# Enneagrammic prism

Enneagrammic prism
Rank3
TypeUniform
Notation
Bowers style acronymStep
Coxeter diagramx x9/2o ()
Elements
Faces9 squares, 2 enneagrams
Edges9+18
Vertices18
Vertex figureIsosceles triangle, edge lengths 2, 2, 2cos(2π/9)
Measures (edge length 1)
Circumradius${\displaystyle {\frac {\sqrt {1+{\frac {1}{\sin ^{2}{\frac {2\pi }{9}}}}}}{2}}\approx 0.92470}$
Volume${\displaystyle {\frac {9}{4\tan {\frac {2\pi }{9}}}}\approx 2.68145}$
Dihedral angles4–4: 100°
4–9/2: 90°
Height1
Central density2
Number of external pieces20
Level of complexity6
Related polytopes
ArmySemi-uniform Ep, edge lengths ${\displaystyle {\frac {1}{2\cos {\frac {\pi }{9}}}}}$ (base), 1 (sides)
RegimentStep
DualEnneagrammic tegum
ConjugatesEnneagonal prism, Great enneagrammic prism
Convex coreEnneagonal prism
Abstract & topological properties
Flag count108
Euler characteristic2
OrientableYes
Genus0
Properties
SymmetryI2(9)×A1, order 36
ConvexNo
NatureTame

The enneagrammic prism, or step, is a prismatic uniform polyhedron. It consists of 2 enneagrams and 9 squares. Each vertex joins one enneagram and two squares. As the name suggests, it is a prism based on an ennearam.

## Vertex coordinates

The coordinates of an enneagrammic prism, centered at the origin and with edge length 2sin(2π/9), are given by:

• (1, 0, \pmsin(2π/9)),
• (cos(2π/9), \pmsin(2π/9), \pmsin(2π/9)),
• (cos(4π/9), \pmsin(4π/9), \pmsin(2π/9)),
• (–1/2, \pm3/2, \pmsin(2π/9)),
• (cos(8π/9), \pmsin(8π/9), \pmsin(2π/9)).