# 12-4 step prism

12-4 step prism
Rank4
TypeIsogonal
SpaceSpherical
Info
SymmetryI2(12)+×2×I, order 24
Elements
Vertex figureTetragonal antiwedge
Cells12 phyllic disphenoids, 3 square antiprisms
Faces12 isosceles triangles, 24 scalene triangles, 3 squares
Edges12+12+12
Vertices12
Measures (circumradius $\sqrt2$ , based on a unit duoprism)
Edge lengths3-valence (12): $\sqrt2 ≈ 1.41421$ 4-valence (12): $\sqrt{5-\sqrt3} ≈ 1.80775$ 3-valence (12): 2
Central density1
Euler characteristic0
Related polytopes
Dual12-4 gyrochoron
Properties
ConvexYes
OrientableYes
NatureTame

The 12-4 step prism is a convex isogonal polychoron and member of the step prism family. It has 3 chiral square antiprisms and 12 phyllic disphenoids as cells, with 4 disphenoids and 2 antiprisms joining at each vertex.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:$\sqrt{\frac{1+\sqrt3}{2}}$ ≈ 1:1.16877.

## Vertex coordinates

Coordinates for the vertices of a 12-4 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(2πk/3), b*cos(2πk/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:$\frac{\sqrt{27}}{3}$ ≈ 1:0.75984.

## Isogonal derivatives

Substitution by vertices of these following elements will produce these convex isogonal polychora:

• Phyllic disphenoid (12): 12-4 step prism
• Scalene triangle (12): 12-4 step prism
• Scalene triangle (24): 24-4 step prism
• Edge (12): 12-4 step prism