Tightly Circumscribed Regular Polygons

TL;DR

This paper investigates the optimal squeezing of concentric regular polygons with different side counts, analyzing limits and extensions in a Babuschka-doll fashion, without intermediate spacing circles, advancing geometric optimization understanding.

Contribution

It introduces a new approach to tightening circumscribed regular polygons by removing the need for intermediate circles, providing new limits and insights into polygon squeezing.

Findings

Derived optimal squeezing limits for concentric polygons

Extended analysis to prime number side counts

Demonstrated increased squeezing without intermediate circles

Abstract

A regular polygon circumscribing another regular polygon (with a different side number) may be tightened to minimize the difference of both areas. The manuscripts computes the optimum result under the restriction that both polygons are concentric, and obtains limits if the process is repeated in a two-dimensional Babuschka-doll fashion with side numbers increasing or decreasing by one or stepping through the prime numbers. The new aspect compared to the circumscription discussed in the literature so far is that further squeezing of the outer polygon is possible as we drop the requirement of drawing intermediate spacing circles between the polygon pairs.