Computational Origami
Ryuhei Uehara
Japan Advanced Institute of Science and Technology (JAIST)
School of Information Scienceuehara@jaist.ac.jp
http://www.jaist.ac.jp/~uehara
2020/01/27

I628E: Information Processing Theory

Today…
1. Basic facts for unfolding
2. Polygons foldable two or more boxes
3. Common unfolding of regular polyhedra (or Platonic solids)
1. Common unfolding of a regular tetrahedron and a cube
2. Common unfolding of a regular tetrahedron and JohnsonZalgaller solids

2020/01/27

I628E: Information
Processing Theory

Construction of Common Unfolding ofa Regular Tetrahedron and a Cube
Reference:•

Toshihiro Shirakawa, Takashi Horiyama,and Ryuhei Uehara: OnCommon Unfolding of a Regular Tetrahedron and a Cube, Japan
Conference on Discrete and Computational Geometry (JCDCG 2011),
2011/11/28-29, Tokyo, Japan.

•

正4面体と立方体の共通の展開図に関する研究(On Common
Unfolding of a Regular Tetrahedron and a Cube)， 白川俊博・堀山貴
史・上原隆平， 『折り紙の科学』， Vol.4, No.1, pp.45-54, 2015.

2011/11/29

3Introduction• Open problem 25.6

(by M. Demaine, F. Hurtado, E. Pegg)Can any

Platonic solid be cut open and unfolded to a polygon that
may be refolded to a different Platonic solid?
For ex., may a cube be so dissected to a tetrahedron ?

Tetrahedron

Cube
Octahedron Dodecahedron(Hexahedron)Icosahedron

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Introduction• Open problem 25.6

(by M. Demaine, F. Hurtado, E. Pegg)Can any

Is there any polygon that can be folded toPlatonic solid be cut open and unfolded to a polygon that
maytwobe refolded
to adifferent
different Platonic(or more)
Platonicsolid?
solids ?
For ex., may a cube be so dissected to a tetrahedron ?

Tetrahedron

Cube
Octahedron Dodecahedron(Hexahedron)Icosahedron

5

Introduction
Is there any polygon that can be folded totwo (or more) different Platonic solids ?

Close call ! [O’Rourke]

Regular Octahedron⇔ Tetramonohedron(tetrahedron that consistsof congruent triangles)

Close call ! ! [Hirata]

Regular Tetrahedron⇔ Box 1 x 1 x 1.232
6Is there any polygon that can be folded totwo (or more) different Platonic solids ?
Introduction
Close calls

Cube (Regular hexahedron)⇔ Tetramonohedron
1 : 0.972 : 0.972

Regular Octahedron⇔ Tetramonohedron
1.0072 : 0.9965 : 0.9965

Regular Icosahedron⇔ Tetramonohedron 1 : 1.145 : 1.25
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Introduction
Our Results
A procedure that produces points s.t.
(1) they seem to converge to a polygon that can fold to a
cube and a regular tetrahedron (with infinitely manypoints)• we solve the open problem in a sense!
• … but we left some conjectures(2)8

Introduction
Our Results

2 3 ±εA procedure that produces points s.t.
(1) they seem to converge to a polygon that can fold to a
cube and a regular tetrahedron (with infinitely manypoints)(2) they certainly form a polygon that folds to a cube andan almost regular tetramonohedron with error
ε<2.89×10-1796
9The Key Theorem• Theorem [Akiyama 2007, Akiyama Nara 2007]
Let P be a development of a regular tetrahedron.
Then, P is a tiling. That is, P fills a plane.

The Key Theorem• Theorem [Akiyama 2007, Akiyama Nara 2007]

P is a development of a regular tetrahedron iff(1) P has a p2 tiling, i.e., tiling by 180°rotations(2) 4 of the rotation centers define
the triangular lattice(3) no two of the 4 rotation centers belong tothe same equivalent class on the tiling

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The Key Theorem• Theorem [Akiyama 2007, Akiyama Nara 2007]

P is a development of a regular tetramonohedron if(1) P has a p2 tiling, i.e., tiling by 180°rotations(2) 4 of the rotation centers define
a nonregular triangular lattice(3) no two of the 4 rotation centers belong to
the same equivalent class on the tiling12

Construction of a development• We modify the unfolding of a cube with thefollowing invariants:• it is a development of a cube
• it is a p2 tiling = a unfolding of a tetramonohedron

Initial unfoldingof a cube

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Construction of a development• make L1 and L2 parallel to c1c2
… c3 and c4 can be symmetrically moved to any position• make each face an isosceles triangle• we have to stretch c1c2 a little to make the triangles toregular ones.
• …so we’ll stretch them horizontally!!

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Construction of a development• We have some “fixed points” that have to be on anedge of the unfolding: they have to make a center of a top/bottom square: they form a vertex of the unit cube

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Idea of move of c1 (and c2)• “shift” the center c1 to c1’ with distance l1• fixed points have their images since they should form ap2 tiling.
• these images of fixed points generate the unfolding of acube and a tetramonohedron with center c1’
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Idea of move of c1 (and c2)• “shift” the center c1 to c1’ with distance l1• l1 is rational: finite points form a certain unfolding• l1 is irrational: infinite points converge to an unfoldingAn example
common unfolding of a cube and analmost regular tetramonohedron

The curves seem to bea kind of “fractal”

We have to tune l1
and l2 to obtain aconnected unfoldingProperty of the curve• [Observation/Conjecture] The “fractal curves’’ aredefined by the value of l1 in continued fraction
1form l =
1

a1 +

1

a2 +

1
a3 + ...

Property of the curve• [Observation/Conjecture] The “fractal curves’’ aredefined by the value of l1 in continued fraction
1form
l1 =

a1 +

1

1
a2 +
a3 + ...
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Summary
Our Results

2 3 ±εA procedure that produces points s.t.
Cnj(1) they seem to converge to a polygon that can fold to acube and a regular tetrahedron (with infinitely manypoints)(2) they certainly form a polygon that folds to a cube andThm
an almost regular tetramonohedron with error
ε<2.89×10-1796

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Future work

Theorem• [Observation/Conjecture]
The “curves’’ are definedby the value of l1 in continued fraction form

• Other Platonic solids：• Easy?: Tetrahedron and Octahedron• Middle?: Tetrahedron and Icosahedron• Hard?: Other pair except tetrahedron(Regular) Cube and(Nonregular) Octahedron
22Simple Exercises• Find “good” common unfolding of a regularoctagon or a regular icosahedron AND atetramonohedron. Estimate how “good” it is.
orAND• Consider why a dodecahedron has no common
unfolding with a tetramonohedron or a non-regular
icosahedron?

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