On Dissecting Polygons into Rectangles

TL;DR

This paper investigates the minimal number of polygon pieces needed to rearrange into a rectangle, providing improved bounds for regular polygons with 3 to 12 sides, and discusses related dissection problems.

Contribution

The authors present new constructions that improve bounds on the number of pieces needed to dissect regular polygons into rectangles, advancing understanding beyond previous bounds for most cases.

Findings

r(n) bounds are at most 2, 1, 4, 3, 5, 4, 7, 4, 9, 5 for n=3 to 12

Construction for 10-gon uses three fewer pieces than previous bounds for s(10)

Only r(3) and r(4) are known exactly for certain

Abstract

What is the smallest number of pieces that you can cut an n-sided regular polygon into so that the pieces can be rearranged to form a rectangle? Call it r(n). The rectangle may have any proportions you wish, as long as it is a rectangle. The rules are the same as for the classical problem where the rearranged pieces must form a square. Let s(n) denote the minimum number of pieces for that problem. For both problems the pieces may be turned over and the cuts must be simple curves. The conjectured values of s(n), 3 <= n <= 12, are 4, 1, 6, 5, 7, 5, 9, 7, 10, 6. However, only s(4)=1 is known for certain. The problem of finding r(n) has received less attention. In this paper we give constructions showing that r(n) for 3 <= n <= 12 is at most 2, 1, 4, 3, 5, 4, 7, 4, 9, 5, improving on the bounds for s(n) in every case except n=4. For the 10-gon our construction uses three fewer pieces than the bound for s(10). Only r(3) and r(4) are known for certain. We also briefly discuss q(n), the minimum number of pieces needed to dissect a regular n-gon into a monotile.