Exact self-tilings of convex polygons: constructions and angle obstructions

Abstract

Every convex trapezoid and every convex quadrilateral with two opposite right angles admits an exact countably infinite tiling by strictly smaller similar closed copies. In contrast, quadrilaterals whose cyclic angles have only the total-angle integer relation admit no finite or countable strict self-tiling; these angles form a full-measure subset of the three-dimensional angle domain. For pentagons, we construct the entire real family of one-corner-truncated parallelograms. In every number of sides, positivity of all adjacent-angle excesses prevents self-tiling. The constructions use a structural equivalence between exact self-tilability and tilability of the closed square or a closed parallelogram, with a sharp least-angle criterion for the parent and arbitrarily small tiles. The proofs cover all boundary and accumulation points; the quadrilateral exclusion uses shrinking rhombus charts and Beltrami uniformization. MSC 2020: Primary 52C20; Secondary 05B45, 30C62. The accompanying files contain the manuscript PDF and a curated LaTeX source archive.