Fermi National Accelerator Laboratory

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From Flapping Birds to Space TelescopesThe
Modern Science of Origami

• Robert J. Lang
• robert_at_langorigami.com

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Fold Lines
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Context
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Background

• Origami Japanese paper-folding.
• Traditional form Decorative abstract shapes
  childs craft
• Modern extension a form of sculpture in which
  the primary means of creating the form consists
  of folding
• Most common version a figure folded from one
  sheet of paper with no cuts.

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Evolution of origami

• Right origami circa 1797.

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Even earlier

• Japanese newspaper from 1734 Crane, boat, table,
  yakko-san
• By 1734, it is already well-developed

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European origami?

• A 1490 text the first paper boat?

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Modern Origami

• The modern art form was reborn in the early 20th
  century through the efforts of a Japanese artist,
  Akira Yoshizawa, who created new figures of
  artistic beauty and developed a written
  instructional language.

A. Yoshizawa, Origami Dokuhon I
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The Design Revolution

• Creative origami caught on worldwide in the
  1950s and 1960s.
• Beginning in the 1970s, many geometric design
  techniques were developed that enabled the
  creation of figures of undreamed-of complexity.
• The mathematical theory of origami was greatly
  expanded in the 1990s, leading to computer-aided
  origami design.

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Origami today

• Black Forest Cuckoo Clock, designed in 1987
• One sheet, no cuts
• 216 steps
• not including repeats
• Several hours to fold

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Ibex
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Dragonfly
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What Changed?

• Origami was discovered by scientists.

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Math in Origami

• The tools of mathematics, as applied to origami,
  have served to advance the art of origami
• But origami can also be used to amplify,
  illustrate, and explain mathematics as well.

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Origami Mathematics

• The mathematics underlying origami addresses
  three areas
• Existence (what is possible)
• Complexity (how hard it is)
• Algorithms (how do you accomplish something)
• The scope of origami math include
• Plane Geometry
• Trigonometry
• Solid Geometry
• Calculus and Differential Geometry
• Linear Algebra
• Graph Theory
• Group Theory
• Complexity/Computability
• Computational Geometry

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Geometry

• Many of the geometric results can be appreciated
  -- and even discovered -- by elementary and
  secondary school students, by
• Inspection
• Manipulation
• Measurement
• of crease patterns and folded forms.

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Properties

• 2-colorability
• Every flat-foldable origami crease pattern can be
  colored so that no 2 adjacent facets are the same
  color. You need only 2 colors.

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An Experiment
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Mountain-Valley Counting

• Maekawa-Justin theorem
• At any interior vertex, M V 2

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Angles Around a Vertex

• Kawasaki-Justin Condition
• Alternate angles around a vertex sum to a
  straight line
• Independently discovered by Kawasaki, Justin, and
  Huffman (Pi condition)

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Targeting Level

• Many concepts applicable to several levels of
  education
• Elementary fold circles, then cut and assemble
  angles
• Secondary demonstrate (or discover) a proof
• College prove related properties for entire
  crease patterns

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Folding on the Cutting Edge

• A leader in current mathematical folding Erik
  Demaine, assistant professor at MIT
• MacArthur Genius Fellow
• First solved (as a student) the one-cut problem

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One-Cut

• From a single sheet of paper, what shapes can be
  formed with a single straight cut across a folded
  sheet of paper?

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One-Cut 2

• Fold on the black and green lines and cut on the
  dotted line.

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One-Cut 3

• A bit more complex.
• Vertex hint
• MgtV, point up
• MltV, point down

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One Cut 4

• Demaine et al proved that any shape or
  combination of shapes could be created with a
  single straight cut
• They did this by giving a general method for
  constructing the required creases
• Mathematically, the problem is equivalent to
  folding the paper so that a given set of lines
  all end up collinear (the cut line).
• Even when you know the creases, getting all the
  folds to go together can be quite a challenge!

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One-Cut 5
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Geometric Constructions

• What shapes and distances can be constructed
  entirely by folding?
• Analogous to compass-and-straightedge, but
  more general

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The Delian Problems

• Trisect an angle
• Double the cube
• Square the circle
• All three are impossible with compass and
  unmarked straightedge
• The first two are possible in origami

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Abes Trisection
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Messers Cube Doubling
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Geometric Constructions

• One-fold-at-a-time origami can solve exactly
• All quadratic equations with rational
  coefficients
• All cubic equations with rational coefficients
• Angle trisection (Abe, Justin)
• Doubling of the cube (Messer)
• Regular polygons for N2i3j2k3l1 if last term
  is prime (Geretschläger)
• All regular N-gons up to N20 except N11

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Foldable Distances

• There are seven distinct ways to form a crease by
  aligning combinations of points and lines
• Six Huzita-Justin operations (or axioms)
• Seventh operation identified by Hatori Koshiro in
  2002
• The seven are now proven to be complete
• See Folding Proportions, in Tribute to a
  Mathemagician, A K Peters, 2004.

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Huzita-Hatori Operations
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Simultaneous Creases

• If you allow forming two creases at one time,
  higher-order equations are possible.
• An angle quinsection!

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Three-dimensional linkages

• The origami flapping bird is a simple mechanical
  linkage
• Linkages allow the simulation of arbitrarily
  complex functions
• Origami gadgets allow the construction of any
  mechanical stick-and-pivot linkage
• There is an origami linkage that signs YOUR name

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Instrumentalists
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Organist
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Applications in Art

• The mathematics of origami has found application
  in representational origami.

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Textures

• Patterns of intersecting pleats can be integrated
  with other folds to create textures and visual
  interest

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The recipient form
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(Scaled Koi)
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Western Pond Turtle
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Rattlesnake
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Flap Generation

• The most extensive and powerful origami tools
  deal with the generation of flaps in a desired
  configuration.
• Why is this useful?

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Origami design

• The fundamental problem of origami design is
  given a desired subject, how do you fold a square
  to produce a representation of the subject?

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A four-step process

• The fundamental concept of design is the base
• The fundamental element of the base is the flap
• From a base, it is relatively straightforward to
  shape the flaps into the appendages of the
  subject.
• The hard step is
• Given a tree (stick figure), how do you fold a
  Base with the same number, length, and
  distribution of flaps as the stick figure?

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Flaps
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What is a Flap?

• A flap is a longish bit of paper that gets turned
  into some feature of the base.
• (Theres a more mathematical description, too.)
• Origami tree theory is all about how to
  construct flaps.

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How to make a flap

• To make a single flap, we pick a corner and make
  it narrower.
• The boundary of the flap divides the crease
  pattern into
• Inside the flap
• Everything else
• Everything else is available to make other flaps

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Limiting process

• What does the paper look like as we make a flap
  skinner and skinnier?
• A circle!

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Other types of flap

• Flaps can come from edges
• and from the interior of the paper.

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Circle Packing

• In the early 1990s, several of us realized that
  we could design origami bases by representing all
  of the flaps of the base by circles overlaid on a
  square.

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Circles are not enough

• But where do the creases go?

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Creases

• The lines between the centers of touching circles
  are always creases.
• But there needs to be more. Fill in the polygons,
  but how?

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Molecules

• Crease patterns that collapse a polygon so that
  all edges lie on a single line are called
  bun-shi, or molecules (Meguro)
• Different bun-shi are known from the origami
  literature.
• Triangles have only one possible molecule.

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Quadrilateral molecules

• There are two possible trees and several
  different molecules for a quadrilateral.
• Beyond 4 sides, the possibilities grow rapidly.

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Four is enough

• It is always possible to add flaps (circles) to a
  base so that the only polygons are triangles and
  quadrilaterals, so these molecules suffice.

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Universal molecule

• Eventually, I found an algorithm that produces
  the crease pattern to collapse an arbitrary valid
  convex polygon into a base whose projection is a
  specified stick figure, dubbed the Universal
  Molecule.

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Uni molecule one-cut

• The Universal Molecule construction is closely
  related to the One-Cut problem solved by
  Demaine, Demaine Lubiw
• The U.M. may have extra gussets in the crease
  pattern that insure the flaps have specified
  lengths

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Circles and Rivers

• Body segments between flaps can also be
  represented by geometric objects Rivers.

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Circle-River Design

• The combination of circle-river packing and
  molecules allows an origami composer to construct
  bases of great complexity using nothing more than
  a pencil and paper.
• But what if the composer had more
• Like a computer?

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Formal Description

• A formal mathematical description of a problem is
  harder to understand for most people
• But it is required in order to apply
  computational techniques to solving the problem.

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Formal Statement of the Problem

• Given an arbitrary weighted tree graph, construct
  a crease pattern that folds into a base whose
  projection is the given tree graph.
• Each edge of the tree corresponds to a flap
• The weight of the edge is the length of the flap.

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Formal Statement of the Solution

• The search for the largest possible base from a
  given square becomes a well-posed nonconvex
  nonlinear constrained optimization
• Linear objective function
• Linear and quadratic constraints
• Nonconvex feasible region
• Solving this system of tens to hundreds of
  equations gives the same crease pattern as a
  circle-river packing

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Computer-Aided Origami Design

• 16 circles (flaps)
• 9 rivers of assorted lengths
• 120 possible paths
• 184 inequality constraints
• Considerations of symmetry add another 16 more
  equalities
• 200 equations total!
• Childs play for computers.
• I have written a computer program, TreeMaker,
  which performs the optimization and construction.

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The crease pattern
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(The folded figure)
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Box-pleating

• Most creases are parallel, more easily foldable
  patterns

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Bull Moose
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The Art
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One art, two expressions

• Two distinct artistic expressions of origami have
  emerged
• Composition
• The design of a new figure akin to the
  composition of a musical score (with its own
  notation)
• Performance
• The realization of a design in folded paper
• Within origami, there are composers (designers),
  and performers (folders)
• performers fold their own compositions or those
  of others

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Dichotomy

• Origami is set apart from many arts by the
  separation between design and execution (and the
  artistic ability required for both).
• Historically, composition (design) has been the
  province of only a few.
• That is now changing.

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Three forms of composition

• Discovery
• The subject resides within the paper and by
  exploration, one eventually finds it
• Many elegant and clever figures are still
  discovered this way
• Intuitive
• Design
• The goal of the composition is to explore a
  design technique or solve a technical challenge
• Typically focused on a specific result
• Engineering
• Synthesis
• The end result is driven by an artistic vision
• Techniques of folding are used to realize the
  vision

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Bull Elephant
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Hummingbird Trumpet Vine
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(Hummingbird Close-up)
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Grizzly Bear
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Roosevelt Elk
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Tree Frog
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Tarantula
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Murex
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Spindle Murex
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12-Spined Shell
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Banana Slug
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Spiral Tessellation
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Egg17 Tessellation
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Continued evolution
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Applications in the Real World

• Mathematical origami has found many applications
  in solving real-world technological problems, in
• Space exploration (telescopes, solar arrays,
  deployable antennas)
• Automotive (air bag design)
• Medicine (sterile wrappings, implants)
• Consumer electronics (fold-up devices)
• and more.

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The Eyeglass Telescope

• Under development at Lawrence Livermore National
  Laboratory
• 25,000 miles above the earth
• 100 meter diameter (a football field)
• Look up see planets around distant stars
• Look down

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The lens and the problem

• The 100-meter lens must fold up to 3 meters
  (shuttle bay)
• Lens must be made from ultra-thin sheets of glass
  with flexures along hinges
• What pattern to use?

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5-meter prototype

• I provided several designs from origami
  techniques, one of which was selected for the
  first prototype.

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Solar Sail

• Japanese Aerospace Exploration Agency
• Mission flown in August 2004
• First deployment of a solar sail in space
• Pleated when furled, expands into sail

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Solar Sail
http//www.isas.jaxa.jp/e/snews/2004/0809.shtml
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NASA

• NASA, too, is developing unfolded and inflatable
  solar sails.

Video courtesy Dave Murphy, AEC-Able Engineering,
developed under NASA contract NAS803043
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Stents
www.tulane.edu/sbc2003/pdfdocs/0257.PDF
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Airbags

• A mathematical algorithm developed for origami
  design turned out to be the proper algorithm for
  simulating the flat-folding of an airbag.

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Airbag Algorithm

• The airbag-flattening algorithm was derived
  directly from the universal molecule algorithm
• More complex airbag shapes (nonconvex) can be
  flattened using derivatives of the DDL one-cut
  algorithm
• No one foresaw these technological applications.

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Resources

• Origami design software TreeMaker (with 170 pp
  manual) can be downloaded from
• http//origami.kvi.nl/programs/treemaker
• or Google-search for TreeMaker
• Version 5.0 (Mac/Linux/Windows) is under
  construction.
• Other origami-related software, including
  ReferenceFinder, is at the same site

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More Resources

• Origami Design Secrets, my new book teaching how
  to design origami (and more), was published by A.
  K. Peters in October 2003.
• For the month of November, FNAL employees get 15
  discount from www.akpeters.com enter FNAL
  promotion code.
• Origami Insects II, my latest, contains a
  collection of fairly challenging insect designs
• Both (and other books) available from the
  OrigamiUSA Source (www.origami-usa.org).
• Further information may be found at
  http//www.langorigami.com, or email me at
  robert_at_langorigami.com