One-Cut Fold via the Art of Origami
Shao-Shiung Lin
Depart of Math, National Taiwan University
April 22, 2012
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Fold and One-cut Theorem
Any straight-line segments drawing on a sheet of
paper may be folded flat so that one straight
scissors cut completely through the folding cuts all
the segments of the drawing, and nothing else.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
History
I
1721 A.D.: Japanese Kan Chu Sen solved some simple
drawings.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
History
I
1922 A.D.: H. Houdini solved regular 5-pointed star drawing
in his book ”Paper Magic”.
Figure: 5-pointed star
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
History
I
1955 A.D.: G. Loe solved some symmetric drawings in his
book ”Paper Capers”.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
History
I
1955 A.D.: G. Loe solved some symmetric drawings in his
book ”Paper Capers”.
I
1960 A.D.: M. Gardner formulated the complete problem in a
” Scientific American” Math. Game Series.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
History
I
1955 A.D.: G. Loe solved some symmetric drawings in his
book ”Paper Capers”.
I
1960 A.D.: M. Gardner formulated the complete problem in a
” Scientific American” Math. Game Series.
I
1999-2002 A.D.: A complete mathematical proof was given by
Demaine, Bern, Lubiw, O’Rourke and Hayes.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Figure: Fold and one-cut
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
D
C
x
y
A
B
Figure: Rectangle creases
Shao-Shiung Lin
Figure: Triangle creases
One-Cut Fold via the Art of Origami
Origami (Oru-folding, Kami-paper)
The traditional Japanese art of paper folding to
transform a flat sheet of material into a finished
sculpture, without the use of cuts and glue.
Figure: Crane
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Figure: Elephants
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Moutain and Valley Folds
Figure: mountain fold
Figure: valley fold
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Crease Patterns
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Crease pattern- A ”crease pattern” on a piece of paper is
the collection of crease which form the final folded state.
I
M-V crease pattern- When each crease is specified either
as a mountain fold or a valley fold, it is called a ”M-V crease
pattern”.
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Vertices- The crease in a crease pattern meet only at
common endpoints, called the ”vertices”.
I
A crease pattern is thus an embedding of a ”graph” in the
paper.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Figure: Crease pattern of flapping bird
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Figure: Crease pattern of fish
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Flat Foldability
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When the parallel layers of the final state of an origami can be
squashed into a plane, then the corresponding M-V crease
pattern is called ”flat foldable”.
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A crease pattern is called ”flat-foldable” if there exists a
mountain and valley folds specification on the creases so that
the resulting M-V crease pattern becomes flat-foldable.
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Whether a crease pattern is flat-foldable is a ”strong
NP-hard problem (Bern and Hayes (1996)).
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Figure: Degree-4 vertex
Shao-Shiung Lin
Figure: Degree-6 vertex
One-Cut Fold via the Art of Origami
Single-Vertex Flat-folding (1)
Given a single-vertex crease pattern with successive angles
θ1 , θ 2 , · · · , θ n
I Kawasaki-Justin Theorem– It is flat-foldable iff n = 2m
is even, and θ1 + θ3 + · · · + θ2m−1 = θ2 + θ4 + · · · + θ2m = π.
Figure: Maekawa Proof
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Single-Vertex Flat-Folding (2)
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Maekawa-Justin Theorem– If it is flat-foldable, then
any flat-foldable mountain and valley folds specification on
the creases must satisfy |M − V | = 2, where M and V are the
number of mountain and valley folds respectively.
I
Local Minimum Theorem– If it is flat-foldable, and
θi−1 > θi = · · · = θi+k−1 < θi+k , then, among the creases
between neighboring pairs of these equal angles, the
specification of mountain and valley folds satisfies
|M − V | = 0 for odd k, but |M − V | = 1 for even k.
I
If θj =
2π
n
for all j, then it is flat-foldable iff |M − V | = 2.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Straight Skeleton
Figure: Triangle skeleton
Figure: Polygon skeleton
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Straight Skeleton
Given an embedded graph on a piece of paper. Another graph
called the straight skeleton is constructed following the
procedures below.
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(a) Simultaneously shrink each face of the graph in such a
way that the edges retain their orientation, and the
perpendicular distance from every shrunken edge to the
corresponding edge is the same for all shrunken edges.
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(b) When a face becomes nonsimple (as a vertex touches an
edge), split the face into two components , and continue
shrinking each piece.
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(c) When an edge e collapses to a point, ignore e and
continue shrinking elsewhere.
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(d) When a face collapses to a point, stop shrinking that face.
The ”straight skeleton” is the union of the trajectories of the
vertices (of original graph) during this shrinking process.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Figure: Turtle
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Two Lemmas
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Lemma 1.
There is an one-to-one correspondence between
graph edges and the skeleton faces that contain them.
Lemma 2. Every skeleton edge is a subsegment of the
bisector of the two graph edges contained in the two skeleton
faces sharing the skeleton edge.
However, the crease pattern given by the skeleton may not be
flat-folding, and even if it is flat-folding, the folded state may not
serve as an one-cut folded state for the original graph.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Figure: Straight skeleton and perpendiculars
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
Perpendiculars
From a given vertex in the straight skeleton, and a skeleton face
incident to this vertex, draw a perpendicular following the
procedures below.
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(a) Draw the perpendicular to the graph edge contained in
this face, and continue this perpendicular.
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(b) If the perpendicular hits a skeleton edge, reflect this
perpendicular through the skeleton edge to obtain the
beginning of a new perpendicular edge.
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(c) Continue the process in (b) until this perpendicular hits
the paper boundary or another skeleton edge, then the process
ends.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
The Fold and One-cut Theorem is almost proved
Assuming that the original graph satisfies certain
conditions, the crease pattern formed by the straight
skeleton, and some perpendiculars has a mountain
and valley specification which is flat-foldable. And
the folded origami contains exactly one folded edge
which is folded from all graph edges, and vertices.
Shao-Shiung Lin
One-Cut Fold via the Art of Origami
http://blog.makezine.com/2011/03/07/folded-metal-bunny/
http://www.youtube.com/watch?v=GAnW-KU2yn4
Shao-Shiung Lin
One-Cut Fold via the Art of Origami