From Flapping Birds to Space Telescopes : The Modern Science of - PDF document

From Flapping Birds to Space Telescopes : The Modern Science of Origami Robert J . Lang Usenix Conference , Boston , MA June , 2008 Background • Origami • Traditional form • Modern extension • Most common version: One Sheet , No Cuts Usenix Conference , Boston , MA June , 2008 1

Evolution of origami • Right: origami circa 1797. • The traditional “ tsuru ” ( crane ) Usenix Conference , Boston , MA June , 2008 Even earlier… • Japanese newspaper from 1734 : Crane , boat , table , “ yakko - san ” Usenix Conference , Boston , MA June , 2008 2

Modern Origami Reborn by Yoshizawa A. Yoshizawa, Origami Dokuhon I Usenix Conference , Boston , MA June , 2008 Origami Today • “ Black Forest Cuckoo Clock ,” designed in 1987 • One sheet , no cuts • 216 steps – not including repeats • Several hours to fold Usenix Conference , Boston , MA June , 2008 3

Ibex Usenix Conference , Boston , MA June , 2008 Klein Bottle Usenix Conference , Boston , MA June , 2008 4

What Changed? • Origami was discovered by mathematicians . • Or rather , mathematical principles • 1950-2000 … – From about 100 … – …to over 36,000 ! ( see http: // www . origamidatabase . com ). Usenix Conference , Boston , MA June , 2008 The Technical Revolution • The connection between art and science is made by mathematics . Usenix Conference , Boston , MA June , 2008 5

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 Di ff erential Geometry Linear Algebra Graph Theory Group Theory Complexity / Computability Computational Geometry Usenix Conference , Boston , MA June , 2008 Geometric Constructions • What shapes and distances can be constructed entirely by folding? • Analogous to “ compass - and - straightedge ,” but more general Usenix Conference , Boston , MA June , 2008 6

The Delian Problems • Trisect an angle • Double the cube • Square the circle • All three are impossible with compass and unmarked straightedge , but: Usenix Conference , Boston , MA June , 2008 Hisashi Abe ’ s Trisection P P P P A D A D A D A D E F E F E F G H G H θ B C B C B C B C 1. Begin with the 2. Make a horizontal 3. Fold line BC up to 4. Fold the bottom desired angle (PBC fold anywhere across line EF and unfold, left corner up so that in this example) the square, defining creating line GH. point E touches line marked within the line EF. BP and the corner, corner of a square. point B, touches line GH. P P P P J J A D A D A D A D θ /3 E E F F E F E θ /3 G G K B B H H G H G H θ /3 C C B C B C 5. With the corner 6. Unfold corner B. 7. Fold along the 8. The two creases still up, fold all crease that runs to BJ and BK divide the layers through the point J, extending it original angle PBC existing crease that to point B. Fold the into thirds. hits the edge at point bottom edge BC up G and unfold. to line BJ and unfold. Usenix Conference , Boston , MA June , 2008 7

Peter Messer ’ s Cube Doubling A D A D A D A D F G E E E E H I B C B C B C B C 1. Make a small fold 2. Make a crease 3. Fold the top edge 4. Fold corner C to halfway up the right connecting points A down along a lie on line AB while side of the paper. and C and another horizontal fold to point I lies on line connecting B and E. touch the crease FG. Only make them intersection and sharp where they unfold. Then fold the D cross each other. bottom edge up to A touch this new crease G and unfold. 3 2 F I C H 1 B 5. Point C divides edge AB into two segments whose proportions are 1 and the cube root of 2. Usenix Conference , Boston , MA June , 2008 Binary Approximation for Distance • Any distance can be approximated to 1/ N using log 2 N folds taken from its binary expansion • Example: 0.7813 ~ 25/32 = .11001 2 .11001 Usenix Conference , Boston , MA June , 2008 8

Generalize Constructions • The binary algorithm is a special answer to a general question: • Starting with a blank square , • for a given point or line , • construct an folding sequence accurate to a speci fi ed error , • de fi ning every fold in the sequence in terms of preexisting points and lines . Usenix Conference , Boston , MA June , 2008 Building Blocks • Points and Lines ( creases ) Usenix Conference , Boston , MA June , 2008 9

Points A point ( mark ) can only be de fi ned as the intersection of two lines . But a line ( fold ) can be made in many ways… Usenix Conference , Boston , MA June , 2008 Lines • For many years , it was thought that there were only six ways to de fi ne a fold . • The six operations are called the Huzita “ Axioms .” Usenix Conference , Boston , MA June , 2008 10

Huzita Axioms 2 Usenix Conference , Boston , MA June , 2008 Hatori ’ s Axiom • In 2002, Koshiro Hatori discovered a seventh “ axiom .” • In 2006, it was observed that Jacques Justin had identi fi ed all 7 in 1989. • It has since been proven that these seven are the only ways to de fi ne a single fold . Usenix Conference , Boston , MA June , 2008 11

Geometric Constructions • One - fold - at - a - time origami can solve exactly: – All quadratic equations with rational coe ffi cients – All cubic equations with rational coe ffi cients – Angle trisection ( Abe , Justin ) – Doubling of the cube ( Messer ) – Regular polygons for N =2 i 3 j {2 k 3 l +1} if last term is prime ( Alperin , Geretschl ä ger ) • All regular N - gons up to N =20 except N =11 Usenix Conference , Boston , MA June , 2008 Simultaneous Creases • If you allow forming E two creases at one time , J higher - order equations D are possible . • An angle quintisection! C H I A F • Quintisections are G impossible with only Huzita ( one - fold - at - a - time ) axioms . • There are over 400 two - fold - at - a - time “ axioms .” Usenix Conference , Boston , MA June , 2008 12

More simultaneous • What about N - at - a - time folding? Usenix Conference , Boston , MA June , 2008 Crease Patterns • The design of an origami fi gure is encoded in the crease pattern • What constraints are there on such patterns? Usenix Conference , Boston , MA June , 2008 13

Properties of Crease Patterns • 2- colorability • Every fl at - foldable origami crease pattern can be colored so that no 2 adjacent facets are the same color with only 2 colors . Usenix Conference , Boston , MA June , 2008 Mountain - Valley Counting • Maekawa Condition: – At any interior vertex , M – V = ±2 Usenix Conference , Boston , MA June , 2008 14

Angles Around a Vertex • Kawasaki Condition: – Alternate angles around a vertex sum to a straight line – Independently discovered by Kawasaki , Justin , and Hu ff man – Generalized to 3 D by Hull & belcastro 3 1 2 2 4 3 4 1 6 5 6 5 Usenix Conference , Boston , MA June , 2008 Layer Ordering • A complete description of a folded form includes the layer ordering among overlapping facets ( M - V is not enough! ) • Four necessary conditions were enumerated by Jacques Justin • Pictorially , these are the “ legal ” layer orderings between layers , folded creases , and unfolded (fl at ) creases CCFO CFUCO C i C i F F i ,1 i ,1 F F i ,2 i ,2 F F j j C j CUUCO CFFCO C i F C i F F F i ,1 i ,1 C i i ,1 i ,2 F F F F i ,2 i ,1 C j j ,1 j ,2 F F C j i ,1 i ,2 F F C j i ,2 i ,2 Usenix Conference , Boston , MA June , 2008 15

Complexity • Satisfying M - V =±2 is “ easy ” • Satisfying alternate angle sums is “ easy ” • Satisfying layer order ( M - V assignment ) is “ hard ” … • How hard? Usenix Conference , Boston , MA June , 2008 Pleats as logical signals • Two parallel pleats must be opposite parity • For a speci fi ed direction , there are 2 allowed crease assignments Valley on right = “true” Valley on left = “false” Usenix Conference , Boston , MA June , 2008 16

Not - All - Equal • A particular crease pattern enforces the condition “ Not - All - Equal ” on its incident pleats Crease Pattern Folded Form Crease Pattern True True ? True True True False Invalid Valid • It is possible to create multiple such conditions , thereby encoding NAE logic problems as crease assignment problems Usenix Conference , Boston , MA June , 2008 Crease Assignment Complexity • Marshall Bern and Barry Hayes showed in 1996 that any NAE -3- SAT problem can be encoded as a crease assignment problem • NAE -3- SAT is NP - complete! • Ergo , “ Origami is hard! ” • But most problems of interest are polynomial ( still hard , but solvable ) Usenix Conference , Boston , MA June , 2008 17

P . S . – Even if you have the complete crease assignment , simply determining a valid layer ordering is still NP - complete! Usenix Conference , Boston , MA June , 2008 Flat - Foldability • A crease pattern is “fl at foldable ” i ff it satis fi es: – Maekawa Condition ( M - V parity ) at every interior vertex – Kawasaki Condition ( Angles ) at every interior vertex – Justin Conditions ( Ordering ) for all facets and creases Within this description , there are many interesting and unsolved problems! Usenix Conference , Boston , MA June , 2008 18