A Trapezium Generates Tiling of Concentric Regular Pentagons

Abstract

A case of tiling using a single tile of golden trapezium is presented and each of the five pentagonal sectors when seen in isolation, reveals planes of translation suggesting periodicity. It is shown that this tile not only tessellates Euclidean plane, expanding to infinity, but also maintains perfect regular pentagonal outline at each completed generation of tiling. The entire tiling suggests a five-fold intergrowth structure of a twinned crystal, which should yield sum of five discrete and superimposed ordered periodic diffraction patterns. Clue to this tiling was primarily provided by a Coccolithophore species: Braarudosphaera bigelowii, which makes an elegant regular pentagonal dodecahedron of mineral Calcite, invisible to unaided human eye. Ke yw ords: Golden Trapezium, Periodic Tiling, Concentric Regular Pentagons, Twinned Crystal, Braarudosphaera bigelowii