Dodecagon

Dodecagon
(OFF file)
Rank	2
Type	Regular
Notation
Bowers style acronym	Dog
Coxeter diagram	x12o ()
Schläfli symbol	{12}
Elements
Edges	12
Vertices	12
Vertex figure	Dyad, length (√2+√6)/2
Measures (edge length 1)
Circumradius	${\frac {{\sqrt {2}}+{\sqrt {6}}}{2}}\approx 1.93185$
Inradius	${\frac {2+{\sqrt {3}}}{2}}\approx 1.86603$
Area	$3(2+{\sqrt {3}})\approx 11.19615$
Angle	150°
Central density	1
Number of external pieces	12
Level of complexity	1
Related polytopes
Army	Dog
Dual	Dodecagon
Conjugate	Dodecagram
Abstract & topological properties
Flag count	24
Euler characteristic	0
Orientable	Yes
Properties
Symmetry	I2(12), order 24
Flag orbits	1
Convex	Yes
Nature	Tame

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

The only non-compound stellation of the dodecagon is the dodecagram. This makes it the largest polygon with a single non-compound stellation. The only other polygons with only one are the pentagon, the octagon, and the decagon.

The dodecagon is not found in the non-prismatic uniform polyhedra nor the Johnson solids. It does appear in regular-faced tilings such as the uniform truncated hexagonal tiling (since it's the uniform truncation of the hexagon) and is the largest polygon to do so.

The dodecagon appears as the face of several paracompact regular skew polyhedra, and is the largest planar polygon to appear in a spherical, euclidean, compact, or paracompact polyhedron in 3D space.

Naming

The name decagon is derived from the Ancient Greek δώδεκα (12) and γωνία (angle), referring to the number of vertices.

Other names include:

• dog, Bowers style acronym, short for "dodecagon".

The combining prefix in BSAs is tw-, as in twip or twaddip.

Vertex coordinates

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

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

Representations

A dodecagon has the following Coxeter diagrams:

• x12o () (full symmetry)
• x6x () (G₂ symmetry, generally a dihexagon)
• xy3yx&#zx (A2 symmetry, y = $1+{\sqrt {3}}$)

Variations

Two main variants of the dodecagon have hexagon symmetry: the dihexagon, with two alternating side lengths and equal angles, and the dual hexambus, with two alternating angles and equal edges. Other less regular variations with chiral hexagonal, square, triangular, rectangular, inversion, mirror, or no symmetry also exist.

Stellations

• 1st stellation: Stellated dodecagon (compound of two hexagons)
• 2nd stellation: Trisquare (compound of three squares)
• 3rd stellation: Tetratriangle (compound of four triangles)
• 4th stellation: Dodecagram

External links

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

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