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Mathematics and Origami MATHEMATICS AND ORIGAMI WITH COMPLIMENTS TO THE SPANISH ORIGAMI ASOCIATION Jesús de la Peña Hernández This book has been written initially by the author for the SPANISH ORIGAMI ASOCIATION. This book may not be reproduced by any means neither totally nor partially without prior written permission of the Author. All rights reserved. © Jesús de la Peña Hernández Publisher: Second edition:. ISBN To my friend Jesús Voyer because of his abnegation to correct and enrich this book. INDEX 1 Origami resources to deal with points, straight lines and surfaces ............................. 1 Symmetry. Transportation. Folding. Perpendicular bisector. Bi- sectrix. Perpendicularity. Parallelism. Euler characteristic applied to the plane. Interaction of straight lines and surfaces. The right angle. Vertical angles. Sum of the angles of a triangle. 2 Haga´s theorem ............................................................................................................ 8 Demonstration, applications. 5 Corollary P................................................................................................................... 11 Demonstration, applications. 6 Obtention of parallelograms ........................................................................................ 12 Square from a rectangle or from other square. Rhomb from rec- tangles or squares. Rhomboid from a paper strip. Rectangles with their sides in various proportions. DIN A from any other rectangle or from any other DIN A. Argentic and auric rectangles. 6.7 Dynamic rectangles. Square roots................................................................................ 20 6.8 A rectangle from an irregular piece of paper ............................................................... 22 6.9 Stellate rectangle ......................................................................................................... 22 7 Geometry in the plane. Cartesian plane. Algebra ....................................................... 24 The area of a rectangle. Binomial product. Squares difference. Area of the other parallelograms and trapezium. Problems in the cartesian plane. Maxima and minimums. 7.7 Resolution of a quadratic equation ............................................................................. 29 Square root of a number. Square of a number. Parabola associated to the folding of a quadratic equation. 7.11 Complete equation of 3rd degree: Its resolution (J. Justin) .......................................... 33 Idem equation of 4th degree. Parabolas associated to the folding of a complete equation of 3rd degree. 7.14 Fundament of orthogonal billiards game (H. Huzita)................................................. 37 Squares and square roots. Cubes y cubic roots. The orthogonal spiral of powers. Resolution of a quadratic equation (H.H). Reso- lution of the complete equation of 3rd degree (H.H). 7.15 Arithmetic and geometric progressions ...................................................................... 46 I 8 Squares. Triangles. Various ......................................................................................... 50 Square with half the area of another one. Isosceles right-angled triangle from a square. Equilateral triangle from a square: Equal sides of triangle and square, or maximum equilateral triangle (4 solutions). Equilateral triangle: from a rectangle; envelope. Stel- late triangle. Square / Set square. Another curiosity. 8.2.8 Singular points in triangles .......................................................................................... 59 Orthocenter. Circumcenter. Baricenter. Incenter. 8.2.8.5 Rumpled and flattened origami.................................................................................... 61 8.2.8.6 Incenter and hyperbola................................................................................................. 63 8.2.8.7 Flattening of a quadrilateral......................................................................................... 65 8.3 Various......................................................................................................................... 67 Homotomic figures. Area of a triangle. Pythagorean theorem. 8.3.4 Pythagorean units......................................................................................................... 70 8.3.5 Unit squares (Jean Johnson) ........................................................................................ 73 9 Division in equal parts ................................................................................................. 75 Of a perigon. A square in two parts of equal area. Trisection of the angle of a square. A square in three equal parts (exact and ap- proximate solutions). A square in three equal parts after Haga´s theorem. 9.7 Trisection ..................................................................................................................... 80 Of a square (Corollary P). Of a square by trisection of its diago- nals. Of any angle. Idem after H. Huzita. 9.11 Division of a square in five equal parts ....................................................................... 84 Other inexact form of division. 9.13 Thales´ theorem: division of a rectangle in n equal parts ............................................ 85 9.14 Division of a square in 7 equal parts........................................................................... 86 Two approximate solutions. 9.15 Division of a square in 9 equal parts........................................................................... 88 Inexact and exact solutions. 9.16 Division in n parts after Corollary P............................................................................ 89 9.17 Division of a paper strip (Fujimoto´s method) ............................................................ 94 In 3 and 5 equal parts. II 9.18 Division of a paper strip by means of binomial numeration....................................... 98 10 Regular convex polygons with more than 4 sides ...................................................... 102 10.1 Pentagon ...................................................................................................................... 102 From an argentic rectangle. From a DIN A 4. From a paper strip made out of argentic rectangles. With a previous folding. Knot type.. 10.2 Hexagon....................................................................................................................... 108 With a previous folding. Knot type. 10.3 Heptagon...................................................................................................................... 109 H. Huzita´s solution. A quasi-perfect solution. Knot type solution. 10.4 Octagon........................................................................................................................ 113 10.5 Enneagon ..................................................................................................................... 114 11 Stellate polygons.......................................................................................................... 116 Pentagon (S. Fujimoto) and Heptagon. Flattening conditions. Hexagonal star (4 versions) 12 Tessellations ................................................................................................................ 123 By Forcher, Penrose, Chris K. Palmer, Alex Bateman and P. Taborda. 13 Conics .......................................................................................................................... 131 13.1 Circumference: Its center. As the envelope of its own tangents (inscribed within a square, or concentric with another one) 13.2 Origami and plückerian coordinates ............................................................................ 133 13.3 Ellipse .......................................................................................................................... 136 Its parameters. As envelope of its own tangents. Directrix. Poles and polars. Inscribed within a rectangle. Poncelet´s theorem. 13.4 Parabola ....................................................................................................................... 143 13.5 Hyperbola..................................................................................................................... 144 13.6 Another curves............................................................................................................. 145 Logarithmic spiral. Cardioid. Nephroid. 14 Topologic evocations................................................................................................... 150 14.1 Möbius´ bands.............................................................................................................. 150 14.2 Flexagons..................................................................................................................... 152 15 From the 2nd to the 3rd dimension ................................................................................ 156 III 16 Flattening: relation between dihedral and plane angles............................................... 162 17 Paper surfaces .............................................................................................................. 165 Real and virtual A conoid of paper ........................................................................................................ 169 A twisted column (salomonic)..................................................................................... 170 18 Polyhedra ....................................................................................................................
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