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Part III: Polyhedra b: Unfolding

Part III: Polyhedra b: Unfolding. Joseph O’Rourke Smith College. Outline: Edge-Unfolding Polyhedra. History (Dürer) ; Open Problem; Applications Evidence For Evidence Against. Unfolding Polyhedra. Cut along the surface of a polyhedron. Unfold into a simple planar polygon without overlap.

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Part III: Polyhedra b: Unfolding

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  1. Part III: Polyhedrab: Unfolding Joseph O’Rourke Smith College

  2. Outline: Edge-Unfolding Polyhedra • History (Dürer) ; Open Problem; Applications • Evidence For • Evidence Against

  3. Unfolding Polyhedra • Cut along the surface of a polyhedron • Unfold into a simple planar polygon without overlap

  4. Edge Unfoldings • Two types of unfoldings: • Edge unfoldings: Cut only along edges • General unfoldings: Cut through faces too

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

  6. Albrecht Dürer, 1425

  7. Melancholia I

  8. Albrecht Dürer, 1425 Snub Cube

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

  10. Open Problem Can every simple polygon be folded (via perimeter self-gluing) to a simple, closed polyhedron? How about via perimeter-halving?

  11. Example

  12. Symposium on Computational Geometry Cover Image

  13. 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. • spanning: to flatten every vertex • forest: cycle would isolate a surface piece • tree: connected by boundary of polygon

  14. Unfolding the Platonic Solids Some nets: http://www.cs.washington.edu/homes/dougz/polyhedra/

  15. Unfolding the Archimedean Solids

  16. Archimedian Solids

  17. Nets for Archimedian Solids

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

  19. Successful Software • Nishizeki • Hypergami  • Javaview Unfold • ... http://www.cs.colorado.edu/~ctg/projects/hypergami/ http://www.fucg.org/PartIII/JavaView/unfold/Archimedean.html

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

  21. Overlapping Unfolding

  22. Volcano Unfolding

  23. Unfolding “Domes”

  24. Cube with one corner truncated

  25. “Sliver” Tetrahedron

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

  27. Sclickenrieder1:steepest-edge-unfold “Nets of Polyhedra” TU Berlin, 1997

  28. Sclickenrieder2:flat-spanning-tree-unfold

  29. Sclickenrieder3:rightmost-ascending-edge-unfold

  30. Sclickenrieder4:normal-order-unfold

  31. Open: Edge-Unfolding Convex Polyhedra (revisited) Does every convex polyhedron have an edge-unfolding to a net (a simple, nonoverlapping polygon)?

  32. 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 [for large F] • ≤ (1/2)F [for large F] • ≤ F < ½?; o(F)?

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