Continuous Flattening of All Polyhedral Manifolds using Countably Infinite Creases

TL;DR

This paper proves that any finite 3D polyhedral manifold can be continuously flattened into 2D with distance preservation and no crossings by extending folding models to include countably infinite creases, solving a long-standing open problem.

Contribution

It introduces a method to flatten all polyhedral manifolds in 3D using countably infinite creases, extending previous results limited to convex or semi-orthogonal polyhedra.

Findings

Any finite polyhedral manifold can be flattened into 2D.

Flattening preserves intrinsic distances and avoids crossings.

The area supporting moving creases can be made arbitrarily small.

Abstract

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.