origami : local foldability

Descriptions of OrigamiBrief Introduction to Flat Origami

Generalizing to Global Flat Foldability

Origami : Local Foldability

Lim, Sung Hak

December 21, 2007

Lim, Sung Hak Origami : Local 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

What is Origami?Mathematical Definition of origami

Brief Example

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

Brief Example

This is origami!

If you want to do more further....

Brief Example

Sailing Ship

Brief Example

Sea Urchin

Brief Example

Housemaid, by Kamiya Ryo

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.

Goal of Origami

Folding

Goal of Origami

Folding

Goal of Origami

Folding

Goal of Origami

Folding

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].

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

Line Angle

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

Line Angle

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

Definition of Origami : Example

O : {(l1, π)} (2)

Definition of Origami : Example

O : {(l1, π)} (2)

How to Analyze Origami?

I We call the last picture ”crease pattern”.

Unfolding

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.

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,