12-2 step prism

12-2 step prism
(OFF file)
Rank	4
Type	Isogonal
Space	Spherical
Elements
Cells	12+12+12+12 phyllic disphenoids, 6 rhombic disphenoids
Faces	24+24+24+24 scalene triangles, 12 isosceles triangles
Edges	6+12+12+12+12+12
Vertices	12
Vertex figure	Ridge-tritriakis notch
Measures (circumradius $\sqrt2$, based on a uniform duoprism)
Edge lengths	10-valence (12): $\sqrt{3-\sqrt3} ≈ 1.12603$
	3-valence (12): 2
	4-valence (6): 2
	4-valence (12): $\sqrt{3+\sqrt3} ≈ 2.17533$
	4-valence (12+12): $\sqrt6 ≈ 2.44949$
Central density	1
Related polytopes
Dual	12-2 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(12)-2), order 24
Convex	Yes
Nature	Tame

The 12-2 step prism is a convex isogonal polychoron and a member of the step prism family. It has 6 rhombic disphenoids and 48 phyllic disphenoids of four kinds as cells, with 18 (2 rhombic and 16 phyllic disphenoids) joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\frac{\sqrt{10+4\sqrt3}}{2}$ ≈ 1:2.05719.

Vertex coordinates

Coordinates for the vertices of a 12-2 step prism inscribed in a dodecagonal duoprism with base lengths a and b are given by:

• (a*sin(πk/6), a*cos(πk/6), b*sin(πk/3), b*cos(πk/3)),

where k is an integer from 0 to 11. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:$\sqrt{2+\sqrt3}$ ≈ 1:1.93185.

External links

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