Edge Tessellation and Stamp Foldings Puzzles

TL;DR

This paper classifies polygons that generate edge tessellations and confirms which triangular stamp shapes are suitable for folding puzzles, providing a comprehensive understanding of geometric tilings and practical puzzle constraints.

Contribution

It proves the complete list of polygons that can generate edge tessellations and verifies the specific triangle shapes suitable for stamp folding puzzles.

Findings

Eight types of polygons generate edge tessellations.

Only three triangle shapes are suitable for stamp folding puzzles.

Complete classification of edge tessellating polygons.

Abstract

An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. We prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right, or 120-isosceles triangle; a 120-rhombus; a 60-90-120 kite; or a regular hexagon. A stamp folding puzzle is a paper folding problem constrained to the perforations on a sheet of postage stamps. We establish the following conjecture due to G. Frederickson: "Although triangular stamps have come in a variety of different triangular shapes, only three shapes seem suitable for [stamp] folding puzzles: equilateral, isosceles right triangles, and 60-right triangles."