Download presentation

Presentation is loading. Please wait.

1
**Part III: Polyhedra b: 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,

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:

16
**Unfolding the Archimedean Solids**

17
Archimedian Solids

18
**Nets for Archimedian Solids**

19
**Unfolding the Globe: Fuller’s map http://www. grunch**

20
**Successful Software Nishizeki Hypergami Javaview Unfold ...**

21
**Prismoids Convex top A and bottom B, equiangular.**

Edges parallel; lateral faces quadrilaterals.

22
**Overlapping Unfolding**

23
Volcano Unfolding

24
Unfolding “Domes”

25
**Cube with one corner truncated**

26
“Sliver” Tetrahedron

27
**Percent Random Unfoldings that Overlap**

[O’Rourke, Schevon 1987]

28
**Sclickenrieder1: steepest-edge-unfold**

“Nets of Polyhedra” TU Berlin, 1997

29
**Sclickenrieder2: flat-spanning-tree-unfold**

30
**Sclickenrieder3: rightmost-ascending-edge-unfold**

31
**Sclickenrieder4: normal-order-unfold**

32
**Open: Edge-Unfolding Convex Polyhedra (revisited)**

Does every convex polyhedron have an edge-unfolding to a net (a simple, nonoverlapping polygon)?

33
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)?

Similar presentations

OK

Solid Figures Solids are 3 dimensional or 3D. Solids with flat surfaces that are polygons are called POLYHEDRONS. There are two types of Polyhedrons.

Solid Figures Solids are 3 dimensional or 3D. Solids with flat surfaces that are polygons are called POLYHEDRONS. There are two types of Polyhedrons.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on series and parallel combination of resistors in series Ppt on earth hour san francisco Ppt on asymptotic notation of algorithms meaning Ppt on road accidents Moving message display ppt on tv Ppt on famous indian entrepreneurs in america Ppt on tata trucks commercial vehicle Ppt on sound navigation and ranging system mechanic professional Ppt on nitrogen cycle and nitrogen fixation example Ppt on history of badminton in china