Paper by Martin L. Demaine
- Zachary Abel, Erik D. Demaine, Martin L. Demaine, Jason S. Ku, Jayson Lynch, Jin-ichi Itoh, and Chie Nara, “Continuous Flattening of All Polyhedral Manifolds using Countably Infinite Creases”, Computational Geometry: Theory and Applications, volume 98, October 2021, Article 101773.
We prove that any finite polyhedral manifold in 3D can be continuously
flattened into 2D while preserving intrinsic distances and avoiding crossings,
answering a 19-year-old open problem, if we extend standard folding models to
allow for countably infinite creases. The most general cases previously known
to be continuously flattenable were convex polyhedra and semi-orthogonal
polyhedra. For non-orientable manifolds, even the existence of an
instantaneous flattening (flat folded state) is a new result. Our solution
extends a method for flattening semi-orthogonal polyhedra: slice the
polyhedron along parallel planes and flatten the polyhedral strips between
consecutive planes. We adapt this approach to arbitrary nonconvex polyhedra
by generalizing strip flattening to nonorthogonal corners and slicing along a
countably infinite number of parallel planes, with slices densely approaching
every vertex of the manifold. We also show that the area of the polyhedron
that needs to support moving creases (which are necessary for closed polyhedra
by the Bellows Theorem) can be made arbitrarily small.
- This paper is also available from ScienceDirect and as arXiv:2105.10774.
- The paper is 14 pages.
- The paper is available in PDF (1390k).
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