Part III: PolyhedraPart III: Polyhedrab: Unfoldingb: Unfolding

Joseph O’RourkeJoseph O’RourkeSmith CollegeSmith College

Outline: Edge-Unfolding Outline: Edge-Unfolding PolyhedraPolyhedra

History (Dürer) ; Open Problem; Applications

Evidence ForEvidence Against

Unfolding PolyhedraUnfolding Polyhedra

Cut along the surface of a polyhedron

Unfold into a simple planar polygon without overlap

Edge UnfoldingsEdge Unfoldings

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

Commercial SoftwareCommercial Software

Lundström Design, http://www.algonet.se/~ludesign/index.html

Albrecht DAlbrecht Düürer, 1425rer, 1425

Melancholia I

Albrecht DAlbrecht Düürer, 1425rer, 1425

Snub Cube

Open: Open: Edge-Unfolding Convex Edge-Unfolding Convex PolyhedraPolyhedra

Does every convex polyhedron have an edge-unfolding to a simple, nonoverlapping polygon?

[Shephard, 1975]

Open ProblemOpen Problem

Can every simple polygon be folded (via perimeter self-gluing) to a simple, closed polyhedron?

How about via perimeter-halving?

ExampleExample

Symposium on Computational Symposium on Computational Geometry Cover ImageGeometry Cover Image

Cut Edges form Spanning Cut Edges form Spanning TreeTree

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.

o spanning: to flatten every vertexo forest: cycle would isolate a surface pieceo tree: connected by boundary of polygon

Unfolding the Platonic SolidsUnfolding the Platonic Solids

Some nets:http://www.cs.washington.edu/homes/dougz/polyhedra/

Unfolding the Archimedean Unfolding the Archimedean SolidsSolids

Archimedian SolidsArchimedian Solids

Nets for Archimedian SolidsNets for Archimedian Solids

Unfolding the Globe: Unfolding the Globe: Fuller’s maphttp://www.grunch.net/synergetics/map/dymax.html

Successful SoftwareSuccessful Software

NishizekiHypergami Javaview Unfold...

http://www.fucg.org/PartIII/JavaView/unfold/Archimedean.html

http://www.cs.colorado.edu/~ctg/projects/hypergami/

PrismoidsPrismoids

Convex top A and bottom B, equiangular.Edges parallel; lateral faces quadrilaterals.

Overlapping UnfoldingOverlapping Unfolding

Volcano UnfoldingVolcano Unfolding

Unfolding “Domes”Unfolding “Domes”

Cube with one corner Cube with one corner truncatedtruncated

““Sliver” TetrahedronSliver” Tetrahedron

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

SclickenriederSclickenrieder11::steepest-edge-unfoldsteepest-edge-unfold

“Nets of Polyhedra”TU Berlin, 1997

SclickenriederSclickenrieder22::flat-spanning-tree-unfoldflat-spanning-tree-unfold

SclickenriederSclickenrieder33::rightmost-ascending-edge-rightmost-ascending-edge-unfoldunfold

SclickenriederSclickenrieder44::normal-order-unfoldnormal-order-unfold

Open: Edge-Unfolding Convex Open: Edge-Unfolding Convex Polyhedra (revisited)Polyhedra (revisited)

Does every convex polyhedron have an edge-unfolding to a net (a simple, nonoverlapping polygon)?

Open: Fewest NetsOpen: 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 ≤ F ≤ (2/3)F [for large F] ≤ (1/2)F [for large F]

< ½?; o(F)?