eJMT Abstract

A polytile or p-tile as an equilateral polygon with turn angles as multiples of 360°/p, for even p=4,6,8…. This paper continues from paper 1a introducing polytiles. Section 2 explores the enumeration for strictly convex p-tiles, and subcounts by sides. Second 4 describes software used to explore polytiles. Section 5 offers tables for all solutions up to 18-tiles. Section 6 enumerations polytile counts by sides and symmetry to 32-tiles. Section 3 shows most of zonogons (even-sided with parallel opposite edges). Equilateral zonogons allow rhombic dissections, and the Generalized Dual Method (GDM) allows translated dual edge lines intersections to define a rhombic dissection. We also define a new class of progons, generalizing zonogons, as Minkowski sums of regular polygons instead of just digons (vectors). Regular polygons have regular k-rays duals (k divisor of p) making k-belts of rhombi. Zonogon and progons can be seen as projective envelopes of prism products of regular polygons. Exotic polytiles appear at 30-tiles which can’t all be decomposed into independent k-belts, requiring adjacent triangles and pentagons to dissect.