Difold ditetraswirlchoron

Difold ditetraswirlchoron
File:Difold ditetraswirlchoron.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	24 phyllic disphenoids, 16 triangular pyramids, 8 triangular gyroprisms
Faces	48 scalene triangles, 48 isosceles triangles, 16 triangles
Edges	8+24+48
Vertices	16
Vertex figure	10-vertex polyhedron with 3 tetragons and 10 triangles
Measures (circumradius 1, based on a 2D regular dodecagonal envelope)
Edge lengths	6-valence (8): $\sqrt{2-\sqrt3} ≈ 0.51764$
	6-valence (24): $\sqrt{\frac{6-\sqrt3}{3}} ≈ 1.19275$
	3-valence (48): $\sqrt3 ≈ 1.41421$
Central density	1
Related polytopes
Dual	Ditetraswirlic hexadecachoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	A3●K2, order 48
Convex	Yes
Nature	Tame

The difold ditetraswirlchoron, also known as the rectifold tetraswirlchoron or disphenoidal-digonal scalenohedral 8-3 double step prism, is one of several isogonal polychoron, formed as a convex hull of two hexadecachora. It consists of 8 triangular gyroprisms, 16 triangular pyramids, and 24 phyllic disphenoids. 3 triangular gyroprisms, 4 triangular pyramids, and 6 phyllic disphenoids join at each vertex.

This polychoron cannot be optimized using the ratio method, because the solution (with intended minimal ratio 1:$\sqrt2$ ≈ 1:1.41421) would yield a tesseract instead.

Vertex coordinates

Coordinates for the vertices of a difold ditetraswirlchoron based on a 2D regular dodecagonal envelope of circumradius 1, centered at the origin, are given by:

• $±\left(0,\,0,\,0,\,1\right),$
• $±\left(\frac{\sqrt6}{3},\,0,\,\frac{\sqrt3}{3},\,0\right),$
• $±\left(\frac{\sqrt6}{6},\,±\frac{\sqrt2}{2},\,-\frac{\sqrt3}{3},\,0\right),$
• $±\left(0,\,0,\,\frac12,\,\frac{\sqrt3}{2}\right),$
• $±\left(0,\,\frac{\sqrt6}{3},\,-\frac12,\,\frac{\sqrt3}{6}\right),$
• $±\left(±\frac{\sqrt2}{2},\,\frac{\sqrt6}{3},\,\frac12,\,-\frac{\sqrt3}{6}\right).$

External links

• Bowers, Jonathan. "Four Dimensional Dice Up To Twenty Sides".