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**11 - 14**of**14**### The Unfolding Problem

, 2005

"... The Unfolding Problem can be succinctly described as “How to peel an orange... no matter what shape the orange is. ” It is the question of how to ‘unwrap ’ a 3D polyhedron, breaking some of its edges or faces so that it can be unfolded into a flat net in the 2D plane. From there the flattened net of ..."

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The Unfolding Problem can be succinctly described as “How to peel an orange... no matter what shape the orange is. ” It is the question of how to ‘unwrap ’ a 3D polyhedron, breaking some of its edges or faces so that it can be unfolded into a flat net in the 2D plane. From there the flattened net of faces might be printed out, cut from paper or steel and folded to recreate the virtual model in the real world. In 1525 the artist Albrecht Dürer used the term ‘net ’ to describe a set of polygons linked together edge-to-edge to form the planar unfoldings of some of the platonic solids and their truncations. Dürer used these unfoldings to teach aspiring artists how to construct elemental forms, but today the applications for solutions to the unfolding problem lie in a broad range of fields, from industrial manufacturing and rapid prototyping to sculpture and aeronautics. In the textiles industry, work has already begun in computing digital representations of fabric and trying to flatten those representations to optimize seam and dart placement [MHC05]. There has been similar work in the fields of paper-folding [MS04] and origami [BM04] and even ship and sail manufacturing. Advances in robotics and folding automation [GBKK98] have brought with them a new need for faster, more robust unfolding methods. If a polyhedron can generate a net which is not self-intersecting, solely by breaking a subset of its edges and flattening the join angles of those which remain, then it is called edge-unfoldable or developable. At present, it is strongly believed–but not yet proven–that all convex surfaces are developable. In counterpoint, examples are easily found of non-convex surfaces which are cannot be edge-unfolded, but no robust solution yet exists for testing whether or not a given mesh will prove to be developable.

### between Polygons and Polytopes

, 2008

"... 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 fol ..."

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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

### Edge-Unfolding Almost-Flat Convex Polyhedral Terrains

, 2013

"... In this thesis we consider the centuries-old question of edge-unfolding convex polyhedra, focusing specifically on edge-unfoldability of convex polyhedral terrain which are “almost flat ” in that they have very small height. We demonstrate how to determine whether cut-trees of such almost-flat terra ..."

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In this thesis we consider the centuries-old question of edge-unfolding convex polyhedra, focusing specifically on edge-unfoldability of convex polyhedral terrain which are “almost flat ” in that they have very small height. We demonstrate how to determine whether cut-trees of such almost-flat terrains unfold and prove that, in this context, any partial cut-tree which unfolds without overlap and “opens ” at a root edge can be locally extended by a neighboring edge of this root edge. We show that, for certain (but not all) planar graphs G, there are cut-trees which unfold for all almost-flat terrains whose planar projection is G. We also demonstrate a non-cut-tree-based method of unfolding which relies on “slice ” operations to build an unfolding of a complicated terrain from a known unfolding of a simpler terrain. Finally, we describe several heuristics for generating cut-forests and provide some computational results of such heuristics on unfolding almost-flat convex polyhedral terrains.