# Dodecagram

The dodecagram is a star polygon with 12 sides. A regular dodecagram has equal sides and equal angles.

Dodecagram Rank2
TypeRegular
SpaceSpherical
Notation
Bowers style acronymDodag
Coxeter diagramx12/5o (     )
Schläfli symbol{12/5}
Elements
Edges12
Vertices12
Measures (edge length 1)
Circumradius$\frac{\sqrt6-\sqrt2}2 ≈ 0.51764$ Inradius$\frac{2-\sqrt3}2 ≈ 0.13397$ Area$3(2-\sqrt3) ≈ 0.80385$ Angle30°
Central density5
Number of external pieces24
Level of complexity2
Related polytopes
ArmyDog, edge length $2-\sqrt3$ DualDodecagram
ConjugateDodecagon
Convex coreDodecagon
Abstract & topological properties
Flag count24
Euler characteristic0
OrientableYes
Properties
SymmetryI2(12), order 24
ConvexNo
NatureTame

This is the fourth stellation of the dodecagon, and the only one that is not a compound. The only other polygons with a single non-compound stellation are the pentagon, the octagon, and the decagon.

It is the uniform quasitruncation of the hexagon, and as such appears as faces in a handful of uniform Euclidean tilings. It is the largest star polygon known to appear in any non-prismatic spherical or Euclidean uniform polytopes.

## Vertex coordinates

Coordinates for a dodecagram of unit edge length, centered at the origin, are all permutations of:

• $\left(±\frac{\sqrt3-1}2,\,±\frac{\sqrt3-1}2\right),$
• $\left(±\frac12,\,±\frac{2-\sqrt3}2\right).$

## Representations

A dodecagram has the following Coxeter diagrams:

• x12/5o (full symmetry)
• x6/5x (     ) (G2 symmetry)