[Paper Review] Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes

Erik D. Demaine, Martin L. Demaine+2 more|ArXiv.org|Jul 13, 2000

Computational Geometry and Mesh Generation9 references14 citations

TL;DR

This paper investigates the combinatorial and geometric enumeration of foldings and unfoldings between convex polygons and convex polytopes, using Aleksandrov's theorem and gluing/cut trees to analyze how many distinct polytopes can be formed from a polygon or vice versa. It establishes that while convex polygons yield infinitely many distinct polytopes via folding, their combinatorially distinct gluings are polynomially bounded, and that some simple polygons and polytopes admit exponentially many foldings or unfoldings.

ABSTRACT

We investigate how to make the surface of a convex polyhedron (a polytope) by folding up a polygon and gluing its perimeter shut, and the reverse process of cutting open a polytope and unfolding it to a polygon. We explore basic enumeration questions in both directions: Given a polygon, how many foldings are there? Given a polytope, how many unfoldings are there to simple polygons? Throughout we give special attention to convex polygons, and to regular polygons. We show that every convex polygon folds to an infinite number of distinct polytopes, but that their number of combinatorially distinct gluings is polynomial. There are, however, simple polygons with an exponential number of distinct gluings. In the reverse direction, we show that there are polytopes with an exponential number of distinct cuttings that lead to simple unfoldings. We establish necessary conditions for a polytope to have convex unfoldings, implying, for example, that among the Platonic solids, only the tetrahedron has a convex unfolding. We provide an inventory of the polytopes that may unfold to regular polygons, showing that, for n>6, there is essentially only one class of such polytopes.

Motivation & Objective

• To understand the number of distinct ways a polygon can be folded into a convex polytope via perimeter gluing.
• To determine how many distinct unfoldings a convex polytope can have into a simple polygon.
• To analyze the constraints and combinatorial structure of convex and regular polygons in folding and unfolding processes.
• To establish bounds on the number of combinatorially distinct foldings and unfoldings, especially for convex and regular cases.
• To investigate the existence and structure of convex unfoldings, particularly for Platonic solids and regular polygons.

Proposed method

• Applies Aleksandrov's theorem to characterize valid foldings of a polygon into a convex polytope by ensuring local angle sum conditions at identified points.
• Introduces cut trees and gluing trees as combinatorial models to represent the structure of unfoldings and foldings respectively.
• Uses geometric distance inequalities and shortest path arguments to eliminate invalid edge candidates in polytope unfoldings.
• Employs symmetry and vertex labeling (e.g., primed and unprimed vertices) to analyze regular and pita polytopes.
• Applies extremal combinatorics to derive upper and lower bounds on the number of distinct gluings and cuttings.
• Leverages the triangle inequality and properties of inscribed regular polygons to compare chord lengths and rule out non-shortest geodesics.

Experimental results

Research questions

• RQ1 How many combinatorially distinct foldings (gluings) can a convex polygon have into a convex polytope?
• RQ2 How many geometrically distinct polytopes can be formed by folding a single convex polygon?
• RQ3 What is the maximum number of distinct unfoldings (cuttings) a convex polytope can have into a simple polygon?
• RQ4 Which convex polytopes admit convex unfoldings, and what structural conditions must they satisfy?
• RQ5 How many distinct polytopes can unfold to a regular polygon, and what are the possible classes for n > 6?

Key findings

• Every convex polygon can be folded into infinitely many distinct polytopes, but only finitely many combinatorially distinct gluings, bounded by O(n³) for convex polygons.
• There exist simple polygons with an exponential number of distinct gluings, specifically 2^Ω(n) combinatorially distinct foldings.
• Some polytopes admit an exponential number of distinct cuttings that lead to simple polygon unfoldings, with lower bounds of 2^Ω(n).
• Among the Platonic solids, only the tetrahedron has a convex unfolding, due to necessary conditions on sharp vertices and combinatorial structure.
• For n > 6, there is essentially only one class of polytopes that can unfold to a regular n-gon, with only two such classes existing for regular polygons.
• Pita polytopes—formed by folding regular polygons with symmetric gluing—have a uniquely determined triangulated mouth structure, with edges and diagonals constrained by shortest path and symmetry arguments.