Octafold tetraswirlchoron

Octafold tetraswirlchoron
File:Octafold tetraswirlchoron.png (OFF file)
Rank	4
Type	Isogonal
Elements
Cells	96 phyllic disphenoids, 48 rhombic disphenoids
Faces	192 scalene triangles, 96 isosceles triangles
Edges	32+48+96
Vertices	32
Vertex figure	Vertical-bisected joined triangular prism
Measures (circumradius 1)
Edge lengths	6-valence (48): ${\sqrt {2-{\sqrt {2}}}}\approx 0.76537$
	6-valence (24): ${\sqrt {\frac {6-2{\sqrt {3}}}{3}}}$ $\approx 0.91940$
	4-valence (96); ${\sqrt {\frac {6-{\sqrt {6}}}{3}}}\approx 1.08789$
Central density	1
Related polytopes
Dual	Tetraswirlic triacontadichoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A3●I2(8), order 192
Convex	Yes
Nature	Tame

The octafold tetraswirlchoron is an isogonal polychoron with 48 rhombic disphenoids, 96 phyllic disphenoids, and 32 vertices. 6 rhombic and 12 phyllic disphenoids join at each vertex. It is the fourth in an infinite family of isogonal tetrahedral swirlchora.

The ratio between the longest and shortest edges is 1:${\frac {\sqrt {18+9{\sqrt {2}}-3{\sqrt {9+6{\sqrt {2}}}}}}{3}}$ ≈ 1:1.42140.

Vertex coordinates

Coordinates for the vertices of an octafold tetraswirlchoron of circumradius 1, centered at the origin, are given by:

• ±(0, 0, sin(kπ/4), cos(kπ/4)),

along with 120° and 240° rotations in the xy axis of:

• ±(√6sin(kπ/4)/3, √6cos(kπ/4)/3, √3cos(kπ/4)/3, √3sin(kπ/4)/3),

where k is an integer from 0 to 3.

Isogonal derivatives

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

• Edge (32): Octafold tetraswirlchoron
• Edge (48): Octafold ambotetraswirlchoron
• Edge (96): Octafold truncatotetraswirlchoron