1 / 33

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.

lyn
Download Presentation

Part III: Polyhedra b: Unfolding

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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)?

More Related