On Dissecting Polygons into Rectangles
N. J. A. Sloane, Gavin A. Theobald

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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsComputational Geometry and Mesh Generation · Manufacturing Process and Optimization · Optimization and Packing Problems
