Dodecagram

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

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

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:

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

Representations

A dodecagram has the following Coxeter diagrams:

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