is lowest (via the ratio method).
 * $$c_1≈0.3191085335012948850450705,$$
 * $$c_2≈0.6310069285250780509169974,$$
 * $$c_3≈0.9720792302282349822564494,$$
 * $$c_4≈1.4440010653634313903563943,$$