geometralle

Geometralle is a planar geometric construct used in tiling theory and computational geometry. It describes a family of compact, simply connected regions whose boundary is piecewise linear or smooth and that can be generated by applying a single morphing rule to a base polygon. The concept is primarily theoretical, serving to examine how boundary deformations affect tiling properties and symmetry.

Construction and properties: A geometralle of order n begins with a regular n-gon. A parametric morphing rule, controlled by a parameter t in [0,1], replaces each edge with a congruent pair of edge segments and connects them with a smooth transition so that the boundary remains closed. For t = 0 the boundary matches the original polygon; as t increases, the shape deforms while preserving the original symmetry group. The resulting region has area A(t) and perimeter P(t) that change continuously with t.

Tilings and applications: When copies of a geometralle are arranged under the same isometries as the base polygon, they can form tilings of the plane for certain parameter values. This makes geometralle a useful model for studying tiling stability, pattern design, and mesh generation in computer graphics. Variants derived from triangles, squares, and pentagons are common in demonstrations of tiling transitions.

See also: tiling, isohedral tiling, shape interpolation, polygon, morphing.