Dodecagram

Dodecagram
(OFF file)
Rank	2
Type	Regular
Notation
Bowers style acronym	Dodag
Coxeter diagram	x12/5o ()
Schläfli symbol	{12/5}
Elements
Edges	12
Vertices	12
Vertex figure	Dyad, length (√6–√2)/2
Measures (edge length 1)
Circumradius	${\frac {{\sqrt {6}}-{\sqrt {2}}}{2}}\approx 0.51764$
Inradius	${\frac {2-{\sqrt {3}}}{2}}\approx 0.13397$
Area	$3(2-{\sqrt {3}})\approx 0.80385$
Angle	30°
Central density	5
Number of external pieces	24
Level of complexity	2
Related polytopes
Army	Dog, edge length $2-{\sqrt {3}}$
Dual	Dodecagram
Conjugate	Dodecagon
Convex core	Dodecagon
Abstract & topological properties
Flag count	24
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(12), order 24
Convex	No
Nature	Tame

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:

• $\left(\pm {\frac {{\sqrt {3}}-1}{2}},\,\pm {\frac {{\sqrt {3}}-1}{2}}\right),$
• $\left(\pm {\frac {1}{2}},\,\pm {\frac {2-{\sqrt {3}}}{2}}\right).$

Representations

A dodecagram has the following Coxeter diagrams:

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

External links

• Bowers, Jonathan. "Regular Polygons and Other Two Dimensional Shapes".

• Wikipedia contributors. "Dodecagram".
• Hartley, Michael. "{12}*24".