Rectifications of Convex Polyhedra

TL;DR

This paper explores the geometric process of rectifying convex polyhedra by analyzing edge midpoints, deriving formulas for regular polygons, and examining rectification sequences of Platonic solids.

Contribution

It introduces a formal approach to rectification of convex polyhedra, derives explicit formulas for regular polygons, and characterizes rectification sequences of Platonic solids.

Findings

Formulas for side lengths and areas of rectified regular polygons

Computed surface areas and volumes of various convex polyhedra

Identified two disjoint rectification sequences for Platonic solids

Abstract

A convex polyhedron, that is, a compact convex subset of $\mathbb{R}^3$ which is the intersection of finitely many closed half-spaces, can be rectified by taking the convex hull of the midpoints of the edges of the polyhedron. We derive expressions for the side lengths and areas of rectifications of regular polygons in plane, and use these results to compute surface areas and volumes of various convex polyhedra. We introduce rectification sequences and show that there are exactly two disjoint pure rectification sequences generated by the platonic solids.