Digonal-square triswirlprism

Digonal-square triswirlprism
File:Digonal-square triswirlprism.png (OFF file)
Rank	4
Type	Isogonal
Elements
Cells	24+24 phyllic disphenoids, 12 rhombic disphenoids, 6 square gyroprisms
Faces	48+48+48 scalene triangles, 6 squares
Edges	12+24+24+24+24
Vertices	24
Vertex figure	9-vertex polyhedron with 2 tetragons and 10 triangles
Measures (based on square prisms of edge length 1)
Edge lengths	Short side edges (24): ${\frac {\sqrt {5-2{\sqrt {3}}}}{2}}\approx 0.61966$
	Medial side edges (24): ${\frac {\sqrt {3}}{2}}\approx 0.86603$
	Long side edges (24): ${\frac {\sqrt {7-2{\sqrt {3}}}}{2}}\approx 0.94020$
	Digon edges (12): 1
	Edges of squares (24): 1
Circumradius	${\frac {\sqrt {3}}{2}}\approx 0.86603$
Central density	1
Related polytopes
Dual	Digonal-square triswirltegum
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	(G2×I2(12))+/3, order 48
Convex	Yes
Nature	Tame

The digonal-square triswirlprism, also known as the 12-2 double step prism or 12-4 double step prism, is a convex isogonal polychoron and member of the duoprismatic swirlprism family that consists of 6 square gyroprisms, 12 rhombic disphenoids, and 48 phyllic disphenoids of two kinds. 2 square gyroprisms, 2 rhombic disphenoids, and 8 phyllic disphenoids join at each vertex. It can be obtained as a subsymmetrical faceting of the hexagonal-dodecagonal duoprism.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\frac {\sqrt {260+104{\sqrt {3}}}}{13}}$ ≈ 1:1.61380.

Vertex coordinates

Coordinates for the vertices of a digonal-square triswirlprism, assuming that the edge length differences are minimized, are given as Cartesian products of the vertices of square S₁ and digon D₂ with length ratio 1:1:

• S₁ × D₂,
• S₃ × D₄ (T₁ rotated 30 degrees and D₂ rotated 60 degrees),
• S₅ × D₆ (T₁ rotated 60 degrees and D₂ rotated 120 degrees).