Triangular tetraswirlprism

Triangular tetraswirlprism
File:Triangular tetraswirlprism.png (OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	36 rhombic disphenoids, 72 phyllic disphenoids, 24 triangular gyroprisms
Faces	144+144 scalene triangles, 24 triangles
Edges	36+36+72+72
Vertices	36
Vertex figure	12-vertex polyhedron with 4 tetragons and 12 triangles
Measures (based on triangular duoprisms of edge length 1)
Edge lengths	Short 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 density	1
Related polytopes
Dual	Triangular tetraswirltegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(I2(12)≀S2)+/4, order 144
Convex	Yes
Nature	Tame

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).