# Triangular-gyroprismatic enneacontahexachoron

Triangular-gyroprismatic enneacontahexachoron
Rank4
TypeNoble
SpaceSpherical
Elements
Cells96 triangular gyroprisms
Faces288 isosceles triangles, 96 triangles
Edges72+288
Vertices72
Vertex figureSquare gyrotegum
Edge lengths4-valence (72): ${\displaystyle \sqrt{2-\sqrt3} ≈ 0.51764}$
3-valence (288): ${\displaystyle \sqrt{\frac{3-\sqrt3}{2}} ≈ 0.79623}$
Central density1
Related polytopes
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryB3●I2(12), order 576
ConvexYes
NatureTame

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 ${\displaystyle \sqrt{\frac{3-\sqrt3}{3}} ≈ 0.65012}$.

The ratio between the longest and shortest edges is 1:${\displaystyle \frac{\sqrt{6+2\sqrt3}}{2}}$ ≈ 1:1.53819.

## Vertex coordinates

Coordinates for the vertices of a triangular-antiprismatic enneacontahexachoron of circumradius 1 (and thus edge lengths ${\displaystyle \sqrt{\frac{3-\sqrt3}{2}}}$ and ${\displaystyle \sqrt{2-\sqrt3}}$), centered at the origin, are given by all permutations of:

• ${\displaystyle \left(0,\,0,\,0,\,±1\right),}$
• ${\displaystyle \left(±\frac12,\,±\frac12,\,±\frac12,\,±\frac12\right),}$

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

• ${\displaystyle \left(0,\,0,\,±\frac12,\,±\frac{\sqrt3}{2}\right),}$

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

• ${\displaystyle \left(\frac{\sqrt3-1}{4},\,\frac{1+\sqrt3}{4},\,\frac{\sqrt3-1}{4},\,\frac{1+\sqrt3}{4}\right),}$

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

• ${\displaystyle \left(\frac{\sqrt3-1}{4},\,\frac{1+\sqrt3}{4},\,\frac{1+\sqrt3}{4},\,\frac{\sqrt3-1}{4}\right),}$

## Isogonal derivatives

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