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arXiv · 2412.15130

Continuous Flattening and Reversing of Convex Polyhedral Linkages

Erik D. Demaine·Martin L. Demaine·Markus Hecher·Rebecca Lin·Victor H. Luo·Chie Nara
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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.

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Erik D. Demaine, Martin L. Demaine, Markus Hecher, Rebecca Lin, Victor H. Luo, Chie Nara. 2024-12-19. Continuous Flattening and Reversing of Convex Polyhedral Linkages. https://arxiv.org/abs/2412.15130

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