Dodecagon

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Dodecagon
Regular dodecagon.svg
Rank2
TypeRegular
SpaceSpherical
Notation
Bowers style acronymDog
Coxeter diagramx12o (CDel node 1.pngCDel 12.pngCDel node.png)
Schläfli symbol{12}
Elements
Edges12
Vertices12
Vertex figureDyad, length (2+6)/2
Measures (edge length 1)
Circumradius
Inradius
Area
Angle150°
Central density1
Number of pieces12
Level of complexity1
Related polytopes
ArmyDog
DualDodecagon
ConjugateDodecagram
Abstract properties
Flag count24
Euler characteristic0
Topological properties
OrientableYes
Properties
SymmetryI2(12), order 24
ConvexYes
NatureTame

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[edit | edit source]

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"

Vertex coordinates[edit | edit source]

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

Representations[edit | edit source]

A dodecagon has the following Coxeter diagrams:

  • x12o (full symmetry)
  • x6x (G2 symmetry, generally a dihexagon)
  • xy3yx&#zx (A2 symmetry, y = 1+√3)

Variations[edit | edit source]

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[edit | edit source]

External links[edit | edit source]