part iii: polyhedra b: unfolding joseph orourke smith college
TRANSCRIPT
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Part III: PolyhedraPart III: Polyhedrab: Unfoldingb: Unfolding
Joseph O’RourkeJoseph O’RourkeSmith CollegeSmith College
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Outline: Edge-Unfolding Outline: Edge-Unfolding PolyhedraPolyhedra
History (Dürer) ; Open Problem; Applications
Evidence ForEvidence Against
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Unfolding PolyhedraUnfolding Polyhedra
Cut along the surface of a polyhedron
Unfold into a simple planar polygon without overlap
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Edge UnfoldingsEdge Unfoldings
Two types of unfoldings: Edge unfoldings: Cut only along edges General unfoldings: Cut through faces too
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Commercial SoftwareCommercial Software
Lundström Design, http://www.algonet.se/~ludesign/index.html
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Albrecht DAlbrecht Düürer, 1425rer, 1425
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Melancholia I
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Albrecht DAlbrecht Düürer, 1425rer, 1425
Snub Cube
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Open: Open: Edge-Unfolding Convex Edge-Unfolding Convex PolyhedraPolyhedra
Does every convex polyhedron have an edge-unfolding to a simple, nonoverlapping polygon?
[Shephard, 1975]
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Open ProblemOpen Problem
Can every simple polygon be folded (via perimeter self-gluing) to a simple, closed polyhedron?
How about via perimeter-halving?
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ExampleExample
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Symposium on Computational Symposium on Computational Geometry Cover ImageGeometry Cover Image
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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
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Unfolding the Platonic SolidsUnfolding the Platonic Solids
Some nets:http://www.cs.washington.edu/homes/dougz/polyhedra/
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Unfolding the Archimedean Unfolding the Archimedean SolidsSolids
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Archimedian SolidsArchimedian Solids
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Nets for Archimedian SolidsNets for Archimedian Solids
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Unfolding the Globe: Unfolding the Globe: Fuller’s maphttp://www.grunch.net/synergetics/map/dymax.html
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Successful SoftwareSuccessful Software
NishizekiHypergami Javaview Unfold...
http://www.fucg.org/PartIII/JavaView/unfold/Archimedean.html
http://www.cs.colorado.edu/~ctg/projects/hypergami/
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PrismoidsPrismoids
Convex top A and bottom B, equiangular.Edges parallel; lateral faces quadrilaterals.
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Overlapping UnfoldingOverlapping Unfolding
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Volcano UnfoldingVolcano Unfolding
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Unfolding “Domes”Unfolding “Domes”
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Cube with one corner Cube with one corner truncatedtruncated
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““Sliver” TetrahedronSliver” Tetrahedron
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Percent Random Unfoldings Percent Random Unfoldings that Overlapthat Overlap [O’Rourke, Schevon 1987]
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SclickenriederSclickenrieder11::steepest-edge-unfoldsteepest-edge-unfold
“Nets of Polyhedra”TU Berlin, 1997
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SclickenriederSclickenrieder22::flat-spanning-tree-unfoldflat-spanning-tree-unfold
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SclickenriederSclickenrieder33::rightmost-ascending-edge-rightmost-ascending-edge-unfoldunfold
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SclickenriederSclickenrieder44::normal-order-unfoldnormal-order-unfold
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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)?
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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)?