9-3 step prism

9-3 step prism
(OFF file)
Rank	4
Type	Isogonal
Elements
Cells	9 phyllic disphenoids, 3 triangular gyroprisms
Faces	18 scalene triangles, 9 isosceles triangles, 3 triangles
Edges	9+9+9
Vertices	9
Vertex figure	Tetragonal antiwedge
Measures (circumradius ${\sqrt {2}}$, based on a uniform duoprism)
Edge lengths	3-valence (9): ${\sqrt {3}}\approx 1.73205$
	4-valence (9): ${\sqrt {3+4\sin ^{2}{\frac {\pi }{9}}}}\approx 1.86223$
	3-valence (9): ${\sqrt {5-2\sin {\frac {\pi }{18}}}}\approx 2.15701$
Central density	1
Related polytopes
Dual	9-3 gyrochoron
Abstract & topological properties
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	S2(I2(9)-3), order 18
Convex	Yes
Nature	Tame

The 9-3 step prism is a convex isogonal polychoron and a member of the step prism family. It has 3 triangular gyroprisms and 9 phyllic disphenoids as cells, with 4 disphenoids and 2 gyroprisms joining at each vertex..

It is one of three isogonal polychora with 9 vertices (the others are the 9-2 step prism and triangular duoprism), as well as the simplest step prism to have cells other than tetrahedra.

Using the ratio method, the lowest possible ratio between the longest and shortest edges is 1:${\sqrt {\frac {3+2\cos {\frac {2\pi }{9}}-2\sin {\frac {\pi }{18}}}{3}}}$ ≈ 1:1.18107.

Vertex coordinates

Coordinates for the vertices of a 9-3 step prism inscribed in an enneagonal duoprism with base lengths a and b are given by:

• (a*sin(2πk/9), a*cos(2πk/9), b*sin(2πk/3), b*cos(2πk/3)),

where k is an integer from 0 to 8. If the edge length differences are to be minimized, the ratio of a:b must be equivalent to 1:${\sqrt {\frac {1+2\cos {\frac {2\pi }{9}}}{3}}}$ ≈ 1:0.91871.

Isogonal derivatives

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

• Phyllic disphenoid (9): 9-3 step prism
• Scalene triangle (9): 9-3 step prism
• Scalene triangle (18): 18-3 step prism
• Edge (9): 9-3 step prism

External links

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