Mathematical Methods In Origami
Mathematical Methodsin OrigamiUniversity of Tokyo, KomabaDay 1 Slides, Dec. 16, 2015Thomas C. Hull, Western New England UniversityLecture notes and assignments available at:http://mars.wne.edu/ thull/tokyo/class2015.html
Overview of Origami-Math origami geometric constructionsOCL3
Overview of Origami-Math origami geometric constructions combinatorial geometry of origamiα1α4 α2α3
Overview of Origami-Math origami geometric constructions combinatorial geometry of origami matrix models and rigid foldability
Overview of Origami-Math origami geometric constructions combinatorial geometry of origami matrix models and rigid foldability computationalcomplexity of origami
Overview of Origami-Math origami geometric constructions combinatorial geometry of origami matrix models and rigid foldability computational complexity of origami folding manifolds(a)(b)γ v1γ v4B(x, µ(x))γ v2xγ v3B(x, µ(x))x
Overview of Origami-Math origami geometric constructions combinatorial geometry of origami matrix models and rigid foldability computational complexity of origami folding manifolds origami design
Overview of Origami-Math origami geometric constructions combinatorial geometry of origami matrix models and rigid foldability computational complexity of origami folding manifolds origami design origami algebra
Overview of Origami-Math origami geometric constructions combinatorial geometry of origami matrix models and rigid foldability computational complexity of origami folding manifolds modular origami origami design applications origami algebra curved folds
Folding a Equilateral Triangle
Folding a Regular PentagonAABABBABABIf the side of the square 2, then AB method by David ChandlerB
Folding a Regular HeptagonP1AP1L1P2(4)0CCCCC 00C 00(7)C 2)C00F(12)method by HullC0L4
Folding a Regular HeptagoniA2A e2πi/7A3A3 1/A3A2 1/A2221A 1/A21/A31/A1/A2R
Origami angle trisectionL3L2p2L1θp12θ3L1θL3L2Proof:AB Ccredit: Hisashi Abe, 1980ODL1
Straightedge and Compassbasic operationsGiven two points P1 and P2, we can drawthe line P1P2.!Given a point P and a line segment oflength r, we can draw a circle centered atP with radius r.!We can locate intersection points, if theyexist, between lines and circles.
What are the BasicOperations of Origami?Given two points P1 and P2, we can foldthe crease line P1P2.!Given two points P1 and P2, we can make acrease that puts P1 onto P2.!Given two lines L1 and L2, we can make acrease that puts L1 onto L2.!and so on.
Folding a point to a lineppLpL
Folding a point to a line
A different way.Given a family of curves F (x, y, t) 0 in R , its envelopeconsists of all points (x, y) such that that for some t R ,! F (x, y, t) 0F (x, y, t) 0 and tThe envelope is a single curve that is tangent to all thecurves in the family.!In our case we have! F (x, y, t) t2 2xt 2y 12 SoF (x, y, t) 0 becomes 2t t2x 0 , or x t .!Combining this with t1 0 gives1 2 1y x 22x 2y222xt 2y1 0or
Lill’s MethodLill’s geometric method forfinding real roots of anypolynomial:!nn-12anx an-1x . a2x a1x a0 0!a3θTa0θa1a4Start at O, go an, turn 90 ,!go an-1, turn 90 , etc, endingat T.a2θa5θOThen shoot from O with an angle θ, bouncing offthe walls at right angles, to hit T.!(Lill, 1867)Then x -tan θ is a root.
Lill’s MethodWhy does Lill’s method work? !PnQn-1 /an tan θ -x!P n -1So PnQn-1 -anx!Pn-1Qn-2 /(an-1-PnQn-1) -x!So Pn-1Qn-2 -x(an-1 anx)!Similarly,!Pn-2Qn-3 -x(an-2 x(an-1 anx))!Continuing.!Q n -2 a n-2θa2Ta0θP1a n -1a1θQ n -1PnanθOa0 P1T - a1x - a2x2 - . - an-1xn-1 - anxn !or, anxn an-1xn-1 . a2x2 a1x a0 0.!(Lill, 1867)
Beloch’s use of Lill to Solve CubicsDraw the turtle path!a3, a2, a1, a0.!ODraw D1 at dist a3 from andparallel to a2.!ψTψD2XYa1Draw D2 at dist a0 from andparallel to a1.!a0a2Oa3TD1Then fold O onto D1 andT onto D2 at the same time.(Beloch, 1936)
Origami can solve any cubic equationLet’s find the roots of the cubic3z- 7z - 6. !D1T-4O14128D2
Origami can solve any cubic equationLet’s find the roots of the cubic3z- 7z - 6. !D1T-4O14128D2Here !tan θ 1, !so!-tan θ -1!is a root.
Origami can solve any cubic equationLet’s find the roots of the cubic3z- 7z - 6. !D1T-4O14128D2Here !tan θ 2, !so!-tan θ -2!is a root.
Origami can solve any cubic equationLet’s find the roots of the cubic3z- 7z - 6. !D1T-4O14128D2Here !tan θ -3, !so!-tan θ 3!is a root.
Origami can solve any cubic equationLet’s find the roots of the cubic3z- 7z - 6. !D1T-4O14128D2
How to fold a regular heptagonP1AL1P2L2B(2)(1)P1P2(3)(4)CC COOCL4COL3(5)C (6)(7)C(8)C C EC OOCFC (9)L3C (10)(11)(12)
Heptagon - cubic equationA2The equationA3iA e2ºi /712º /7has as a solutionA 1/A2A4 1/A3A5 1/A2Therefore, solving this equation will give us1A6 1/AR
Task 2: Solveby origamiFoldonto the x-axis!andonto the !y-axis at the same time.Claim: When we do thisfold, P1 gets folded to!on the x-axis.L21P1 (0,1)-11P 2 (-1,-1/2)-1stL1
Task 2: Solveby origamiStart with P1 (0,1).!Make a “turtle diagram” from !P1 using the coefficients/2.!(Go, turn 90 , repeat.)L2P1 (0,1)x3-2xx21-1P 2 (-1,-1/2)-1stL1
Task 2: Solveby origamiHow did we pick P1 and P2?Then we “shoot” from P1 byangle θ, bounce off the x2and -2x lines, to hit P2.!L2P11x3 θ-2x-1This gives us three similartriangles.!Also note that thehypothenuse of the biggesttriangle is the crease line.P2x2θ1-1θ-1stL1
Task 2: Solveby origamiNotice how the trianglemade by P1, t, and the originverifies that !t tan θ x 2cos(2π/7).L21P1θθ-1P21θ-1stL1
How we fold thisP1P1L1P2L2(3)P2L3(4)Step 3 has P1 (0,1) being folded to the x-axis and !P2 (-1,-1/2) being folded to the y-axis.!In step 4 we mark P1’s location to get t 2cos(2π/7).
Does it make sense now?P1AL1P2L2B(2)(1)P1P2(3)(4)C2C COOC2cos(2π)7L4COL3(5)C (6)(7)C(8)C C EC OOCFC (9)L3C (10)(11)(12)
The Algebraic PerspectiveThe set of constructible numbers under SE&C is the smallestsubfield of C (complex #s) that is closed under square roots.!or !nC is SE&C constructible if and only if! [Q( ) : Q] 2for some nis algebraic over Q and0 . In other words,the degree if its minimal polynomial over Q is a power of 2 .!Origami version:!LetC be algebraic over Q , and let L Q be thesplitting field of the minimal polynomial ofover Q . Thenis origami constructible from our list of BOOs if and only if !a b for some integers a, b0.[L : Q] 2 3
Folding KnotsTake a strip of paper and tie it in a knot.!What do you get?
Folding KnotsThe knot forms a perfect pentagon!!Can you make a hexagon knot?
Answer: NO!!!But you can make a hexagon !from two strips of paper:!And you can make a heptagon (7 sides) !knot from one:
Apermitting the simultaneous creationnments,it is 1. Startwithlong stripto1solveunit high and 5–6unitslong.AngleEABpossibleD withEthe 7 Huzita-HatoriA more folds that satisfy2. Fold line FG downB to lie aof two orher-orderequations,as this Make a verticalis the angleto be quintisected.creaseabout1/3unitCopyright 2004.axioms, each of which defines a1. Start with aoflongstrip 1 unit high and 5–6 units long. Angle EABvarious combinationspoint/linemplefromillustrates.the right side.All Rightssinglefold byReserved.simultaneousDnAgofisleGriByonERobert Lang’s Angle Quintisectionis theto betoquintisected.Make a vertical crease about 1/3 unitalignments,it isanglepossiblesolvealignmentof pointsisandlines.ofByDEfromequations,the right side.Angle quintisectiondivisionhigher-orderas thisAGBGFC7. Mountain-foldcorner D behind.permittingsimultaneousFan arbitrarytheangleintoThisexample illustrates.A fifths.creationline FG down to lie along edge AB.oftwo orsolutionmore foldssatisfy 3. Fold point F over2.toFoldrequiresof nsof ispoint/lineCquintic Aequationand thusnotB3. Foldpoint F over to point A.alignments,is possibleto solvepossible withitthe7 Huzita-HatoriCBDEhigher-orderas thisaaxioms, each equations,of which definesAsingle foldillustrates.by simultaneousGBexample3. Fold point F over toDEDEalignmentof points and lines. By2. Fold line FG down to lie alongedge AB.AGBpermitting the simultaneous creation B A DDEE2. Fold line FG down to lie along edge AB.1. Start with a long strip 1 unit highunitsfoldslong.thatAngleEABof andtwo 5–6or moresatisfyis theangle to be quintisected. Makea verticalcrease about1/3 unitvariouscombinationsof point/lineDAEDEDFfrom the right side.alignments, it is possible to solve3. Fold point F over to point A.DEhigher-order equations, as thisCGGFAexample illustrates.AFDEFC7. Mountain-fold corner D behind.4. Fold and unfold.AFC and unfold.4. FoldBACGF DAFGEGB3. Fold point F over to point A.AFCBCBAGABCB5. Squash-fold on5. Squash-foldtheon existing D E4. existingFold and unfold.thecreases.CneeGd.E GE 4.BCCGGFAFold and unfold.BAFBGDCEGDFAD. Make5.a horizontalfold alignedpointwithC. Epoint C.Make a horizontalfoldwithalignedBCABCE5. Squash-fold onthe existingcreases.CCGFA5. Squash-foldGADF on ECGthe existing4.F Fold and unfold.B creases.3. Fold point F overto point A. CADCCBGBEBBFAEEDBFAFAGBCCBnot look exactly like this, dependingon the angleyou used and the length of your strip. Unfold toGstep 7.AFCBEDDAFDDBDEFAFcreases.8. Here’sit looks3. Fold pointF overwhatto pointA. like folded. Yours may2. FoldD lineE FG down to lie along edge AB.DGGAFDEGDCFA5. Squash-fold onthe existingCBB8. Here’sC whatcreases.it looks like folded. Yoursnot look exactly like this, depending on theFAyouusedand the length of your strip. UnfoGE 6. Fold point C to point A and unfold, makingD creaseE10.FoldAK down6.toFold point C to p5. MakeahorizontalfoldalignedwithpointC.6. Fold point C to point A and unfold,step 7. makingAM and unfold.a second longer hora second longerhorizontalcrease.a secondlongerhorizontal crease.CBBC
IGARobert Lang’s Angle Quintisection3. Fold point F overE to point A.7. Mountain-fold corner D behind.7. Here’s where it all happens. Fold edge AE down alongJAJ.Atthetime,fold theleft flap up so that 9.8. Here’swhatcreaseitElookslikeedgefolded.Yours7. Here’sE whereit all happens.FoldAEsamedownmayalongDpointF thetouchescreaseHIat the same point that edge AEnot looklikethis,dependingonupthesoangleJ exactlycrease AJ. At thesametime,foldleftthatJ flapyouused andthelengthyourstrip.toAE AJ. You will have to adjustD creaseD ofandpointCtouchespoint F touchesHI doesat thesamepointthatUnfoldedgecreasestep 7.I E happen at once.both foldsto makeall thealignmentsdoes and point C touches creaseAJ.YouwillhavetoadjustEKCBC foldsbothto make all the alignments happenatA once.J2. Fold line FG down to lie along edge AB.J7. Mountain-foldDE corner D behind.DCBH CFABGIBG5. DSquash-foldon A.3. Fold pointF over to pointthe existingcreases.GJGI DCBEEBDJDFAKJKIIELMBGCI Yours may8. Here’s what it looks like folded.FAnotGIlook exactly like this, depending on the angleFAyou used and the length of your strip. Unfold toG7. Here’s where it all happens. Fold edge AEalongstepdown7.EKCBcrease AJ. At the same 10.time,foldtheAKleftdownflaptoup so thatFoldcreaseJEKwhere it all pointhappens.FoldedgeAEdownalongDAMandunfold.JF touches crease HI at the same point that edge AEDLcrease AJ. At the sametime,left flapupAJ.so Youthatwill have todoesand foldpoint theC touchescreaseCB adjustbothto samemake allthe alignments happen at once. CtouchescreaseHIfoldsatYourstheEK8. pointHere’sFwhatit lookslike folded.may point that edge AE B9. BisectF A angle EAJ.GIJ8. Here’s what it looks likefolded.Yoursmay9. Bisectangle EAJ.andCtouchescreaseYou will haveto adjustnotdoeslookexactlylikethis,dependingon theAJ.angle5.pointSquash-foldonFADGEKnot look exactly like this,youdependingonthetheexistingangle of your strip. Unfold tousedandthelengthboth folds to make all the alignments happen at once.JJyou Cused and the lengthofyourstrip.Unfoldtocreases.Dstep 7.Dstep 7.IEEK6. Fold pointF A C to point A andEunfold,K makinga second longer horizontal crease.JEIDAGFAG9. Bisect angle EAJ.BFAIGCKJDMFAGLNMMJ12. Angle EAM is nowDdivided into fifths.LL10. Fold crease AK down toCAM and unfold.IGFAIJBC11.KKI DFABCMECJGCIKL8. Here’s what it looks like folded. Yours maynot look exactly like this, depending on the angleK of your strip. Unfold toyou used and the Jd with point6. Fold point C to point A and unfold, makingstepC.7.DAM and crease.unfold.AM anda secondunfold.LlongerhorizontalEEBJDLBCBEHFAIG’s where it all happens. Fold edge AE downI E along AAJ. At the same time, fold theA left flap up so that7. Mountain-fold corner D behind.touches crease HI at the same point that edge AEdDpointC touches crease AJ. You will have7.toHere’sadjustbehind.lds to make all the alignments happen at once.BJG DJunfold.HFAFAIFAMBGM CFA9. Bisect angle EAJ.EKJDK11. Bisect angleLAM.JLDEKLJ11. Bisect angle LAM.12. Angle EAM i
Robert Lang’s Angle QuintisectionDraw C’F’ to be the image of CF under the folding.
Origami Constructions TheoremHow to solve quintics with a 3-fold and Lill’s Method:
Origami Constructions TheoremTheorem: (Alperin and Lang, 2009)Every polynomial equation of degree n with real solutionscan be solved by an (n-2)-fold.
Overview of Origami-Math origami geometric constructions combinatorial geometry of origami matrix models and rigid foldability computational complexity of origami folding manifolds B(x,µ(x)) B(x,µ(x)) x x (a) (b) γ v 1 γ v 2 γ v γ 3 v4
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Origami is a paper folding art that emerged in Japan (Yoshioka, 1963). Origami has two types, classical origami and modular origami (Tuğrul & Kavici, 2002). A single piece of paper is used in classic origami. Different items, animal figures and two-dimensional geometric shapes can be made with classic origami.
of rigid-foldable origami into thick panels structure with kinetic mo-tion, which leads to novel designs of origami for various engineering purposes including architecture. 1 Introduction Rigid-foldable origami or rigid origami is a piecewise linear origami that is continuously transformable without the deformation of each facet. There-
Origami is the art of paper folding and has its origin in Japan. The word Origami comes from the two words, oru which means \to fold" and kami which means paper [2]. Although Origami is an ancient art, the interest from a mathematical point of view has increased during the last century. Speci cally, the area of Computational Origami began.
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For origami tutorials, please click on each title: Origami Butterfly Origami Sunglasses Origami Boat that Floats Origami Flapping Bird. Wednesday, March 18, 2020. Reflection: How can I build the Kingdom of God? Song: “The Wise Man Built His House Upon the Rock” .
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