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Unfolding Polyhedral Surfaces Joseph ORourke Smith College

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Outline What is an unfolding? What is an unfolding? Why study? Why study? Main questions: status Main questions: status Convex: edge unfolding Convex: edge unfolding Convex: general unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Nonconvex: general unfolding Orthogonal polyhedra Orthogonal polyhedra

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ten1.mov

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What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

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What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

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

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What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

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Cube with one corner truncated

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Sliver Tetrahedron

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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.

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Polygons: Simple vs. Weakly Simple

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Nonsimple Polygons

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Andrea Mantler example

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Cut edges: strengthening 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, ORourke, Snoeyink 01]

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What is an unfolding? Cut surface and unfold to a single nonoverlapping piece in the plane.

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Cuboctahedron unfolding [Matthew Chadwick]

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[Biedl, Lubiw, Sun, 2005]

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Outline What is an unfolding? What is an unfolding? Why study? Why study? Main questions: status Main questions: status Convex: edge unfolding Convex: edge unfolding Convex: general unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Nonconvex: general unfolding Orthogonal polyhedra Orthogonal polyhedra

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Lundström Design,

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Outline What is an unfolding? What is an unfolding? Why study? Why study? Main questions: status Main questions: status Convex: edge unfolding Convex: edge unfolding Convex: general unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Nonconvex: general unfolding Orthogonal polyhedra Orthogonal polyhedra

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Status of main questions Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible ???

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Status of main questions Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible ???

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

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Albrecht Dürer, 1425 Melancholia I

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Albrecht Dürer, 1425 Snub Cube

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Unfolding the Platonic Solids Some nets:

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Archimedian Solids [Eric Weisstein]

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Sclickenrieder 1 : steepest-edge-unfold Nets of Polyhedra TU Berlin, 1997

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Sclickenrieder 2 : flat-spanning-tree-unfold

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Sclickenrieder 3 : rightmost-ascending-edge-unfold

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Sclickenrieder 4 : normal-order-unfold

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Percent Random Unfoldings that Overlap [ORourke, Schevon 1987]

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Classes of Convex Polyhedra with Edge-Unfolding Algorithms Prisms Prisms Prismoids Prismoids Domes Domes Bands Bands Radially monotone lattice quadrilaterals Radially monotone lattice quadrilaterals Prismatoids??? Prismatoids???

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

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Overlapping Unfolding

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Volcano Unfolding

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Unfolding Domes

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Proof via degree-3 leaf truncation [Benton, JOR, 2007] dodec.wmv

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Lattice Quadrilateral Convex Caps

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Classes of Convex Polyhedra with Edge-Unfolding Algorithms Prisms Prisms Prismoids Prismoids Domes Domes Bands Bands Radially monotone lattice quadrilaterals Radially monotone lattice quadrilaterals Prismatoids??? Prismatoids???

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Unfolding Smooth Prismatoids [Benbernou, Cahn, JOR 2004]

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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[Spriggs] (1/2)F [Dujmenovic, Moran, Wood] (3/8)F [Pincu, 2007]

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Outline What is an unfolding? What is an unfolding? Why study? Why study? Main questions: status Main questions: status Convex: edge unfolding Convex: edge unfolding Convex: general unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Nonconvex: general unfolding Orthogonal polyhedra Orthogonal polyhedra

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Status of main questions Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible ???

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Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). Theorem: Every convex polyhedron has a general nonoverlapping unfolding (a net). General Unfoldings of Convex Polyhedra 1)Source unfolding [Sharir & Schorr 86, Mitchell, Mount, Papadimitrou 87] 2)Star unfolding [Aronov & JOR 92] 3)Quasigeodesic unfolding [Itoh, JOR, Vilcu, 2007] [Poincare?]

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Shortest paths from x to all vertices

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Source Unfolding

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Star Unfolding

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Star-unfolding of 30-vertex convex polyhedron

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[Alexandrov, 1950]

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Geodesics & Closed Geodesics Geodesic: locally shortest path; straightest lines on surface Geodesic: locally shortest path; straightest lines on surface Simple geodesic: non-self-intersecting Simple geodesic: non-self-intersecting Simple, closed geodesic: Simple, closed geodesic: Closed geodesic: returns to start w/o corner Closed geodesic: returns to start w/o corner Geodesic loop: returns to start at corner Geodesic loop: returns to start at corner (closed geodesic = simple, closed geodesic)

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Lyusternick-Schnirelmann Theorem Theorem: Every closed surface homeomorphic to a sphere has at least three, distinct closed geodesics. Birkoff 1927: at least one closed geodesic Birkoff 1927: at least one closed geodesic LS 1929: at least three LS 1929: at least three gaps filled in 1978 [BTZ83] gaps filled in 1978 [BTZ83] Pogorelov 1949: extended to polyhedral surfaces Pogorelov 1949: extended to polyhedral surfaces

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Quasigeodesic Aleksandrov 1948 Aleksandrov 1948 left(p) = total incident face angle from left left(p) = total incident face angle from left quasigeodesic: curve s.t. quasigeodesic: curve s.t. left(p) left(p) right(p) right(p) at each point p of curve. at each point p of curve.

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Closed Quasigeodesic [Lysyanskaya, ORourke 1996]

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Grayson Curve Shortening [Lysyanskaya, ORourke 1996]

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Open: Find a Closed Quasigeodesic Is there an algorithm polynomial time or efficient numerical algorithm for finding a closed quasigeodesic on a (convex) polyhedron?

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Exponential Number of Closed Geodesics Theorem: 2 (n) distinct closed quasigeodesics. [Aronov & JOR 2002]

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Status of main questions Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible ???

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Edge-Ununfoldable Orthogonal Polyhedra [Biedl, Demaine, Demaine, Lubiw, JOR, Overmars, Robbins, Whitesides, 98]

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Spiked Tetrahedron [Tarasov 99] [Grünbaum 01] [Bern, Demaine, Eppstein, Kuo 99]

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Unfoldability of Spiked Tetrahedron Theorem: Spiked tetrahedron is edge-ununfoldable Theorem: Spiked tetrahedron is edge-ununfoldable (BDEKMS 99)

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Overlapping Star-Unfolding

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Outline What is an unfolding? What is an unfolding? Why study? Why study? Main questions: status Main questions: status Convex: edge unfolding Convex: edge unfolding Convex: general unfolding Convex: general unfolding Nonconvex: edge unfolding Nonconvex: edge unfolding Nonconvex: general unfolding Nonconvex: general unfolding Orthogonal polyhedra Orthogonal polyhedra

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Status of main questions Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible ???

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Overlapping Source Unfolding [Kineva, JOR 2000]

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Status of main questions Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible ??? Orthogonalpolyhedra Yes: always possible

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Orthogonal Polygon / Polyhedron

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Grid refinement: Orthogonal Polyhedra [DIL04] [DM04]

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Types of Unfoldings

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Gridding Hierarchy Edge Unfolding 1 x 1 k1 x k2 O(1) x O(1) 2 O(n) x 2 O(n) polycubes/ lattice

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All genus-0 o-polyhedra o-terrains; o-convex o-stacks All genus-0 o- polyhedra o-stacks: 1 x 2 Manhattan towers: 4 x 5 o-stacks Not always possible Edge-UnfVertex-Unf

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Orthogonal Terrain

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Four algorithmic techniques 1) Strip/Staircase unfoldings 2) Recursive unfoldings 3) Spiraling unfoldings 4) Nested spirals

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Four algorithmic techniques 1) Strip/Staircase unfoldings 2) Recursive unfoldings 3) Spiraling unfoldings 4) Nested spirals [Damian, Flatland, JOR 05] 4x5 grid unfolding Manhattan towers

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All genus-0 o-polyhedra o-terrains o-stacks: 1 x 2 Manhattan towers: 4 x 5 o-stacks Not always possible Edge-UnfVertex-Unf All genus-0 o-polyhedra

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Manhattan Tower

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Base & Recursion Tree

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Single Box Unfolding

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Suturing two spirals

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Suture Animation Animation by Robin Flatland & Ray Navarette, Siena College rdsuture_4x.wmv

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Four algorithmic techniques 1) Strip/Staircase unfoldings 2) Recursive unfoldings 3) Spiraling unfoldings 4) Nested spirals [Damian, Flatland, JOR 06b] -unfolding orthogonal polyhedra

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All genus-0 o-polyhedra o-terrains o-stacks: 1 x 2 Manhattan towers: 4 x 5 o-stacks Not always possible Edge-UnfVertex-Unf All genus-0 o-polyhedra

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Dents vs. Protrusions

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Band unfolding tree (for extrusion)

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Visiting front children

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Visiting back children

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Retrace entire path back to entrance point

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Deeper recursion

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All strips thickened to fill surface

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4-block example b 0 b 1 b 2 b 3 b 4

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Result Arbitrary genus-0 Orthogonal Polyhedra have a general unfolding into one piece, Arbitrary genus-0 Orthogonal Polyhedra have a general unfolding into one piece, which may be viewed as a 2 n x 2 n grid unfolding which may be viewed as a 2 n x 2 n grid unfolding (and so is, in places, 1/2 n thin). (and so is, in places, 1/2 n thin). Arbitrary (non-orthogonal) polyhedra: still open. Arbitrary (non-orthogonal) polyhedra: still open.

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Status of main questions Shapes Edge Unfolding? General Unfolding? Convex polyhedra ??? Yes: always possible Nonconvex polyhedra No: not always possible ???

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