Zipper Unfoldings of Polyhedral Complexes. Erik Demaine Martin Demaine Anna Lubiw Arlo Shallit Jonah Shallit

Transcription

1 Zipper Unfoldings of Polyhedral Complexes Erik Demaine Martin Demaine Anna Lubiw Arlo Shallit Jonah Shallit 1

2 Unfolding Polyhedra Durer 1400 s Durer, 1498 snub cube 2

3 Unfolding Polyhedra Octahedron all unfoldings 3

4 4

5 Zipper Unfoldings of Polyhedra Octahedron 5

6 Zippers separating zipper multiple toggles (cosmetic) 6

7 Zippers 1891 patent by Whitcomb Judson novel, but not practical ( If skirt is to be washed, remove fastener. ) named zipper by B.F. Goodrich company in 1920 s ubiquitous but superfluous 7

8 Edge Cuts versus Face Cuts edge cuts face cuts Zipper edge cuts = Hamiltonian unfolding [Shepherd 75] 8

9 Zipper Edge Cuts (Hamiltonian Unfolding) What is a zipper unfolding of a polyhedron? on the polyhedron the cut is a simple path on the polygon this is forbidden Nick Chase 9

10 Outline of Talk Convex Polyhedra Platonic Solids Archimedean Solids Polyhedral Manifolds Polyhedral Complexes 10

11 Platonic Solids 12

12 Platonic Solids These are doubly Hamiltonian the cut is a path and faces are joined in a path. 13

13 Archimedean Solids great rhombicosidodecahedron truncated tetrahedron truncated dodecahedron truncated cube truncated icosahedron truncated octahedron great rhombicuboctahedron small rhombicosidodecahedron snub cube small rhombicuboctahedron cuboctahedron icosidodecahedron snub dodecahedron 14

14 Archimedean Solids great rhombicosidodecahedron truncated tetrahedron truncated dodecahedron truncated cube truncated icosahedron truncated octahedron great rhombicuboctahedron small rhombicosidodecahedron snub cube small rhombicuboctahedron cuboctahedron icosidodecahedron snub dodecahedron 15

15 Archimedean Solids great rhombicosidodecahedron 16

16 Archimedean Solids great rhombicosidodecahedron truncated tetrahedron truncated dodecahedron truncated cube truncated icosahedron truncated octahedron great rhombicuboctahedron small rhombicosidodecahedron snub cube small rhombicuboctahedron cuboctahedron icosidodecahedron snub dodecahedron 18

17 What next? these have zipper unfoldings?? Peda Software Polyhedron Poster 19

18 Not all convex polyhedra have Hamiltonian unfoldings rhombic dodecahedron net its graph has no Hamiltonian path OPEN: find a convex polyhedron with no Hamiltonian unfolding but whose graph has a Hamiltonian path. 20

19 Unfolding Convex Polyhedra unfolding zipper unfolding face cuts YES Every convex polyhedron has an unfolding star, source unfolding. OPEN Does every convex polyhedron have a zipper unfolding? edge cuts OPEN Does every convex polyhedron have an edge unfolding? NO Not every convex polyhedron has a Hamiltonian unfolding. 21

20 Polyhedral Manifolds polyhedral manifold a finite union of planar polygons in 3D s.t. every point has a neighbourhood homeomorphic to a disk may be non-convex may have genus 0 Magnus Wenninger 22

21 ers of the tetrahedron. So we have these four corners s inside the hats. Viewing each of these four paths, we can see that it is impossible to avoid a cycle, 1 arcs. This violates Lemma , and establishes e unfolding to a net. hedron can be unfolded without restricting the cuts tegy (for a differently parametrized version of the kes safely out to the boundary of the tetrahedron Polyhedral Manifolds Unfolding nnot be unfolded without overlap: a nice non-convex polyhedron with no edge unfolding some higher genus polyhedral manifolds with edge unfoldings Donoso & O Rourke an ununfoldable triangulated polyhedron. Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink, Császár torus (net by Lutz, picture by Polthier) OPEN: Does every polyhedral manifold have a [general] unfolding? x C 23

22 Polyhedral Manifolds Zipper Unfoldings use separating zipper 24

23 Polyhedral Manifolds Zipper Unfoldings Theorem. If P is a polyhedral manifold that has a zipper unfolding to a planar polygonal region F, then either P is a polyhedron and F is a polygon, or in the case of a separating zipper P is a torus polyhedron and F is an annulus (with outer perimeter = inner perimeter). not possible Nick Chase 25

24 Polyhedral Manifolds Zipper Unfoldings These have no zipper edge unfoldings. Lemma. A zipper edge unfolding of a torus polyhedron is an annulus with faces forming a cycle with trees attached. 26

25 Polyhedral Complexes polyhedral complex a finite union of planar polygons in 3D s.t. intersections are edge-to-edge joins only one face at an edge 3 faces meeting at an edge Weaire Phelan structure 27

26 Polyhedral Complexes Unfolding cannot be unfolded v Bern, Demaine, Eppstein, Kuo, Mantler, Snoeyink, Liu & Tai, in Computer-Aided Design,

27 Polyhedral Complexes Unfolding What does unfolding mean when multiple faces meet at an edge? valid (c) Valid spanning tree. invalid (b) Invalid spanning tree. Liu & Tai, in Computer-Aided Design,

28 Polyhedral Complexes Zipper Unfolding not allowed on the polygon 30

29 Polyhedral Complexes Zipper Unfolding 31

30 Polyhedral Complexes Zipper Unfolding Polyhedra Sharing Faces 2 tetrahedra what will this zip into? 32

31 Polyhedral Complexes Zipper Unfolding Polyhedra Sharing Faces 2 cubes seems impossible 2 squashed cubes 33

32 Polyhedral Complexes Zipper Unfolding Polyhedra Sharing Edges chain of tetrahedra sharing adjacent edges can extend to n tetrahedra 34

33 Polyhedral Complexes Zipper Unfolding 35

34 Open Problems Does every convex polyhedron have an edge unfolding? Does every polyhedron have a [general] unfolding? Every polyhedral manifold? New: Does every convex polyhedron have a zipper unfolding? Show it s NP-hard to recognize torus polyhedra with zipper edge unfoldings. Which polyhedral complexes have [zipper/edge] unfoldings? 36

35 Zipper Unfoldings of Polyhedral Complexes Erik Demaine Martin Demaine Anna Lubiw Arlo Shallit Jonah Shallit 37