GEOMETRY IN ART
Hilton Andrade de Mello
Translated from the Portuguese, “Geometria nas Artes”, by
Marcelo R. M. Crespo da Silva, Ph.D.
Professor Emeritus of Aerospace and Mechanical Engineering, Rensselaer Polytechnic Institute
Copyright ©2010 by Hilton Andrade de Mello. All rights reserved.
To Leonardo, Rafael, Ana Clara and Dominique, whose joy and creativity inspired me throughout this work.
And to the memory of Paula, that enlightened and extraordinary spirit that guided our paths with joy.
HAM (Hamello)
Do not let enter anyone who has no knowledge of geometry
In 387 BC Plato started his academy of philosophy in Athens, which existed until closed by Emperor Justinian in 529 AD. The words above were at the entrance to the academy.
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Table of Contents
Acknowledgments viii About the author ix Introduction x Chapter:
1 Abstraction and Geometry 1
2 Planar geometric forms 9
2.1 Euclid of Alexandria 10
2.2 The axiomatic method of Aristotle 10
2.3 Primitive concepts of Euclidean Geometry 11
2.4 Polygons 13
2.4.1 Definition 13
2.4.2 Polygon types 14
2.4.3 Polygon names 16
2.4.4 Triangles 16
2.4.5 Quadrilaterals 17
2.4.5.1 Parallelograms 17
2.4.5.2 Trapezoids 20
2.5 Circumference and circle 20
2.6 Inscribed and circumscribed polygons 22
2.7 Stars 23
2.8 Rosaceae 27
2.9 Spirals 30
2.9.1 Archimedes’ spiral 30
2.9.2 Bernoulli’s spiral 30
2.9.3 Spirals in our lives 31
3 Polyhedrons 33
3.1 Introduction 34
3.2 Convex and concave polyhedrons 35
3.3 Interesting families of polyhedrons 35
3.3.1 Platonic solids 35
3.3.2 Archimedean solids 36
3.3.3 Star solids 37
3.4. Other polyhedrons 39
3.4.1 Pyramid 39
3.4.2. Truncated pyramid 41
3.4.3 Straight prism 42
3.5 Polyhedrons and the great masters 43
3.6 Polyhedrons and informatics 45
3.7 Closing comments on polyhedrons 45
4 Other spatial figures 46
4.1 Sphere 47
4.2 Cone of revolution 49
4.3 Cylinder of revolution 51
4.4 Conics 52
4.4.1 Generalities 52
4.4.2 Ellipse and ellipsoid of revolution 53
4.4.3 Parabola and paraboloid of revolution 55
4.4.4 Hyperbola and hyperboloid of revolution 56
4.5 The conics and Paul Cézanne 58
4.6 Helices and helicoids 59
4.6.1 Helices 59
4.6.2 Helicoids 59
5 Composing a canvas with polygons: Tesselations 62
5.1 General concepts 63
5.2 Tessellations with regular polygons 63
5.3 Tessellations with irregular polygons 65
5.4 Tessellations and nature 66
5.5 Penrose tessellations (or tilings) 66
5.6 Final considerations 68
6 Perspective 69
6.1 Introduction 70
6.2 After all, what is a linear perspective? 71
6.3 Perspective with two vanishing points 74
6.4 Perspective with a larger number of vanishing points 75
6.5 Linear perspective and color perspective 79
6.6 Final considerations 79
7 The golden ratio 80
7.1 Fundamentals of the golden ratio 81
7.2 Golden geometric figures 82
7.2.1 Golden rectangles 82
7.2.2 Golden triangles 82
7.3 Mathematics of the golden ratio 83
7.4 The golden ratio in art and in nature 84
7.4.1 The golden ratio in architeture and in art 84
7.4.2 The golden ratio in nature 89
7.5 The golden ratio in aesthetics 89
7.6 Conclusions 90
8 Symmetry and specular image 91
8.1 Symmetry 92
8.2 Specular image 94
8.3 Symmetry in nature 95
9 Geometry and Symbolisms 97
9.1 The “Vesica Piscis” 98
9.2 Mandalas 100
9.3 Yantras 102
9.4 Labyrinths 103
9.5 The star of David, pentagrams, and hexagrams 104
9.6 Spirals and Symbolisms 106
9.7 Additional symbols to be investigated 106
9.8 Polyhedrons and symbolism: Platonic solids and the elements 107
10 Informatics and the arts 108
10.1 Introduction 109
10.2 Searching information on the Internet 109
10.3 Geometry software 109
11 Bibliography 110
12 Virtual references 113
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Acknowledgments
This project was made possible by the generosity and understanding of many artists who gave me permission to use images of their work.
In Brazil I am grateful to Abraham Palatnik, Almir Mavignier, Aluísio Carvão (Aluísio Carvão Filho), Antonio Maluf (Rose Maluf), Ascânio MMM, Decio Vieira (Dulce Maria Holzmeister), Geraldo de Barros (Lenora de Barros), Helio Oiticica (Cesar Oiticica), Ivan Serpa (Yves Henrique Cardoso Serpa), João Carlos Galvão, Lothar Charoux (Adriana Charoux), Luiz Sacilotto (Valter Sacilotto), Paiva Brasil and Rubem Ludolf.
Obviously, no justice is done to the importance of their work by viewing only one or two of their artwork, as they are, without any doubt, the exponents of the art in Brazil.
I am indebted to João Carlos Galvão, for making it possible for me to establish the contacts I have in the art world.
From abroad, I acknowledge the significant contributions I received from Dick Termes, Barry Stevens, George Hart, Philippe Hurbain, Russel Towle, Steve Frisby, and Dar Freeland.
My thanks to all, hoping that I have contributed to the dissemination of geometric art.
HAM (Hamello)
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About the author
Hilton Andrade de Mello is a graduate of the Federal University of Rio de Janeiro, Brazil, from which he received his electrical/electronics engineering degree, and his Nuclear Engineering degree, both in 1962. He also holds graduate degrees from Stanford University, U.S.A., where he majored in Electrical Engineering, concentrating in electronics. His scientific career has been, since graduation, at the National Nuclear Energy Commission (“Comissão Nacional de Energia Nuclear”), in Brazil, and he is the author and co-author of several electronics textbooks (listed in www.hamello.com.br). At present he is devoted to the Plastic Arts, and has many paintings to his credit.
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