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WITH COMPLIMENTS TO THE SPANISH ORIGAMI ASOCIATIONMATHEMATICSANDORIGAMIJesús de la Peña Hernández

This book has been written initially by the author for the SPANISH ORIGAMIASOCIATION.

This book may not be reproduced by any meansneither totally nor partially without prior writtenpermission of the Author. All rights reserved. Jesús de la Peña HernándezPublisher:Second edition:.ISBN

To my friend Jesús Voyerbecause of his abnegation to correctand enrich this book.

INDEX1Origami resources to deal with points, straight lines and surfaces .Symmetry. Transportation. Folding. Perpendicular bisector. Bisectrix. Perpendicularity. Parallelism. Euler characteristic appliedto the plane. Interaction of straight lines and surfaces. The rightangle. Vertical angles. Sum of the angles of a triangle.12Haga s theorem .Demonstration, applications.85Corollary P . 11Demonstration, applications.6Obtention of parallelograms . 12Square from a rectangle or from other square. Rhomb from rectangles or squares. Rhomboid from a paper strip. Rectangles withtheir sides in various proportions. DIN A from any other rectangleor from any other DIN A. Argentic and auric rectangles.6.7Dynamic rectangles. Square roots. 206.8A rectangle from an irregular piece of paper . 226.9Stellate rectangle . 227Geometry in the plane. Cartesian plane. Algebra . 24The area of a rectangle. Binomial product. Squares difference.Area of the other parallelograms and trapezium. Problems in thecartesian plane. Maxima and minimums.7.7Resolution of a quadratic equation . 29Square root of a number. Square of a number. Parabola associatedto the folding of a quadratic equation.7.11Complete equation of 3rd degree: Its resolution (J. Justin) . 33Idem equation of 4th degree. Parabolas associated to the foldingof a complete equation of 3rd degree.7.14Fundament of orthogonal billiards game (H. Huzita). 37Squares and square roots. Cubes y cubic roots. The orthogonalspiral of powers. Resolution of a quadratic equation (H.H). Resolution of the complete equation of 3rd degree (H.H).7.15Arithmetic and geometric progressions . 46I

8Squares. Triangles. Various . 50Square with half the area of another one. Isosceles right-angledtriangle from a square. Equilateral triangle from a square: Equalsides of triangle and square, or maximum equilateral triangle (4solutions). Equilateral triangle: from a rectangle; envelope. Stellate triangle. Square / Set square. Another curiosity.8.2.8Singular points in triangles . 59Orthocenter. Circumcenter. Baricenter. Incenter.8.2.8.5Rumpled and flattened origami. 618.2.8.6Incenter and hyperbola. 638.2.8.7Flattening of a quadrilateral . 658.3Various. 67Homotomic figures. Area of a triangle. Pythagorean theorem.8.3.4Pythagorean units. 708.3.5Unit squares (Jean Johnson) . 739Division in equal parts . 75Of a perigon. A square in two parts of equal area. Trisection of theangle of a square. A square in three equal parts (exact and approximate solutions). A square in three equal parts after Haga stheorem.9.7Trisection . 80Of a square (Corollary P). Of a square by trisection of its diagonals. Of any angle. Idem after H. Huzita.9.11Division of a square in five equal parts . 84Other inexact form of division.9.13Thales theorem: division of a rectangle in n equal parts . 859.14Division of a square in 7 equal parts. 86Two approximate solutions.9.15Division of a square in 9 equal parts. 88Inexact and exact solutions.9.16Division in n parts after Corollary P . 899.17Division of a paper strip (Fujimoto s method) . 94In 3 and 5 equal parts.II

9.18Division of a paper strip by means of binomial numeration. 9810Regular convex polygons with more than 4 sides . 10210.1Pentagon . 102From an argentic rectangle. From a DIN A 4. From a paper stripmade out of argentic rectangles. With a previous folding. Knottype.10.2Hexagon . 108With a previous folding. Knot type.10.3Heptagon. 109H. Huzita s solution. A quasi-perfect solution. Knot type solution.10.4Octagon. 11310.5Enneagon . 11411Stellate polygons. 116Pentagon (S. Fujimoto) and Heptagon. Flattening conditions.Hexagonal star (4 versions)12Tessellations . 123By Forcher, Penrose, Chris K. Palmer, Alex Bateman and P.Taborda.Conics . 131Circumference: Its center. As the envelope of its own tangents(inscribed within a square, or concentric with another one)Origami and plückerian coordinates . 1331313.113.213.3Ellipse . 136Its parameters. As envelope of its own tangents. Directrix. Polesand polars. Inscribed within a rectangle. Poncelet s theorem.13.4Parabola . 14313.5Hyperbola. 14413.6Another curves. 145Logarithmic spiral. Cardioid. Nephroid.14Topologic evocations. 15014.1Möbius bands. 15014.2Flexagons . 15215From the 2nd to the 3rd dimension . 156III

16Flattening: relation between dihedral and plane angles . 16217Paper surfaces . 165Real and virtualA conoid of paper . 169A twisted column (salomonic). 17018Polyhedra . 17418.1A kneading-trough . 17518.2Pyramids . 17918.2.1Triangular pyramids. 179Tetrahedral. With tri-rightangled vertex.18.2.2Quadrangular pyramids. 180Virtual. Triangle-equilateral.18.2.3Pentagonal pyramid . 18218.2.4Hexagonal pyramid. 18318.2.5Rhombic pyramid . 18418.3Prisms . 18518.4Truncated prism . 18518.5Prism torsion: Obtention of prismoids. 185Triangled, quadrangled and pentagonal18.6Regular polyhedra. 191Relations within: dodecahedron, icosahedron and stellate pentagon.18.7Tetrahedron. 196Pyramidal. Wound up. Bi-truncated. Ex-triangle. Skeletonlike18.8Cube. 197Ex-rectangle. Cube of the sum of two numbers. Magic cube (Jeremy Shafer). With half (or double) volume. Laminar. Diophantinecubes.18.9Octahedron. 206Bipyramidal. Wound up. Ex-tetrahedron. Made of two interlockeddomes. Skeletonlike.18.10Perforated pentagonal-dodecahedron. 208IV

18.11Icosahedron . 21018.1218.12.1Stellate regular polyhedraNumber 1 . 21118.12.2Number 2 . 21518.12.3Number 3 . 21618.12.4Number 4 . 21718.13Pseudorregular polyhedra . 22018.13.1Rhombic-dodecahedron . 22018.13.2Trapezohedron . 22318.14Macles. 225Tetrahedral. Made of cubes. Aragonite. Cube-octahedral. Pyritohedra. The iron cross.19Round solids . 233Sphere. Cylinder. Cone.20Paper flexibility.20.1Hooke s law . 23620.2The p number. 23821Quadrics . 24121.1Elliptic ellipsoid. 24121.2Cycli

Origami receives more sophisticated geometrical help to design folding bases. In passing, I tackle this matter when dealing with the triangle s incenter and its related hyperbola. But one who masters this subject is our ingenious creator Anibal Voyer. Not long ago, he lectured on that in the conference held at the Spanish Institute of Engineering under the title of Engineering, origami and .