MATHEMATICS AND ORIGAMI

273 Pages·2002·2.12 MB·English

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I Jesús de la Peña Hernández
W
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..................................................................................................................... 174
18.1 A kneading-trough....................................................................................................... 175
18.2 Pyramids...................................................................................................................... 179
18.2.1 Triangular pyramids..................................................................................................... 179
Tetrahedral. With tri-rightangled vertex.
18.2.2 Quadrangular pyramids................................................................................................ 180
Virtual. Triangle-equilateral.
18.2.3 Pentagonal pyramid..................................................................................................... 182
18.2.4 Hexagonal pyramid...................................................................................................... 183
18.2.5 Rhombic pyramid........................................................................................................ 184
18.3 Prisms.......................................................................................................................... 185
18.4 Truncated prism........................................................................................................... 185
18.5 Prism torsion: Obtention of prismoids......................................................................... 185
Triangled, quadrangled and pentagonal
18.6 Regular polyhedra........................................................................................................ 191
Relations within: dodecahedron, icosahedron and stellate penta-
gon.
18.7 Tetrahedron.................................................................................................................. 196
Pyramidal. Wound up. Bi-truncated. Ex-triangle. Skeletonlike
18.8 Cube............................................................................................................................. 197
Ex-rectangle. Cube of the sum of two numbers. Magic cube (Jer-
emy Shafer). With half (or double) volume. Laminar. Diophantine
cubes.
18.9 Octahedron................................................................................................................... 206
Bipyramidal. Wound up. Ex-tetrahedron. Made of two interlocked
domes. Skeletonlike.
18.10 Perforated pentagonal-dodecahedron........................................................................... 208
IV
18.11 Icosahedron.................................................................................................................. 210
18.12 Stellate regular polyhedra
18.12.1 Number 1..................................................................................................................... 211
18.12.2 Number 2.................................................................................................................... 215
18.12.3 Number 3..................................................................................................................... 216
18.12.4 Number 4..................................................................................................................... 217
18.13 Pseudorregular polyhedra............................................................................................ 220
18.13.1 Rhombic-dodecahedron............................................................................................... 220
18.13.2 Trapezohedron............................................................................................................. 223
18.14 Macles.......................................................................................................................... 225
Tetrahedral. Made of cubes. Aragonite. Cube-octahedral. Pyrito-
hedra. The iron cross.
19 Round solids................................................................................................................ 233
Sphere. Cylinder. Cone.
20 Paper flexibility.
20.1 Hooke´s law................................................................................................................. 236
20.2 The p number.............................................................................................................. 238
21 Quadrics....................................................................................................................... 241
21.1 Elliptic ellipsoid........................................................................................................... 241
21.2 Cyclic sections deformable ellipsoid (cyclic ellipsoid)............................................... 242
21.3 Hyperboloid................................................................................................................. 247
21.4 Hyperbolic paraboloid................................................................................................. 253
Acknowledgement, Bibliography................................................................................ 257
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