Partitioning Regular Polygons into Circular Pieces I: Convex Partitions

TL;DR

This work investigates convex partitions of regular polygons into pieces whose shapes are as circular as possible by minimizing the maximum aspect ratio ${\gamma}$, defined via circumcircle and incircle radii. It proves that for all regular $k$-gons with $k\ge 5$, the unpartitioned polygon is optimal ${\gamma}^*={\gamma}_1$, while the equilateral triangle requires infinitely many pieces to approach the lower bound ${\gamma}_{60^\circ}=3/2$, and the square lies in between with ${\gamma}^*\in[1.28868,1.29950]$, leaving a small gap. The analysis develops constructive partitions using corner pieces, 80°-quadrilaterals, and 80°-curves to approach the lower bound on ${\gamma}$ for the triangle, while boundary- and interior-filling strategies yield progressively better squares partitions up to ${\gamma}=1.29950$; a rigorous lower bound for the square is ${\gamma}^*\ge 1.28868$, with stronger evidence under tangent-indisks constraints. Overall, the paper establishes the optimal convex partition behavior for most regular polygons, highlights the square as the main open case, and connects geometric packing/covering ideas to polygon partitioning with practical implications for fast geometric computations.

Abstract

We explore an instance of the question of partitioning a polygon into pieces, each of which is as ``circular'' as possible, in the sense of having an aspect ratio close to 1. The aspect ratio of a polygon is the ratio of the diameters of the smallest circumscribing circle to the largest inscribed disk. The problem is rich even for partitioning regular polygons into convex pieces, the focus of this paper. We show that the optimal (most circular) partition for an equilateral triangle has an infinite number of pieces, with the lower bound approachable to any accuracy desired by a particular finite partition. For pentagons and all regular k-gons, k > 5, the unpartitioned polygon is already optimal. The square presents an interesting intermediate case. Here the one-piece partition is not optimal, but nor is the trivial lower bound approachable. We narrow the optimal ratio to an aspect-ratio gap of 0.01082 with several somewhat intricate partitions.

Figures (24)

• Figure 1: Aspect Ratio of Regular Polygons.
• Figure 2: One-angle lower bound.
• Figure 4: A polygonal approximation in the vicinity of the point of tangency $x$ between the indisk $C_0$ and the circumcircle $C_1$ must either go exterior to $C_1$ (left) or interior to $C_0$ (right).
• Figure 5: The range of rectangles fitting between the in-circle and out-circle have ${\gamma}_1 < 3/2$.
• Figure 6: $k_b=6$ and $k_a=2$; ${\varepsilon}=0.15$. The bottom row shows ${\delta} \times {\delta}$ squares, leaving a gap less than ${\delta}$, and the top row show ${\delta}(1+{\varepsilon}_1) \times {\delta}$ squares, with ${\varepsilon}_1 < {\varepsilon}$, that fully fill the strip.

Theorems & Definitions (20)

• Lemma 1
• Lemma 2
• Theorem 3
• Lemma 4
• Lemma 5
• Lemma 6
• Lemma 7
• Lemma 8
• Lemma 9
• Lemma 10