part iii: polyhedra b: unfolding joseph orourke smith college
TRANSCRIPT
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)?