origami : local foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Origami : Local Foldability

Lim, Sung Hak

KAIST

December 21, 2007

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Table of Contents

Descriptions of OrigamiWhat is Origami?Mathematical Definition of origami

Brief Introduction to Flat OrigamiDefinition of Flat OrigamiConditions for the Structure is Flat

Generalizing to Global Flat Foldability

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

This is origami! If you want to do more further....

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

This is origami! If you want to do more further....

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

This is origami! If you want to do more further....

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

This is origami! If you want to do more further....

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

This is origami!

If you want to do more further....

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

This is origami! If you want to do more further....

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

Crane

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

Arrow

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

Sailing Ship

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

Sea Urchin

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

Girl

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Brief Example

Housemaid, by Kamiya Ryo

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Goal of Origami

I Making three-dimensional structure from a paper by folding, without cutting orgluing the paper.

I However, sometimes the restriction is ignored to give a variety to the structure.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Goal of Origami

I Making three-dimensional structure from a paper by folding, without cutting orgluing the paper.

I However, sometimes the restriction is ignored to give a variety to the structure.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Goal of Origami

Folding

I Making three-dimensional structure from a paper by folding, without cutting orgluing the paper.

I However, sometimes the restriction is ignored to give a variety to the structure.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Goal of Origami

Folding

I Making three-dimensional structure from a paper by folding, without cutting orgluing the paper.

I However, sometimes the restriction is ignored to give a variety to the structure.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Goal of Origami

Folding

I Making three-dimensional structure from a paper by folding, without cutting orgluing the paper.

I However, sometimes the restriction is ignored to give a variety to the structure.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Goal of Origami

Folding

I Making three-dimensional structure from a paper by folding, without cutting orgluing the paper.

I However, sometimes the restriction is ignored to give a variety to the structure.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Definition of Origami

Line Angle

Definition (Origami : Simple Model)Origami is a set of lines on finite plane with corresponding angle parameter. i.e.

O : {(l1, θ1), (l2, θ2), (l3, θ3), · · · , (ln, θn)} (1)

where li are lines and θi are corresponding angle parameter within [−π, π].

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Definition of Origami

Line

Angle

Definition (Origami : Simple Model)Origami is a set of lines on finite plane with corresponding angle parameter. i.e.

O : {(l1, θ1), (l2, θ2), (l3, θ3), · · · , (ln, θn)} (1)

where li are lines and θi are corresponding angle parameter within [−π, π].

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Definition of Origami

Line Angle

Definition (Origami : Simple Model)Origami is a set of lines on finite plane with corresponding angle parameter. i.e.

O : {(l1, θ1), (l2, θ2), (l3, θ3), · · · , (ln, θn)} (1)

where li are lines and θi are corresponding angle parameter within [−π, π].

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Definition of Origami

Line Angle

Definition (Origami : Simple Model)Origami is a set of lines on finite plane with corresponding angle parameter. i.e.

O : {(l1, θ1), (l2, θ2), (l3, θ3), · · · , (ln, θn)} (1)

where li are lines and θi are corresponding angle parameter within [−π, π].

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Definition of Origami : Example

π

O : {(l1, π)} (2)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

Definition of Origami : Example

π

O : {(l1, π)} (2)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

How to Analyze Origami?

I We call the last picture ”crease pattern”.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

How to Analyze Origami?

I We call the last picture ”crease pattern”.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

How to Analyze Origami?

Unfolding

I We call the last picture ”crease pattern”.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

How to Analyze Origami?

Unfolding

I We call the last picture ”crease pattern”.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

What is Origami?Mathematical Definition of origami

How to Analyze Origami?

Unfolding

I We call the last picture ”crease pattern”.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Definition of Flat Origami

Definition (Flat Origami)Flat origami is an origami with only angle parameters −π and π.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Definition of Flat Origami

Definition (Flat Origami)Flat origami is an origami with only angle parameters −π and π.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Definition of Flat Origami

Definition (Flat Origami)Flat origami is an origami with only angle parameters −π and π.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Definition of Flat Origami

Definition (Flat Origami)Flat origami is an origami with only angle parameters −π and π.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Definition of Flat Origami

Definition (Flat Origami)Flat origami is an origami with only angle parameters −π and π.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Definition of Flat Origami

Definition (Flat Origami)Flat origami is an origami with only angle parameters −π and π.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Definition of Flat Origami

Definition (Flat Origami)Flat origami is an origami with only angle parameters −π and π.

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Degree of a Vertex

I Could you find something in common?I Degrees of each vertices are even.

TheoremIf vertex v is locally flat-foldable then,

deg(v) = 2n n = 1, 2, · · · (3)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Degree of a Vertex

I Could you find something in common?I Degrees of each vertices are even.

TheoremIf vertex v is locally flat-foldable then,

deg(v) = 2n n = 1, 2, · · · (3)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Degree of a Vertex

I Could you find something in common?

I Degrees of each vertices are even.

TheoremIf vertex v is locally flat-foldable then,

deg(v) = 2n n = 1, 2, · · · (3)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Degree of a Vertex

I Could you find something in common?I Degrees of each vertices are even.

TheoremIf vertex v is locally flat-foldable then,

deg(v) = 2n n = 1, 2, · · · (3)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Degree of a Vertex

I Could you find something in common?I Degrees of each vertices are even.

TheoremIf vertex v is locally flat-foldable then,

deg(v) = 2n n = 1, 2, · · · (3)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Sum of Angles

I Anything else?

Theorem (Kawasaki’s Theorem)Let v be a vertex of degree 2n in a single vertex fold and let α1, α2, , · · · , α2n be theconsecutive angles between the creases. Then v is a flat vertex fold if and only if

α1 − α2 + α3 − α4 + · · · − α2n = 0 (4)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Sum of Angles

I Anything else?

Theorem (Kawasaki’s Theorem)Let v be a vertex of degree 2n in a single vertex fold and let α1, α2, , · · · , α2n be theconsecutive angles between the creases. Then v is a flat vertex fold if and only if

α1 − α2 + α3 − α4 + · · · − α2n = 0 (4)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Sum of Angles

I Anything else?

Theorem (Kawasaki’s Theorem)Let v be a vertex of degree 2n in a single vertex fold and let α1, α2, , · · · , α2n be theconsecutive angles between the creases. Then v is a flat vertex fold if and only if

α1 − α2 + α3 − α4 + · · · − α2n = 0 (4)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Example : Crease Pattern of Crane

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Fold Number Difference

I Anything else?

Theorem (Maekawa’s Theorem)Let M be the number of mountain creases and V be the number of valley creasesadjacent to a vertex in a single vertex fold. If a vertex v is locally flat-foldable then,

M − V = ±2 (5)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Fold Number Difference

I Anything else?

Theorem (Maekawa’s Theorem)Let M be the number of mountain creases and V be the number of valley creasesadjacent to a vertex in a single vertex fold. If a vertex v is locally flat-foldable then,

M − V = ±2 (5)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Fold Number Difference

I Anything else?

Theorem (Maekawa’s Theorem)Let M be the number of mountain creases and V be the number of valley creasesadjacent to a vertex in a single vertex fold. If a vertex v is locally flat-foldable then,

M − V = ±2 (5)

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Example : Crease Pattern of Crane

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Quiz

Q : Can you find a non-flat foldable point in this crease pattern?

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Definition of Flat OrigamiConditions for the Structure is Flat

Quiz

Q : Can you find a non-flat foldable point in this crease pattern?

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Locally Flat Foldable but not Globally

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Global Flat Foldability

I Because global flat foldability is depends on the whole structure not only thepoint, you shoud consider some area, not a point.

I If you want to check it, seehttp://www.merrimack.edu/∼thull/papers/flatsurvey.pdf

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Global Flat Foldability

I Because global flat foldability is depends on the whole structure not only thepoint, you shoud consider some area, not a point.

I If you want to check it, seehttp://www.merrimack.edu/∼thull/papers/flatsurvey.pdf

Lim, Sung Hak Origami : Local Foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Global Flat Foldability

I Because global flat foldability is depends on the whole structure not only thepoint, you shoud consider some area, not a point.

I If you want to check it, seehttp://www.merrimack.edu/∼thull/papers/flatsurvey.pdf

Lim, Sung Hak Origami : Local Foldability