Continuous flattening of all polyhedral manifolds using countably infinite creases
Introduction
We crush polyhedra flat all the time, such as when we recycle cereal boxes or store airbags in a steering wheel. But is this actually possible without tearing or stretching the material? This problem was first posed in 2001 [7] (see [9, Chapter 18]): does every polyhedron have a continuous motion that preserves the metric (intrinsic shortest paths), avoids crossings, and ends in a flat folded state? This problem is Open Problem 18.1 of the book Geometric Folding Algorithms [9]. In this paper, we solve this 19-year-old open problem with a positive answer: every polyhedron can be continuously flattend. Specifically, we prove for a broad definition of polyhedron: any compact polyhedral 2-manifold (possibly with boundary) embedded in 3D and having finitely many polygonal faces. However, our result is arguably in a model not intended by the original problem: our folding has countably infinitely many creases at all times.
A necessary first step is to show that every polyhedron has a flat folded state (the end of the desired flattening motion). This problem was also first posed in 2001 [7], where it was solved for convex and semi-orthogonal polyhedra.1 Later, Bern and Hayes [3] solved the problem for orientable polyhedral manifolds, generalizing a previous solution for sphere or disk topology [9]. This result solved Open Problem 18.2 of [9] (also originally posed in 2001 [7]), except for non-orientable polyhedral manifolds, which we solve here.
Continuous flattening necessarily requires continuously moving/sliding the creases on the surface over time (for polyhedra enclosing a volume): if all creases remained fixed throughout the motion (and the set of creases is finite), then the Bellows Theorem [4] tells us that the volume would remain fixed, so could not decrease to zero. A natural question, though, is how much area of the surface needs to be flexible in the sense of supporting moving creases, and how much can be made of rigid panels connected by hinges. Abel et al. [1] showed that a surprisingly small but finite slit suffices for continuous flattening of a regular tetrahedron. Matsubara and Nara [13] recently showed that an arbitrarily small area of flexibility suffices for α-trapezoidal polyhedra. In this paper, we show that an arbitrarily small area of flexibility suffices for any polyhedral manifold.
Several previous results constructed continuous flattenings of special classes of polyhedra. Itoh and Nara [10] solved Platonic solids while preserving two faces, and later with Vîlcu [11] solved convex polyhedra using Alexandrov surgery (which is difficult to compute). At SoCG 2014, Abel et al. [2] solved convex polyhedra using a simple algorithm that respects the straight skeleton gluing, corresponding to the intuitive way to flatten a polyhedron, and solving Open Problem 18.3 of [9] (the last open problem of Chapter 18, also originally posed in 2001 [7]). Unfortunately, this approach seems difficult to extend to nonconvex polyhedra. More recently, a slicing approach (dating back to [7]) was shown to continuously flatten semi-orthogonal polyhedra [6]. In this paper, we extend this slicing approach in several ways to solve arbitrary polyhedral manifolds.
We generalize the slicing approach of [6], which conceptually cuts the polyhedron along parallel planes through every vertex, and several additional planes in between so that the resulting slabs (portions of the polyhedron between consecutive planes) are “short”. In [6], each slab is an orthogonal band, which is relatively easy to flatten continuously. The key difference in our case is that the slabs are much more general: in general, a slab in a polyhedron is a prismatoid (excluding the top and bottom faces), that is, a polyhedron whose vertices lie in two parallel planes, whose faces are triangles and trapezoids spanning both planes. Unfortunately, prismatoids seem extremely difficult to flatten continuously, as original polyhedron vertices are particularly difficult to handle in the general case.
To circumvent this challenge, we instead target the flattening of prismoids: prismatoids whose spanning faces are only trapezoids having parallel top and bottom edges (i.e., no triangular spanning faces), where every vertex is incident to at most two spanning trapezoids. We will use the term cylindrical prismoid to refer to the spanning faces of a prismoid, without the top and bottom face. A key innovation in our approach is to divide a polyhedral manifold using countably infinitely many parallel planar cuts, with slabs approaching zero height as we approach polyhedron vertices. As a result, all slabs consist of disjoint cylindrical prismoids. The key property is that original polyhedron vertices do not appear on the boundary of any slab, because any such slab would get divided in half through countably infinite recursion.
Note that since we allow polyhedral manifolds with boundary, a component within a slab may only be a subset of a cylindrical prismatoid, we call a prismoidal wall; this generalization is discussed in Section 4.
We implement the approach described above in a bottom-up fashion. First, Section 2 formally defines our model of folding. Next, Section 3 shows how to collapse prismoid edges and faces by constructing generalized In-Out and Out-Out gadgets. Then, Section 4 shows how to slice the input polyhedral manifold so that we can flatten subsets of it using the methods from Section 3. Finally, Section 5 puts these algorithms together to prove the following theorem:
Theorem 1 Any compact polyhedral 2-manifold (possibly with boundary) embedded in 3D and having finitely many polygonal faces can be continuously flattened while preserving intrinsic distances and avoiding crossings. A flattening motion exists such that at all times during the flattening motion (except the beginning), the folded form consists of countably infinitely many creases, with finitely many accumulation lines. Furthermore, the area supporting moving creases can be made arbitrarily small.
Section snippets
Model
The standard model of folding 2D surfaces in 3D [9, Chapter 11] assumes finitely many creases, as that is the primary case of interest for origami. A full definition supporting countably infinitely many creases is likely possible, but difficult, as it is no longer possible to focus on well-behaved positive-area neighborhoods. For the purposes of this paper, we define a limited model of folding with countably infinite creases, where the folding decomposes into components separated by horizontal
Flattening prismoids
In this section, we show how to flatten prismoids which have a small height relative to their other features. We will then use this technique to flatten arbitrary polyhedral manifolds after slicing them into a countable set of such prismoids, as detailed in Section 4. Section 3.1 describes an overview of our approach, and the rest of this section describes the details of how to locally flatten the edges and faces of a prismoid.
Slicing
In this section, we show how to slice our polyhedral manifold into prismoids so techniques from the previous section can be applied. Once sliced, we can collapse the subset in each slab separately. Because we are not restricting our input to be homeomorphic to a sphere, components within a slab might not be prismoids, but instead subsets of prismoids (i.e., missing faces). To deal with this generalization, we define a prismoidal wall to be a (non-strict) subset of the spanning faces of some
Flattening polyhedral manifolds
Knowing how to locally flatten prismoids using the gadgets from Section 3, and how to split polyhedral manifolds into projection-disjoint prismoidal slabs whose geometry is well-separated relative to slab height from Section 4, the proof of Theorem 1 follows directly:
Proof of Theorem 1 Slice the prismoidal manifold into a countably infinite set of prismoidal slabs using the construction in Lemma 4. Then decompose each prismoidal slab into projection disjoint prismoidal slabs with well-separated geometry according
Conclusion
In this paper, we showed how to continuously flatten finite polyhedral manifolds using countably many creases with only finitely many accumulation lines. The obvious open problem is whether continuous flattening is possible with only finitely many creases at each time (still, of course, with movable creases that slide over a 2D region of points over time). In the other direction, perhaps our approach could be generalized to polyhedral manifolds with countably many vertices, edges, and faces.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
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