12 Alperin,Roger & J. Lang, Robert. (2009). One,Two, and Multi-Fold Origami Axioms. Origami4: 4th International Meeting ofOrigami Science, Mathematics and Education, 4OSME 2006.

2017.11 RobertJ. Lang, ‘Angle Quintisection’, Robert J.

Lang Origami website, n.d., 10 RobertJ. Lang, ‘Origami Diagramming Conventions’, Robert J. Lang Origamiwebsite, n.

d.,

com/article/origami-diagramming-conventions>Accessed 31 Oct. 2017. 9 Wikipedia,’Lill’s method’ website, n.d.,

org/wiki/Lill%27s_method#Description_of_the_method>Accessed 24 Nov. 2017. 8 RobertJ. Lang, ‘Origami Diagramming Conventions’, Robert J. Lang Origamiwebsite, n.

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com/article/origami-diagramming-conventions>Accessed 31 Oct. 2017. 7 RobertJ. Lang, Origami 4, 1st edn, CRC Press, Boca Raton, 2009, pg. 371-394. 6British Origami Society, Origami In Education And Therapy: Proceedings OfThe First International Conference In Origami In Education And Therapy (COET’91), 2nd edn, CreateSpaceIndependent Publishing Platform, n.p.

, 2016, pg. 37-70. 5Jacques Justin, ‘Résolution Par Le Pliage De L’Équation Du Troisième Degré EtApplications Géométriques’, L’ouvert online journal, 42, Mar. 1986,pg. 9-19,

unistra.fr/fileadmin/upload/IREM/Publications/L_Ouvert/n042/o_42_9-19.pdf>Accessed 31 Oct. 2017. 4 RobertJ. Lang, Origami Design Secrets, 1st edn, CRC Press, Boca Raton, 2003,pg. 13. 3 ThomasC.

Hull, ‘Solving Cubics With Creases: The Work Of Beloch And Lill’, TheAmerican Mathematical Monthly online journal, 118/4, Apr. 2011, pg. 307,

118.04.307-hull.pdf> Accessed 25 Oct. 2017.

2Tandalam S. Row, Wooster W. Beman & David E. Smith, T. Sundara Row’sGeometric exercises in paper folding, 3rd edn, Open Court Pub. Co, Chicago,1917, intro. xiv. 1 Joseph Wu,’Origami: A Brief History of the Ancient Art of Paperfolding’, Joseph WuOrigami website, 15 July 2006, para.

1,

We concluded with Lang’s mind-blowing phenomenon in which aquintic equation is actually possible to solve with the method of angle quintisectionand essentially challenging the work of Galois. To end, we proved Pythagoras’Theorem using the Huzita-Hatori axioms and the common folds. We magnificentlyremoved 4 identical triangles from both pieces of paper, parting us with twoareas: and . As the triangles were identical, the same total area wasremoved, which showed that .

Conclusion (This diagram is a new revision of theone which was made as a collective in the CRM group project.)We begin this diagrammaticproof with two square pieces of paper, and proceed as follows:Proof: a2 + b2 = c2Let T be a right angledtriangle, with hypotenuse length c and remaining two sides length a andb. Pythagoras’ theorem states that the 3 sides of T are relatedby the following equation:Theorem: Section5: The Pythagorean TheoremSection4: Quintic equations are solvable by Origami Theseven Huzita-Hatori axioms which were mentioned in the first section definewhat is physically feasible to construct by making successive single foldsformed by aligning groupings of points and lines. It has been mathematicallyproven that there are only the seven axioms, and that those folds authorize theconstruction of solutions to quadratics, cubics and even quartics, but it stopsthere. 11 Whilethe Huzita-Hatori axioms only permit the solution of equations up to degree 4,by elevating beyond the limitations that define that set, it is possible tosolve equations of much higher degree through origami. The crucial aspect ofthe Huzita-Hatori axioms is they individually describe an action in which asingle fold is defined by positioning numerous combinations of points andlines.

11 Onthe other hand, it is also allowable to define two, three, or more simultaneousfolds by forming several alignment combinations. Some of these alignments canbe broken down into sequences of Huzita-Hatori folds, but others, which we callnon-separable, cannot be broken down into simpler units. These more complicatedalignments form an entirely new class of origami “axioms”, whichpotentially can solve equations of considerably higher order. 11 Theorem: Everypolynomial equation of degree with real solutions can be solved by simultaneous folds. 12 Thefollowing figure below shows the key step in performing an origami anglequintisection (division into fifths with the use of folds only):11 If you would like moreinformation on angle quintisection then refer to the appropriate referencewhich directs to Robert Lang’s website at the end of this paper where you canfind his detailed diagrammatic folding sequence for the angle quintisection inPDF form. Section3: Cubic equations are solvable by Origami 3.1: The Beloch Fold (Axiom A6) 3 A way to noticewhat the Beloch Fold is demonstrating is to consider one of the point-linepairs. If we fold a point to a line , the resulting crease line is tangential to theparabola withfocus and directrix (theequidistant set of lines from and ).

This can be demonstrated by the following activity: Proof: (See Step 1 above) 3Afterfolding a point on to , draw a line perpendicular to the folded image of , on the folded flap of paper from to the creaseline. If is the pointwhere this drawn line intersects the crease line, then we see when unfoldingthe paper that the point is equidistantfrom the point and the line . Any other point on the crease line will beequidistant from and and thus willnot have the same distance to the line .

Therefore, the crease line is tangent to theparabola with focus and directrix 3. Thisessentially means that folding a point to a line implies locating a point on acertain parabola and is comparable to solving a quadratic equation. The BelochFold discovers a common tangent to two parabolas. Two parabolas drawn in theplane can have a maximum of three distinct common tangents, hence this origamifold is equivalent to solving a cubic equation.

Ruler and compass constructions,on the other hand are only capable of solving quadratic equations. 3.2: The Beloch Square 3 Given twopoints A and B and two lines and in the plane,construct a square with twoadjacent corners and lying on and , respectively, and the sides and , or their extensions, passing through and , respectively. The Beloch Square is shown below: 3 3 3.3: Constructing Take to be the -axis and to be the -axis of the plane. Let and .

Then we construct the lines to be and to be . Folding onto and onto using theBeloch fold will make a crease which crosses at a point and s at apoint . Consulting the Beloch square diagram, if we let be the origin,then notice that , , and are all similarright triangles. This follows from the fact that isperpendicular to and . 3 Sowe have , where denotes the length of the segment.

Put and yields and so ), (Discovered by Martin). 3.4: Solving cubicequations using Lill’s method 9 Lill’s method 9:”Toemploy the method adiagram is drawn starting at the origin O. A line segment is drawn rightwardsby the magnitude of the first coefficient (the coefficient of the highest-powerterm) (so that with a negative coefficient the segment will end left of theorigin). From the end of the first segment another segment is drawn upwards bythe magnitude of the second coefficient, then left by the magnitude of thethird, and down by the magnitude of the fourth, and so on.

The sequence ofdirections (not turns) is always rightward, upward, leftward, downward, thenrepeating itself. Thus each turn is counterclockwise. The process continues forevery coefficient of the polynomial including zeroes, with negativecoefficients “walking backwards”.

The final point reached, at the endof the segment corresponding to the equation’s constant term, is the terminus T9.” “A line is then launched from the origin at some angle , reflected off of each line segmentat a right-angle (not necessarily the “natural” angle of reflection),and refracted at a right angle through the linethrough each segment (including a line for the zero coefficients) when theangled path does not hit the line segment on that line. The vertical and horizontallines are reflected off or refracted through in the following sequence: theline containing the segment corresponding to the coefficient of , then of , etc. Choosing ? so that the pathlands on the terminus, the negative of the tangent of ? is a root of thispolynomial. For every real zero of the polynomial there will be one uniqueinitial angle and path that will land on the terminus. A quadratic with tworeal roots, for example, will have exactly two angles that satisfy the above conditions9.

” Twoexamples of Lill’s method being applied to a quintic is shown below for avisual understanding 3: Problem:Given a polynomial , locate a real root of .Claim: is a root of 9 Proof: 3Let be the length of the side opposite the angle in the triangle whose side adjacent to is part of the segment length. Case1: All coefficients are positive -, =,=, …,but is indeed a root of Case2: All coefficients are negative -,) =)),, …,but is indeed a root of Case3: All coefficients are zero (we prove for completeness, but it is trivial that for this case) -,,, … is indeed a root of Inthe cubic case, the line of refraction (angled path) for the general cubicpolynomial will have four sides, so our line of refraction (angled path) willhave three sides. If we think of as the point , and as the point , and we think of the linescontaining the -side and the -side as the lines and respectively, then a Beloch square withadjacent corners on and and opposite passing through and will give us a line of refraction. This isshown in the figure 3 below.

We havefinally demonstrated that paper folding can be used to perform Lill’s method inthe cubic case and thus solve general polynomials of degree three.3 Origami and Geometry: Solving Polynomial Equations & proving Pythagoras’ Theorem using Origami Faculty of Mathematical Sciences University of Southampton Student ID: 28229428 AbstractOrigami is defined to mean”fold paper” in Japanese and is renowned as a delicate art form, however italso comprises of geometric concepts which relate to a variety of fields inmathematics. This report demonstrates an understanding of Margharita P.Beloch’s and Lill’s work in which we can solve general cubic polynomials andextend this to higher order polynomials to also solve the quintic equationusing origami which was explored by Robert Lang. We conclude with a visualproof of the Pythagorean Theorem to illustrate the contrast how origami can beused not only to solve algebraic problems such as polynomials but alsogeometric problems.IntroductionThe art of paper folding hasbeen around since the 6th Century 1, however the firstmathematical publication on Origami was released in 1893 when Tandalam S.

Rao published his book”Geometric Exercises in Paper Folding”. In this paper, Rao used paper foldingto demonstrate proofs of geometric constructions. 2 Roughly 40years later, Margharita P. Beloch then went on to show that it was possible touse Origami to solve the general cubic equation. 3 Section 1 of this reportintroduces the seven Huzita-Hatori axioms, which are the key foundations forpaper folding.

The common folds of Origami appear in Section 2, along with theYoshizawa-Randlett system. The purpose of this system is to define thedifferent folds in a way that straightforwardly exemplifies the necessarynotions of paper folding and has since become the default universal system.4 Section 3 of this report discusses Beloch’s use of Lill’s method forsolving cubic polynomial equations. Section 4 extends to solving the quinticequation with the assistance of 3-fold origami. To end, we take advantage oforigami and geometry to prove a well-renowned mathematical theorem.

Section 5 of thisreport contains an illustrative proof for the Pythagorean Theorem, which statesthat in a right angled triangle, the square of the hypotenuse is equal to thesum of the squares of the two remaining sides. Our approach method involvedusing Pureland Origami, Kirigami and simple geometry to transform a squarepiece of paper into a visual representation of the theorem. Section1: Huzita-Hatori Axioms The Huzita-Hatori axioms, occasionally called the Huzita-Justin axioms,were first discovered in 1986 by Jacques Justin 5,then rediscovered and reported at the First International Conference on Origamiin Education and Therapy by Humiaki Huzita in 1991. 6These axioms describe, using a combination of pre-existing points and lines,every possible situation in which a single fold can be carried out. 7 Section2: Yoshizawa-Randlett System and the Common Folds 2.

1: The Yoshizawa-Randlett System Origami comprises oftransforming a 2D flat sheet of paper into an arbitrary 3D structure, so it isassistive to make use of symbols 8 and annotations to describe theiterations. The Yoshizawa-Randlett system below is universallyrecognised as the universal standard origami diagramming system. Thesenotations are commonly used in step by step origami guides, along with some oforigami’s common folds. Additionally, we use a straight line to represent acrease.

2.2: Common Folds