Square-gyroprismatic heptacontadichoron

The square-antiprismatic heptacontadichoron, also known as the cubeswirl 72, is a noble swirlchoron with 72 square antiprisms for cells and 96 vertices. 6 cells join at each vertex.

It is the first in an infinite family of isogonal cubic swirlchora (the dodecafold cubiswirlchoron) and also the third in an infinite family of isochoric cubic swirlchora (the cubiswirlic heptacontadichoron).

Each cell of this polychoron is a chiral variant of the square antiprism. If the edges of the base squares are of length 1, half the side edges are also of length 1, while the other half are of length $$\sqrt{\frac{3-\sqrt3}{2}} ≈ 0.79623$$.

The ratio between the longest and shortest edges is 1:$$\frac{\sqrt{9+3\sqrt3}}{3}$$ ≈ 1:1.25593.

Vertex coordinates
Coordinates for the vertices of a square-antiprismatic heptacontadichoron of circumradius 1 (thus, edge lengths $$\sqrt{\frac{3-\sqrt3}{3}}$$ and $$\sqrt{2-\sqrt3}$$) centered at the origin, are given by reflections through the x=y and z=w hyperplanes of: along with reflections through the x=y and z=w hyperplanes and with all even sign changes of: along with reflections through the x=y and z=w hyperplanes and with all odd sign changes of:
 * $$\left(0,\,±\sqrt{\frac{3-\sqrt3}{6}},\,±\sqrt{\frac{3+\sqrt3}{12}},\,±\sqrt{\frac{3+\sqrt3}{12}}\right),$$
 * $$\left(\sqrt{\frac{3-\sqrt3}{24}},\,\sqrt{\frac{3-\sqrt3}{8}},\,\sqrt{\frac{9+5\sqrt3}{24}},\,\sqrt{\frac{3-\sqrt3}{24}}\right),$$
 * $$\left(\sqrt{\frac{3\sqrt3}{24}},\,\sqrt{\frac{3-\sqrt3}{8}},\,\sqrt{\frac{3-\sqrt3}{24}},\,\sqrt{\frac{9+5\sqrt2}{24}}\right).$$

Isogonal derivatives
Substitution by vertices of these following elements will produce these convex isogonal polychora:
 * Square antiprism (72): Triangular-antiprismatic enneacontahexachoron
 * Square (72): Triangular-antiprismatic enneacontahexachoron
 * Edge (96): Square-antiprismatic heptacontadichoron