decagonal

Decagonal is an adjective used in geometry and related fields to describe things pertaining to a decagon, a ten-sided polygon, or to tenfold symmetry. It is commonly applied to shapes, tilings, and symmetry patterns that involve tenfold repetition or structure.

In polygonal geometry, a decagon has ten sides. A regular decagon has all sides and angles equal, and each interior angle measures 144 degrees. A regular decagon can be inscribed in a circle, and its area, given a side length s, can be expressed as A = (5/2) s^2 cot(π/10). These properties place decagons among the many regular and irregular polygons studied in planar geometry.

Decagonal symmetry refers to tenfold rotational symmetry around a point. In the plane, true tenfold symmetry cannot yield a periodic lattice, so decagonal patterns often arise in non-periodic tilings and in models of quasicrystals. Decagonal tilings exhibit invariance under rotation by 36 degrees and are typically generated by projection methods from higher-dimensional lattices or by inflation–deflation rules.

In crystallography and materials science, decagonal phases describe a class of quasicrystals with tenfold symmetry about a unique axis. These materials are quasiperiodic in two dimensions but can be periodic along the axial direction, producing characteristic diffraction patterns that reflect the decagonal symmetry.

Other uses include decagonal prisms, a type of polyhedron with two parallel decagonal bases and ten lateral faces, and notions in number theory. The nth decagonal number is Dn = 4n^2 − 3n, while centered decagonal numbers follow Cn = 5n^2 − 5n + 1, illustrating the term’s reach beyond pure geometry.