# Part III: Polyhedra b: Unfolding

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Part III: Polyhedra b: Unfolding
Joseph O’Rourke Smith College

Outline: Edge-Unfolding Polyhedra
History (Dürer) ; Open Problem; Applications Evidence For Evidence Against

Unfolding Polyhedra Cut along the surface of a polyhedron
Unfold into a simple planar polygon without overlap

Edge Unfoldings Two types of unfoldings:
Edge unfoldings: Cut only along edges General unfoldings: Cut through faces too

Commercial Software Lundström Design,

Albrecht Dürer, 1425

Melancholia I

Albrecht Dürer, 1425 Snub Cube

Open: Edge-Unfolding Convex Polyhedra
Does every convex polyhedron have an edge-unfolding to a simple, nonoverlapping polygon? [Shephard, 1975]

Open Problem Can every simple polygon be folded (via perimeter self-gluing) to a simple, closed polyhedron? How about via perimeter-halving?

Example

Symposium on Computational Geometry Cover Image

Cut Edges form Spanning Tree
Lemma: The cut edges of an edge unfolding of a convex polyhedron to a simple polygon form a spanning tree of the 1-skeleton of the polyhedron. spanning: to flatten every vertex forest: cycle would isolate a surface piece tree: connected by boundary of polygon

Unfolding the Platonic Solids
Some nets:

Unfolding the Archimedean Solids

Archimedian Solids

Nets for Archimedian Solids

Unfolding the Globe: Fuller’s map http://www. grunch

Successful Software Nishizeki Hypergami  Javaview Unfold ...

Prismoids Convex top A and bottom B, equiangular.

Overlapping Unfolding

Volcano Unfolding

Unfolding “Domes”

Cube with one corner truncated

“Sliver” Tetrahedron

Percent Random Unfoldings that Overlap
[O’Rourke, Schevon 1987]

Sclickenrieder1: steepest-edge-unfold
“Nets of Polyhedra” TU Berlin, 1997

Sclickenrieder2: flat-spanning-tree-unfold

Sclickenrieder3: rightmost-ascending-edge-unfold

Sclickenrieder4: normal-order-unfold

Open: Edge-Unfolding Convex Polyhedra (revisited)
Does every convex polyhedron have an edge-unfolding to a net (a simple, nonoverlapping polygon)?

Open: Fewest Nets For a convex polyhedron of n vertices and F faces, what is the fewest number of nets (simple, nonoverlapping polygons) into which it may be cut along edges? ≤ F ≤ (2/3)F [for large F] ≤ (1/2)F [for large F] ≤ F < ½?; o(F)?

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