Dodecagon

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

The combining prefix is tw(a)-, as in twip or twaddip.

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.

Regular dodecagons generally do not occur in higher spherical polytopes aside from prisms, though some Euclidean tilings with hexagonal tiling symmetry do use dodecagonal faces, as the dodecagon is the uniform truncation of the hexagon.

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

Other names include:


 * dog, Bowers style acronym, short for "dodecagon"

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


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

Representations
A dodecagon has the following Coxeter diagrams:


 * x12o (full symmetry)
 * x6x (G2 symmetry, generally a dihexagon)
 * xy3yx&#zx (A2 symmetry, y = 1+√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