Edge Unfoldings of Platonic Solids Never Overlap

Transcription

1 Edge Unfoldings of Platonic Solids Never Overlap Takashi Horiyama (Saitama Univ.) joint work with Wataru Shoji 1

2 Unfolding Simple polygon unfolded by cutting along the surface of a polyhedron Two kinds of unfolding Edge unfolding: cut only along the edges General unfolding: also allowed to cut through the faces Dürer Unterweysung der Messung 1525 Question (English translation Painter s manual ) Is every unfolding of a convex polyhedron overlap-free? 2

3 Question Is every unfolding of a convex polyhedron overlap-free? Self-overlapping unfoldings of convex polyhedra Namiki, Fukuda, 1993 Edge unfolding Mitani, Uehara, 2008 General unfolding for an orthogonal box 3

4 Question Is every unfolding of a convex polyhedron overlap-free? Edge unfoldings of an unconvex polyhedron Polyhedra: every edge unfolding has overlap [ Biedl et al, 1998 ] General unfoldings of a convex polyhedron At least one overlap-free unfolding [ Mount, 1985 ][ Sharir, Schorr, 1986 ] [ Aronov, O Rourke, 1992 ] 4

5 Our Question Question Is every edge unfolding of every Platonic solid overlap-free? (Regular polyhedra) Archimedean solids: (Semi-regular polyhedra) Self-overlapping edge unfolding [ Croft et al, 1995 ] Snub dodecahedron 5

6 Our Question Question Is every edge unfolding of every Platonic solid overlap-free? (Regular polyhedra) Archimedean solids: (Semi-regular polyhedra) Self-overlapping edge unfolding [Our result 1] [ Croft et al, 1995 ] 6

7 Our Question Question Is every edge unfolding of every Platonic solid overlap-free? (Regular polyhedra) Archimedean solids: (Semi-regular polyhedra) Self-overlapping edge unfolding Truncated icosahedron 7

8 Our Question Question Is every edge unfolding of every Platonic solid overlap-free? (Regular polyhedra) Archimedean solids: (Semi-regular polyhedra) Self-overlapping edge unfolding Truncated dodecahedron 8

9 Our Question Question Is every edge unfolding of every Platonic solid overlap-free? General unfoldings Tetrahedron: Never overlap [ Akiyama, 2007 ] Other platonic solids [Our result 2]: 9

10 Our Question Question Is every edge unfolding of every Platonic solid overlap-free? Tetrahedron Cube Octahedron Dodecahedron Icosahedron unfoldings ,380 43,380 [ Bouzette, Vandamme ] [ Hippenmeyer 1979 ] 10

11 Theorem No edge unfolding of a Platonic solid has self-overlap Strategy of proof: Enumerate all edge unfoldings of Platonic solids Construct a ZDD that represents all edge unfoldings Eliminate mutually isomorphic unfoldings Check whether each of the unfoldings overlaps Circumscribed circles of the faces overlap or not (except neighboring pairs of faces) There are no overlapping pairs 11

12 0-suppresed Binary Decision Diagram (ZDD) [Minato 1993] Directed acyclic graph representing a family of sets Ex ) { { x 1 x 2 x 4 x 5 }, { x 1 x 3 x 6 }, { x 2 x 3 x 6 } } A 1-path corresponds to a set Every node

13 0-suppresed Binary Decision Diagram (ZDD) [Minato 1993] Directed acyclic graph representing a family of sets Ex ) { { x 1 x 2 x 4 x 5 }, { x 1 x 3 x 6 }, { x 2 x 3 x 6 } } 1 { { x 2 x 3 x 6 } } { { x 2 x 4 x 5 }, { x 3 x 6 } } A 1-path corresponds to a set Every node reporesents a family of sets 13

14 0-suppresed Binary Decision Diagram (ZDD) [Minato 1993] Directed acyclic graph representing a family of sets Ex ) { { x 1 x 2 x 4 x 5 }, { x 1 x 3 x 6 }, { x 2 x 3 x 6 } } Good properties Unique canonical form when the variable order is fixed We share { x 3, x 6 } Compact representation Efficient algorithms for set algebra Applications: CAD of logic circuits Machine learning Data mining 14

15 Iff Condition for Edge Unfoldings Edge unfolding Cut-edges...?? 15

16 Iff Condition for Edge Unfoldings Edge unfolding Cut-edges form a spanning tree [ Folklore ] (1) Every vertex has at least one cut-edge (2) Cut-edges do not contain a cycle (3) Cut-edges are connected 16

17 Variables for Edges Ex) { x 2, x 3, x 4, x 7, x 10, x 11, x 12 } gives a spanning tree x 4 x 1 x 8 x x 3 7 x 2 x 5 x 6 x 9 x 12 x 10 x 11 17

18 Construction of a ZDD of Spanning Trees Top-down construction DP-like approach (Dynamic Programming) 18

19 Construction of a ZDD of Spanning Trees start delete e 1 contract e 1 no e 1 adopt e 1 go to 0 share 19

20 Theorem No edge unfolding of a Platonic solid has self-overlap Strategy of proof: Enumerate all edge unfoldings of Platonic solids Construct a ZDD that represents all edge unfoldings Eliminate mutually isomorphic unfoldings Check whether each of the unfoldings overlaps Circumscribed circles of the faces overlap or not (except neighboring pairs of faces) There are no overlapping pairs 20

21 Result Tetrahedron Cube (2 unfoldings) (11 unfoldings) Octahedron (11 unfoldings) 21

22 Result (Partial list) Dodecahedron 43,380 unfoldings 22

23 Result (Partial list) Icosahedron 43,380 unfoldings 23

24 Conclusion Theorem No edge unfolding of a Platonic solid has self-overlap Enumerate all edge unfoldings of Platonic solids Construct a ZDD that represents all edge unfoldings Eliminate mutually isomorphic unfoldings Check whether each of the unfoldings overlaps or not Circumscribed circles overlap or not (expect neighboring pair of faces) Future Work Archimedean solids? 24

25 [Our result 3] Archimedian Solids [AKL+ 11] #(Edge unfoldings) #(Essentially different edge unfoldings) 331,776 6, ,971,104,256,000 1,741,425,868,800 [AKL+ 11] [BMP+ 91] [BMPR 96] 6, ,154,816 [PEK+ 11] estimated 2.3M 2,108,512 32,400, , ,291,866,372,898,816,000 3,127,432,220,939,473,920 4,982,259,375,000,000,000 41,518,828,261,687, ,056,000,000 6,272,012, ,550,864,919,150,779,950,956,544,000 1,679,590,540,992,923,166,257,971,200 12,418,325,780,889, ,715,122,137,472 21,789,262,703,685,125,511,464,767,107,171,876,864, ,577,189,197,376,045,928,994,520,239,942,164,480 [BMPR 96] 89,904,012,853,248 3,746,001,752, ,201,295,386,966,498,858,139,607,040,000,000 7,303,354,923,116,108,380,042,995,304,896,000