Edge Unfoldings of Platonic Solids Never Overlap

Takashi Horiyama (Saitama Univ.)

1

joint work with Wataru Shoji

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 (English translation “Painter’s manual” )

2

Question Is every unfolding of a convex polyhedron overlap-free ?

∃Self-overlapping unfoldings of convex polyhedra

Namiki, Fukuda, 1993

Mitani, Uehara, 2008

3

Question Is every unfolding of a convex polyhedron overlap-free ?

Edge unfolding

General unfolding for an orthogonal box

Edge unfoldings of an unconvex polyhedron ∃Polyhedra: every edge unfolding has overlap

General unfoldings of a convex polyhedron ∃At least one overlap-free unfolding

4

[ Biedl et al, 1998 ]

[ Aronov, O’Rourke, 1992 ] [ Mount, 1985 ][ Sharir, Schorr, 1986 ]

Question Is every unfolding of a convex polyhedron overlap-free ?

Archimedean solids: (Semi-regular polyhedra) Self-overlapping edge unfolding

[ Croft et al, 1995 ]

Our Question

5

Question Is every edge unfolding of every Platonic solid overlap-free?

(Regular polyhedra)

Snub dodecahedron

Archimedean solids: (Semi-regular polyhedra) Self-overlapping edge unfolding

[ Croft et al, 1995 ]

Our Question

6

Question Is every edge unfolding of every Platonic solid overlap-free?

(Regular polyhedra)

[Our result 1]

Archimedean solids: (Semi-regular polyhedra) Self-overlapping edge unfolding

Our Question

7

Question Is every edge unfolding of every Platonic solid overlap-free?

(Regular polyhedra)

Truncated icosahedron

Archimedean solids: (Semi-regular polyhedra) Self-overlapping edge unfolding

Our Question

8

Question Is every edge unfolding of every Platonic solid overlap-free?

(Regular polyhedra)

Truncated dodecahedron

Our Question

9

General unfoldings Tetrahedron: Never overlap [ Akiyama, 2007 ] Other platonic solids [Our result 2]:

Question Is every edge unfolding of every Platonic solid overlap-free?

Question Our Question

10

Tetrahedron Cube Octahedron Dodecahedron Icosahedron

2 1 1 unfoldings 1 1 43,380 43,380 [ Bouzette, Vandamme ] [ Hippenmeyer 1979 ]

Is every edge unfolding of every Platonic solid overlap-free?

11

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

Theorem No edge unfolding of a Platonic solid has self-overlap

0-suppresed Binary Decision Diagram (ZDD)

Directed acyclic graph representing a family of sets

Ex ) { { x1 x2 x4 x5 }, { x1 x3 x6 }, { x2 x3 x6 } }

12

[Minato 1993]

1

5

3

6

1 0

2

4

2

A 1-path corresponds to a set

Every node ...

0-suppresed Binary Decision Diagram (ZDD)

Directed acyclic graph representing a family of sets

Ex ) { { x1 x2 x4 x5 }, { x1 x3 x6 }, { x2 x3 x6 } }

13

[Minato 1993]

1

5

3

6

1 0

2

4

2

A 1-path corresponds to a set

Every node reporesents a family of sets

{ { x2 x4 x5 }, { x3 x6 } } { { x2 x

3 x6 } }

0-suppresed Binary Decision Diagram (ZDD)

14

[Minato 1993]

Good properties

Applications:

Unique canonical form when the variable order is fixed

Compact representation Efficient algorithms for set algebra

CAD of logic circuits Machine learning Data mining

We share { x3 , x6 }

1

5

3

6

1 0

2

4

2

Directed acyclic graph representing a family of sets

Ex ) { { x1 x2 x4 x5 }, { x1 x3 x6 }, { x2 x3 x6 } }

Iff Condition for Edge Unfoldings Edge unfolding

⇔ Cut-edges ... ??

15

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

Variables for Edges

Ex) { x2, x3, x4, x7, x10, x11, x12 } gives a spanning tree

17

x4 x1 x3

x5 x6

x2 x7 x8

x9

x10 x11

x12

Construction of a ZDD of Spanning Trees

18

Top-down construction

DP-like approach (Dynamic Programming)

Construction of a ZDD of Spanning Trees

19

adopt e1 no e1 contract e1 delete e1

go to 0 share

start

20

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

Theorem No edge unfolding of a Platonic solid has self-overlap

Result

(2 unfoldings)

(11 unfoldings)

(11 unfoldings)

21

Tetrahedron

Cube

Octahedron

Result (Partial list)

43,380 unfoldings

Dodecahedron

22

Result (Partial list) Icosahedron

23

43,380 unfoldings

Conclusion

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

Theorem No edge unfolding of a Platonic solid has self-overlap

#(Edge unfoldings) #(Essentially different edge unfoldings)

331,776 6,912

208,971,104,256,000 1,741,425,868,800

6,000 261

101,154,816 2,108,512

32,400,000 675,585

375,291,866,372,898,816,000 3,127,432,220,939,473,920

4,982,259,375,000,000,000 41,518,828,261,687,500

301,056,000,000 6,272,012,000

201,550,864,919,150,779,950,956,544,000 1,679,590,540,992,923,166,257,971,200

12,418,325,780,889,600 258,715,122,137,472

21,789,262,703,685,125,511,464,767,107,171,876,864,000 181,577,189,197,376,045,928,994,520,239,942,164,480

89,904,012,853,248 3,746,001,752,064

438,201,295,386,966,498,858,139,607,040,000,000 7,303,354,923,116,108,380,042,995,304,896,000

Archimedian Solids

[AKL+ 11]

[AKL+ 11]

[BMP+ 91]

[BMPR 96]

[BMPR 96]

[PEK+ 11] estimated 2.3M

[Our result 3]