Triangular-gyroprismatic enneacontahexachoron

Triangular-gyroprismatic enneacontahexachoron
(OFF file)
Rank	4
Type	Noble
Elements
Cells	96 triangular gyroprisms
Faces	288 isosceles triangles, 96 triangles
Edges	72+288
Vertices	72
Vertex figure	Square gyrotegum
Measures (circumradius 1)
Edge lengths	4-valence (72): ${\sqrt {2-{\sqrt {3}}}}\approx 0.51764$
	3-valence (288): ${\sqrt {\frac {3-{\sqrt {3}}}{2}}}\approx 0.79623$
Central density	1
Related polytopes
Dual	Square-antiprismatic heptacontadichoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3●I2(12), order 576
Convex	Yes
Nature	Tame

The triangular-gyroprismatic enneacontahexachoron, also known as the dodecafold octaswirlchoron or octswirl 96, is a noble swirlchoron with 96 triangular gyroprisms for cells and 72 vertices. 8 cells join at each vertex.

It is the third in an infinite family of isogonal octahedral swirlchora (the dodecafold octaswirlchoron) and also the first in an infinite family of isochoric octahedral swirlchora (the octaswirlic enneacontahexachoron).

Each cell of this polychoron is a chiral variant of the triangular antiprism. If the edges of the base triangles are of length 1, half the side edges are also of length 1, while the other half are of length ${\sqrt {\frac {3-{\sqrt {3}}}{3}}}\approx 0.65012$.

The ratio between the longest and shortest edges is 1:${\frac {\sqrt {6+2{\sqrt {3}}}}{2}}$ \approx 1:1.53819.

Vertex coordinates

Coordinates for the vertices of a triangular-antiprismatic enneacontahexachoron of circumradius 1 (and thus edge lengths ${\sqrt {\frac {3-{\sqrt {3}}}{2}}}$ and ${\sqrt {2-{\sqrt {3}}}}$), centered at the origin, are given by all permutations of:

• $\left(0,0,0,\pm 1\right),$
• $\left(\pm {\frac {1}{2}},\pm {\frac {1}{2}},\pm {\frac {1}{2}},\pm {\frac {1}{2}}\right),$

defining an icositetrachoron, along with reflections through the x=y and z=w hyperplanes of:

• $\left(0,0,\pm {\frac {1}{2}},\pm {\frac {\sqrt {3}}{2}}\right),$

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

• $\left({\frac {{\sqrt {3}}-1}{4}},\,{\frac {1+{\sqrt {3}}}{4}},\,{\frac {{\sqrt {3}}-1}{4}},\,{\frac {1+{\sqrt {3}}}{4}}\right),$

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

• $\left({\frac {{\sqrt {3}}-1}{4}},\,{\frac {1+{\sqrt {3}}}{4}},\,{\frac {1+{\sqrt {3}}}{4}},\,{\frac {{\sqrt {3}}-1}{4}}\right),$

Isogonal derivatives

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

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

External links

• Klitzing, Richard. "trap-96".