# Triangular tetraswirlprism

Triangular tetraswirlprism
File:Triangular tetraswirlprism.png
Rank4
TypeIsogonal
SpaceSpherical
Elements
Cells36 rhombic disphenoids, 72 phyllic disphenoids, 24 triangular gyroprisms
Faces144+144 scalene triangles, 24 triangles
Edges36+36+72+72
Vertices36
Vertex figure12-vertex polyhedron with 4 tetragons and 12 triangles
Measures (based on triangular duoprisms of edge length 1)
Edge lengthsShort side edges (36): $\frac{3-\sqrt3}{3} ≈ 0.42265$ Medium side edges (36): $\frac{\sqrt6}{3} ≈ 0.81650$ Long side edges (72): $\sqrt{\frac{4-\sqrt3}{3}} ≈ 0.86947$ Edges of triangles (72): 1
Circumradius$\frac{\sqrt6}{3} ≈ 0.81650$ Central density1
Related polytopes
DualTriangular tetraswirltegum
Abstract & topological properties
Euler characteristic0
OrientableYes
Properties
Symmetry(I2(12)≀S2)+/4, order 144
ConvexYes
NatureTame

The triangular tetraswirlprism is a convex isogonal polychoron and member of the duoprismatic swirlprism family that consists of 24 triangular antiprisms, 36 rhombic disphenoids, and 72 phyllic disphenoids. 4 triangular gyroprisms, 4 rhombic disphenoids, and 8 phyllic disphenoids join at each vertex. It can be obtained as a subsymmetrical faceting of the dodecagonal duoprism. It is the fourth in an infinite family of isogonal triangular dihedral swirlchora.

The ratio between the longest and shortest edges is 1:$\frac{3+\sqrt3}{2}$ ≈ 1:2.36603.

## Vertex coordinates

Coordinates for the vertices of a triangular tetraswirlprism constructed as the convex hull of four triangular duoprisms of edge length 1, are given as Cartesian products of the vertices of triangle T1:

• T1 × T1,
• T2 × T2 (T1 rotated 30 degrees),
• T3 × T3 (T1 rotated 60 degrees).
• T4 × T4 (T1 rotated 90 degrees).