The Least-Perimeter Partition of a Sphere into Four Equal Areas

TL;DR

This paper proves that dividing a sphere into four equal-area regions with the least total perimeter results in a tetrahedral partition, confirming a geometric optimality for such divisions.

Contribution

It establishes that the minimal perimeter partition of a sphere into four equal areas is uniquely a tetrahedral configuration, resolving a geometric optimization problem.

Findings

Tetrahedral partition minimizes perimeter for four equal areas on a sphere.

Proof confirms the optimality of the tetrahedral shape for this partition.

Provides a rigorous mathematical proof of the minimal perimeter configuration.

Abstract

We prove that the least-perimeter partition of the sphere into four regions of equal area is a tetrahedral partition.