Square-gyroprismatic heptacontadichoron

Square-gyroprismatic heptacontadichoron
(OFF file)
Rank	4
Type	Noble
Elements
Cells	72 square gyroprisms
Faces	288 isosceles triangles, 72 squares
Edges	96+288
Vertices	96
Vertex figure	Triangular gyrotegum
Measures (circumradius 1)
Edge lengths	3-valence (288): ${\sqrt {2-{\sqrt {3}}}}\approx 0.51764$
	3-valence (288): ${\sqrt {\frac {3-{\sqrt {3}}}{3}}}\approx 0.65012$
Central density	1
Related polytopes
Dual	Triangular-gyroprismatic enneacontahexachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3●I2(12), order 576
Convex	Yes
Nature	Tame

The square-gyroprismatic heptacontadichoron, also known as the dodecafold cubiswirlchoron or cubeswirl 72, is a noble swirlchoron with 72 square gyroprisms 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 ${\sqrt {\frac {3-{\sqrt {3}}}{2}}}\approx 0.79623$.

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

Vertex coordinates

Coordinates for the vertices of a square-antiprismatic heptacontadichoron of circumradius 1 (thus, edge lengths ${\sqrt {\frac {3-{\sqrt {3}}}{3}}}$ and ${\sqrt {2-{\sqrt {3}}}}$) centered at the origin, are given by reflections through the x=y and z=w hyperplanes of:

• $\left(0,\,\pm {\sqrt {\frac {3-{\sqrt {3}}}{6}}},\,\pm {\sqrt {\frac {3+{\sqrt {3}}}{12}}},\,\pm {\sqrt {\frac {3+{\sqrt {3}}}{12}}}\right),$

along with reflections through the x=y and z=w hyperplanes and with all even sign changes of:

• $\left({\sqrt {\frac {3-{\sqrt {3}}}{24}}},\,{\sqrt {\frac {3-{\sqrt {3}}}{8}}},\,{\sqrt {\frac {9+5{\sqrt {3}}}{24}}},\,{\sqrt {\frac {3-{\sqrt {3}}}{24}}}\right),$

along with reflections through the x=y and z=w hyperplanes and with all odd sign changes of:

• $\left({\sqrt {\frac {3-{\sqrt {3}}}{24}}},\,{\sqrt {\frac {3-{\sqrt {3}}}{8}}},\,{\sqrt {\frac {3-{\sqrt {3}}}{24}}},\,{\sqrt {\frac {9+5{\sqrt {2}}}{24}}}\right).$

Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Square antiprism (72): Triangular-gyroprismatic enneacontahexachoron
• Square (72): Triangular-gyroprismatic enneacontahexachoron
• Edge (96): Square-gyroprismatic heptacontadichoron

External links

• Klitzing, Richard. "squap-72".