Continuous Flattening and Reversing of Convex Polyhedral Linkages

TL;DR

This paper proves that convex polyhedral linkages can be flattened or reversed through continuous transformations after subdividing edges, with bounds on the number of subdivisions needed, revealing complexities in geometric linkages.

Contribution

It introduces methods for flattening and reversing convex polyhedral linkages via edge subdivisions, establishing bounds and algorithms for these transformations.

Findings

Convex polyhedra can be flattened with linear subdivisions.

Equilateral linkages can be reversed through subdivisions.

Reversing nonequilateral tetrahedra may require many subdivisions.

Abstract

We prove two results about transforming any convex polyhedron, modeled as a linkage L of its edges. First, if we subdivide each edge of L in half, then L can be continuously flattened into a plane. Second, if L is equilateral and we again subdivide each edge in half, then L can be reversed, i.e., turned inside-out. A linear number of subdivisions is optimal up to constant factors, as we show (nonequilateral) examples that require a linear number of subdivisions. For nonequilateral linkages, we show that more subdivisions can be required: even a tetrahedron can require an arbitrary number of subdivisions to reverse. For nonequilateral tetrahedra, we provide an algorithm that matches this lower bound up to constant factors: logarithmic in the aspect ratio.