Dodekagonal

Dodekagonal is an adjective used in geometry to describe things related to a dodecagon, a twelve‑sided polygon, or to twelvefold symmetry. The term derives from Greek dodeka meaning twelve and gonia meaning angle or corner. In geometry, dodekagonal can refer to shapes, tilings, or patterns that involve twelve angles or twelve-sided figures.

A regular dodekagon has all sides equal and all interior angles equal. Each interior angle measures 150 degrees, and the central angle subtended by each side is 30 degrees. The figure possesses dihedral symmetry with 12 rotational symmetries and 12 reflections, denoted by the symmetry group D12.

In tiling and symmetry studies, dodekagonal shapes appear in planar tilings that feature twelvefold symmetry. One example is the uniform tiling with vertex configuration 3.4.12.12, which uses triangles, squares, and dodecagons around each vertex. More generally, twelvefold (dodecagonal) symmetry is a subject in the theory of quasicrystals and aperiodic tilings, yielding patterns that exhibit twelvefold rotation but are not periodically repeating.

In three dimensions, dodekagonal prisms and related polyhedra use dodecagonal bases. A dodekagonal prism has two parallel regular dodecagonal bases connected by rectangular side faces. Dodekagonal shapes also appear in model-building and crystallography, where twelvefold symmetry can arise in certain quasicrystalline contexts.

In number theory, dodekagonal numbers form the polygonal number sequence for twelve sides, with the nth term given by P12(n) = 5n^2 − 4n. Early terms are 1, 12, 33, 72, and 129.