Download - 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