Hinged Dissection of Polyominoes and Polyforms

TL;DR

This paper introduces a universal hinged dissection method for transforming and connecting various polyominoes and polyforms, enabling flexible folding into multiple shapes with a minimal number of pieces.

Contribution

It presents a new hinged dissection construction applicable to all edge-to-edge gluings of congruent polygons, reducing the number of pieces needed for regular polygons and extending to specific polyforms.

Findings

Hinged dissection for all edge-to-edge gluings of congruent polygons.

Reduction in the number of pieces for regular polygons.

Universal hinged dissection for polyominoes, polyiamonds, and polyhexes.

Abstract

A hinged dissection of a set of polygons S is a collection of polygonal pieces hinged together at vertices that can be folded into any member of S. We present a hinged dissection of all edge-to-edge gluings of n congruent copies of a polygon P that join corresponding edges of P. This construction uses kn pieces, where k is the number of vertices of P. When P is a regular polygon, we show how to reduce the number of pieces to ceiling(k/2)*(n-1). In particular, we consider polyominoes (made up of unit squares), polyiamonds (made up of equilateral triangles), and polyhexes (made up of regular hexagons). We also give a hinged dissection of all polyabolos (made up of right isosceles triangles), which do not fall under the general result mentioned above. Finally, we show that if P can be hinged into Q, then any edge-to-edge gluing of n congruent copies of P can be hinged into any edge-to-edge gluing of n congruent copies of Q.