# Difold ditetraswirlchoron

Difold ditetraswirlchoron
File:Difold ditetraswirlchoron.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells24 phyllic disphenoids, 16 triangular pyramids, 8 triangular gyroprisms
Faces48 scalene triangles, 48 isosceles triangles, 16 triangles
Edges8+24+48
Vertices16
Vertex figure10-vertex polyhedron with 3 tetragons and 10 triangles
Measures (circumradius 1, based on a 2D regular dodecagonal envelope)
Edge lengths6-valence (8): ${\displaystyle \sqrt{2-\sqrt3} ≈ 0.51764}$
6-valence (24): ${\displaystyle \sqrt{\frac{6-\sqrt3}{3}} ≈ 1.19275}$
3-valence (48): ${\displaystyle \sqrt3 ≈ 1.41421}$
Central density1
Related polytopes
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
SymmetryA3●K2, order 48
ConvexYes
NatureTame

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:${\displaystyle \sqrt2}$ ≈ 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:

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