Icositetrafold octaswirlchoron

Icositetrafold octaswirlchoron
File:Icositetrafold octaswirlchoron.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	288 rhombic disphenoids, 192 triangular gyroprisms
Faces	1152 scalene triangles, 192 triangles
Edges	144+288+576
Vertices	144
Vertex figure	Edge-vertical bisected square gyrotegum
Measures (circumradius 1)
Edge lengths	8-valence (144): $\sqrt{\frac{4-\sqrt2-\sqrt6}{2}} ≈ 0.26105$
	4-valence (288): $\sqrt{2-\sqrt2} ≈ 0.76537$
	3-valence (576): $\sqrt{\frac{3-\sqrt3}{2}} ≈ 0.79623$
Central density	1
Related polytopes
Dual	Cubiswirlic hecatontetracontatetrachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	B3●I2(24), order 1152
Convex	Yes
Nature	Tame

The icositetrafold octaswirlchoron is an isogonal polychoron with 192 triangular gyroprisms, 288 rhombic disphenoids, and 144 vertices. 8 triangular gyroprisms and 8 rhombic disphenoids join at each vertex. It is the sixth in an infinite family of isogonal octahedral swirlchora.

The ratio between the longest and shortest edges is 1:$\frac{\sqrt{12+6\sqrt2+4\sqrt{9+6\sqrt2}}}{2}$ ≈ 1:3.05006.

Vertex coordinates

Coordinates for the vertices of an icositetrafold octaswirlchoron of circumradius 1, centered at the origin, are given by all permutations of:

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

defining an icositetrachoron, along with all permutations of:

• $\left(0,\,0,\,±\frac{\sqrt2}{2},\,±\frac{\sqrt2}{2}\right),$

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

• $\left(0,\,0,\,±\frac{\sqrt6-\sqrt2}{4},\,±\frac{\sqrt2+\sqrt6}{4}\right),$
• $\left(0,\,0,\,±\frac12,\,±\frac{\sqrt3}{2}\right),$

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

• $\left(\frac{\sqrt3-1}{4},\,\frac{1+\sqrt3}{4},\,\frac{\sqrt3-1}{4},\,\frac{1+\sqrt3}{4}\right),$
• $\left(\frac{\sqrt2}{4},\,\frac{\sqrt6}{4},\,\frac{\sqrt2}{4},\,\frac{\sqrt6}{4}\right),$

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

• $\left(\frac{\sqrt3-1}{4},\,\frac{1+\sqrt3}{4},\,\frac{1+\sqrt3}{4},\,\frac{\sqrt3-1}{4}\right),$
• $\left(\frac{\sqrt2}{4},\,\frac{\sqrt6}{4},\,\frac{\sqrt6}{4},\,\frac{\sqrt2}{4}\right).$

Isogonal derivatives

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

• Triangular gyroprism (192): Icositetrafold cubiswirlchoron
• Triangle (192): Icositetrafold cubiswirlchoron
• Edge (144): Icositetrafold octaswirlchoron