FoldingFolding && Unfolding:Unfolding: UnfoldingUnfolding PolyhedraPolyhedra JosephJoseph O’RourkeO’Rourke SmithSmith CollegeCollege FoldingFolding andand UnfoldingUnfolding TalksTalks Linkage folding Tuesday Erik Demaine Paper folding Wednesday Erik Demaine Folding polygons into convex Saturday1 Joe O’Rourke polyhedra Unfolding Saturday Joe O’Rourke polyhedra 2 OutlineOutline Edge-Unfolding Polyhedra Geodesics & Closed Geodesics Unrestricted Unfoldings OutlineOutline11 Edge-Unfolding Polyhedra History (Dürer) ; Open Problem; Applications Evidence For Evidence Against Ununfoldable Polyhedra OutlineOutline22 Geodesics & Closed Geodesics Lyusternick-Schnirelmann Theorem Gage-Hamilton-Grayson Curve Shortening Exponential Number of Closed Geodesics OutlineOutline33 Unrestricted Unfoldings Vertex Unfolding Orthogonal Polyhedra Open: Nonoverlapping Unfolding for Nonconvex Polyhedra UnfoldingUnfolding PolyhedraPolyhedra Cut along the surface of a polyhedron Unfold into a simple planar polygon without overlap EdgeEdge UnfoldingsUnfoldings Two types of unfoldings: Edge unfoldings: Cut only along edges General unfoldings: Cut through faces too CommercialCommercial SoftwareSoftware Lundström Design, http://www.algonet.se/~ludesign/index.html AlbrechtAlbrecht DDüürerrer,, 14251425 Melancholia I AlbrechtAlbrecht DDüürerrer,, 14251425 Snub Cube Open:Open: Edge-UnfoldingEdge-Unfolding ConvexConvex PolyhedraPolyhedra Does every convex polyhedron have an edge- unfolding to a simple, nonoverlapping polygon? [Shephard, 1975] CutCut EdgesEdges formform SpanningSpanning 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 vertex o forest: cycle would isolate a surface piece o tree: connected by boundary of polygon CutCut EdgesEdges (revisited)(revisited) 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. NonsimpleNonsimple PolygonsPolygons (a) (b) (c) (d) AndreaAndrea MantlerMantler exampleexample (b) (d) (a) (c) (e) Javaview CutCut edges:edges: strengtheningstrengthening Lemma: The cut edges of an edge unfolding of a convex polyhedron to a single, connected piece form a spanning tree of the 1-skeleton of the polyhedron. [Bern, Demaine, Eppstein, Kuo, Mantler, O’Rourke, Snoeyink 01] OutlineOutline11 Edge-Unfolding Polyhedra History (Dürer) ; Open Problem; Applications Evidence For Evidence Against Ununfoldable Polyhedra ArchimedianArchimedian SolidsSolids NetsNets forfor ArchimedianArchimedian SolidsSolids SuccessfulSuccessful SoftwareSoftware Nishizeki Hypergami Javaview Unfold ... http://www.fucg.org/PartIII/JavaView/unfold/Archimedean.html PrismoidsPrismoids Convex top A and bottom B, equiangular. Edges parallel; lateral faces quadrilaterals. A B OverlappingOverlapping UnfoldingUnfolding A B B SplaySplay UnfoldingUnfolding (top(top view)view) ci+1 ai+1 b'i+1 ai A ci b'i SplaySplay UnfoldingUnfolding B A OutlineOutline11 Edge-Unfolding Polyhedra History (Dürer) ; Open Problem; Applications Evidence For Evidence Against Ununfoldable Polyhedra CubeCube withwith oneone cornercorner truncatedtruncated (a) (b) “Sliver”“Sliver” TetrahedronTetrahedron cd (a) ab b c cd (b) ab cd (c) ab c d PercentPercent RandomRandom UnfoldingsUnfoldings thatthat OverlapOverlap [O’Rourke, Schevon 1987] 100 80 60 40 20 0 Percent Overlapping Unfoldings Overlapping Percent 0 1020304050607080 Number of Vertices SclickenriederSclickenrieder1:: “Nets of Polyhedra” steepest-edge-unfoldsteepest-edge-unfold TU Berlin, 1997 SclickenriederSclickenrieder2:: flat-spanning-tree-unfoldflat-spanning-tree-unfold SclickenriederSclickenrieder3:: rightmost-ascending-edge-unfoldrightmost-ascending-edge-unfold SclickenriederSclickenrieder4:: normal-order-unfoldnormal-order-unfold Open:Open: Edge-UnfoldingEdge-Unfolding ConvexConvex PolyhedraPolyhedra (revisited)(revisited) Does every convex polyhedron have an edge- unfolding to a net (a simple, nonoverlapping polygon)? Open:Open: FewestFewest NetsNets 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? C≤ F CSimplicial polyhedra: ≤ F/2 CSimple polyhedra: ≤ (2/3)(F-2) OutlineOutline11 Edge-Unfolding Polyhedra History (Dürer) ; Open Problem; Applications Evidence For Evidence Against Ununfoldable Polyhedra Edge-Edge-UnunfoldableUnunfoldable OrthogonalOrthogonal PolyhedraPolyhedra Biedl, Demaine, Demaine, Lubiw, O’Rourke, Overmars, Robbins, Whitesides [CCCG98] TopologicallyTopologically ConvexConvex PolyhedraPolyhedra (Bern, Demaine, Eppstein, Kuo ’99) A polyhedron is topologically convex if its 1-skeleton is that of a convex polyhedron Steinitz’s theorem: iff 3-connected and planar Natural question: Can all topologically convex polyhedra be edge unfolded? Subclass: Convex-faced polyhedra (every face is convex) Schevon (1987): Are they all edge-unfoldable? TriangulatedTriangulated HatHat 9 triangles 6 base triangles, lying just above a plane 3 spike triangles, with base angle > 60° so that middle vertices have negative curvature tip Planar drawing corner corner corner SpikedSpiked TetrahedronTetrahedron Place a hat on each face of a regular tetrahedron SpikedSpiked TetrahedronTetrahedron JavaView UnfoldabilityUnfoldability ofof SpikedSpiked TetrahedronTetrahedron (BDEKMS ’99) Theorem: Spiked tetrahedron is edge-ununfoldable 2 4 2 4 1 3 1 3 OutlineOutline22 Geodesics & Closed Geodesics Lyusternick-Schnirelmann Theorem Gage-Hamilton-Grayson Curve Shortening Exponential Number of Closed Geodesics GeodesicsGeodesics && ClosedClosed GeodesicsGeodesics Geodesic: locally shortest path; straightest lines on surface Simple geodesic: non-self-intersecting Simple, closed geodesic: Closed geodesic: returns to start w/o corner Geodesic loop: returns to start at corner (closed geodesic = simple, closed geodesic) Lyusternick-SchnirelmannLyusternick-Schnirelmann TheoremTheorem Theorem: Every closed surface homeomorphic to a sphere has at least three, distinct closed geodesics. C Birkoff 1927: at least one closed geodesic C LS 1929: at least three C “gaps” filled in 1978 [BTZ83] C Pogorelov 1949: extended to polyhedral surfaces QuasigeodesicQuasigeodesic Aleksandrov 1948 Define left(p) and right(p) turn angle at point p on curve left(p) = – total incident face angle from left quasigeodesic: curve s.t. left(p) ≥ 0 right(p) ≥ 0 at each point p of curve. ClosedClosed QuasigeodesicQuasigeodesic Open:Open: FindFind aa ClosedClosed QuasigeodesicQuasigeodesic Is there an algorithm polynomial time or efficient numerical algorithm for finding a closed quasigeodesic on a (convex) polyhedron? ExponentialExponential NumberNumber ofof ClosedClosed GeodesicsGeodesics 10 9 1 4 3 7 6 2 11 7 8 0 7 8 8 (n) 5 8 5 6 2 6 3 0 11 7 Theorem: 2 9 5 9 5 1 9 4 6 A 4 10 A A 10 A 4 A 10 A 4 6 1 10 3 7 0 9 5 2 11 5 11 11 2 3 3 distinct closed 8 6 11 4 8 3 0 1 0 2 7 9 0 2 10 5 11 3 1 2 1 4 6 0 8 9 10 4 1 3 7 9 7 10 1 5 0 quasigeodesics. 2 6 8 11 B B B 8 2 6 B 11 0 1 7 9 5 3 10 7 10 6 0 4 1 4 8 11 5 9 3 2 1 2 10 7 6 7 6 11 5 4 7 6 3 9 8 5 8 5 0 6 8 3 5 8 0 4 9 4 9 4 7 11 9 2 4 A A 7 A A 1 A A 5 10 1 10 3 10 3 8 10 3 2 6 6 9 11 11 2 11 2 9 0 2 7 8 5 3 0 0 1 2 0 1 1 4 10 11 3 1 9 6 0 5 7 8 11 10 4 10 1 0 4 8 7 11 9 2 5 B 11 B 0 3 B 6 9 B 10 8 3 6 2 10 5 1 7 7 9 4 Aronov & O’Rourke 8 1 11 4 2 6 0 5 3 2002 GageGage && HamiltonHamilton CurveCurve ShorteningShortening Each point p evolves along normal to curve, at speed proportional to curvature at p. GraysonGrayson CurveCurve ShorteningShortening Lysyanskaya, O’Rourke 1996 FacesFaces CrossedCrossed byby ClosedClosed GeodesicGeodesic GeodesicGeodesic OverlapOverlap b c cd ab QuasigeodesicsQuasigeodesics C B D A (a) D A (b) D A (c) Open:Open: NonoverlappingNonoverlapping FacesFaces crossedcrossed byby ClosedClosed QuasigeodesicQuasigeodesic For a given closed quasigeodesic , is it true that the set of faces whose interior is touched by unfold along without overlap? OutlineOutline33 Unrestricted Unfoldings Vertex Unfolding Orthogonal Polyhedra Open: Nonoverlapping Unfolding for Nonconvex Polyhedra GeneralGeneral UnfoldingsUnfoldings ofof ConvexConvex PolyhedraPolyhedra Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). Source unfolding (Sharir & Schorr ’86, Mitchell, Mount, Papadimitrou ’87) Star unfolding (Aronov & O’Rourke ’92) Star-unfoldingStar-unfolding ofof 30-vertex30-vertex convexconvex polyhedronpolyhedron OverlappingOverlapping SourceSource UnfoldingUnfolding [Kineva, O’Rourke 2000] OverlappingOverlapping Star-UnfoldingStar-Unfolding a' .75 a 2.125 d' d 8.42o x 1.42 2.56 3 2 o 45.10 45.22o b b' 2 (a) 3 131.81o c o .75 48.19 angle incident to c: 2(45.22o+131.81o) = 354.06o c' angle incident to x: 2(8.42o) + 3(89.56o) + 3(89.80o) = 554.92o Trapezoid: 2(48.19o + 131.81o) = 360o a' d' a d xx x x x b c (b) b' c' x VertexVertex UnfoldingUnfolding ((EppsteinEppstein,, Erickson,Erickson, Hart,Hart, OO’’RourkeRourke 2002)2002) Cube: Vertex unfolding: AlgorithmAlgorithm OverviewOverview 2-Manifold → Facet-Path → Strip Layout of Triangles : Vertex-Unfolding VertexVertex UnfoldingUnfolding ((EppsteinEppstein,, Erickson,Erickson, Hart,Hart, OO’’RourkeRourke 2002)2002) Theorem: Every triangulated manifold has a vertex-unfolding. 16 36 56 Open:Open: Vertex-UnfoldingVertex-Unfolding Does every convex polyhedron have a nonoverlapping