Geometrins

Geometrins are a class of two-dimensional geometric figures defined by an iterative substitution rule applied to a starting polygonal motif. Each geometrin type specifies a fixed replacement pattern that is applied to every edge and vertex in successive generations. The result is a closed region whose boundary is a concatenation of segments from the motif and whose interior remains a single connected set. Geometrins are studied within substitution tilings and fractal geometry and can display self-similarity and varying boundary complexity depending on the rule.

Construction and classification: A geometrin is generated by a formal grammar or L-system that prescribes how a motif is replaced at each step. The n-th iteration yields a scaled copy of the original pattern, with a linear scale factor s^n. Area typically grows roughly with the square of the linear scale, while perimeter grows roughly with the linear scale, though exact growth depends on the motif geometry. Some geometrins tile the plane periodically; others produce nonperiodic tilings.

Examples and uses: Commonly studied geometrins include those defined by simple replacement rules that yield triangular or polygonal boundary shapes. In practice, geometrins are used in mathematical art, computer graphics, and theoretical investigations of isoperimetric properties in self-similar shapes. The framework helps compare different substitution schemes and their geometric consequences, without committing to a single canonical form.

See also: substitution tiling, fractal geometry, self-similarity, aperiodic tiling.