**Reference**:- Erik Demaine, Martin Demaine, Anna Lubiw, and Joseph O'Rourke, “Examples, Counterexamples, and Enumeration Results for Foldings and Unfoldings between Polygons and Polytopes”, Technical Report 069, Smith College, July 2000.
**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 polyon, 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, show that, for

*n*> 6, there is essentially only one class of such polytopes. **Comments**:- This paper is also available as arXiv:cs.CG/0007019 of the Computing Research Repository (CoRR).
**Length**:- The paper is 54 pages.
**Availability**:- The paper is available in PostScript (1520k) and gzipped PostScript (384k).
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**Related papers**:- Aleks_GC2002 (Enumerating Foldings and Unfoldings between Polygons and Polytopes)
- JCDCG2000c (Enumerating Foldings and Unfoldings between Polygons and Polytopes)
**Related webpages**:- Folding Polygons into Convex Polyhedra

See also other papers by Erik Demaine.

Last updated January 4, 2022 by Erik Demaine.