Mathematics in the Visual Arts

1 Cover

2 Title page Series Editor Marie-Christine Maurel

3 Copyright First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2020 The rights of Ruth Scheps and Marie-Christine Maurel to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2020942150 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-681-4

4 Introduction

5 1 Infinity of God and Space of Men in Painting, Conditions of Possibility for the Scientific Revolution 1.1. A brief introduction to infinity 1.2. Infinity in painting and the invention of mathematical space 1.3. Geometrical optics and the subject in projective space 1.4. The limit of time, calculus and algebra 1.5. Rational spaces: from trade to physics 1.6. Setting a prioriconditions of representation and knowledge 1.7. Spaces of possibilities for the evolution of life? 1.8. Conclusion and opening: heterogeneous spaces of biological evolution

6 2 Geometry and the Life of Forms 2.1. Introduction 2.2. Taking form 2.3. Art and geometry 2.4. Beyond geometry

7 3 Among the Trees: Iterating Geneses of Forms, in Art and Nature

8 4 The Passion of Flight: From Leonardo da Vinci to Jean Letourneur 4.1. Introduction: from legend to reality 4.2. Leonardo da Vinci and the basis of the theory of flight 4.3. Pioneers of the air and the first fluid movement visualizations 4.4. From Henri Werlé to Jean Letourneur, the sculptor of fluid movement 4.5. Conclusion 4.6. Appendix: additions to the chapter entitled “Why Can’t Man Fly?”, which refers to the article by Marielle Vergès and Kamil Fadel (see footnote 15)

9 5 Sculptor of Fluid Movement 5.1. References

10 6 Internal Geometry of “Salvator Mundi” (The “Cook Version”, Attributed to Leonardo da Vinci) 6.1. Introduction 6.2. Distinctive features of the works of Leonardo da Vinci 6.3. Presentation of the Salvator Mundi, Cook version 6.4. Investigating the compositional mesh 6.5. Compositional format 6.6. Elements of the internal geometry of the Salvator Mundi, Cook version 6.7. A detailed look at the ellipses of the head of the Salvator Mundi 6.8. Visual consonance 6.9. Properties of the type 1 ellipse 6.10. Other applications of the type 1 ellipse 6.11. The decoration of two intersecting bands of the stole 6.12. The internal geometry of the Salvator Mundi(Ganay version) 6.13. Conclusion 6.14. References

11 7 Internal Geometry of a Night Scene by Georges de La Tour: “ The Apparition of the Angel to St. Joseph” 7.1. Introduction 7.2. Methodology 7.3. Distinctive features of the work of Georges de La Tour 7.4. Internal geometry of The Appearance of the Angel to St. Joseph 7.5. The search for the compositional mesh 7.6. Compositional format 7.7. The compositional architecture 7.8. The ellipse of light 7.9. Curved or elliptical forms 7.10. Internal geometry of the two protagonists’ heads 7.11. Discussion 7.12. Compositional construction 7.13. Conclusion 7.14. References

12 8 Emergilience, an Art Research Project 8.1. Background of the project Emergilience 8.2. Description of the Emergilienceproject 8.3. Let us finish with a conclusion that looks to the future 8.4. References

13 List of Authors

14 Index

15 End User License Agreement

1 Chapter 1Figure 1.1. Giotto di Bondone, Life of St. Francis, fresco, around 1290. Assisi,...Figure 1.2. Ambrogio Lorenzetti, Annunciation, tempera on wood, Siena, Pinacotec...Figure 1.3. Antonello da Messina, St. Sebastian, tempera on wood transposed onto...Figure 1.4. Brunelleschi’s experiment with linear perspective, looking through t...Figure 1.5. Andrea Mantegna, Study for the Dead Christ, pen and brown wash on pa...Figure 1.6. Paolo Uccello, Battle of San Romano , detail, 1456, tempera on wood. ...Figure 1.7. Andrea Mantegna, The Dead Christ, tempera on canvas, Milan, Pinacote...Figure 1.8. Piero della Francesca, Polyptych of Perugia, 1470, Perugia, National...Figure 1.9. Polyptych of Perugia, upper register (see Figure 1.8) Figure 1.10. De Martone, art. cit. Figure 1.11. Paolo Uccello, Battle of San Romano, 1456, tempera on wood. London,...

2 Chapter 2Figure 2.1. Reuven Berman Kadim, Hovering Object #1 . Digital image, 1997 11Figure 2.2. Reuven Berman Kadim, Paving B , Digital Image, 1996 12Figure 2.3. Emmanuel Van der Meulen, Quadrum, 2017, acrylic on canvas, 130 × 130...Figure 2.4. Emmanuel Van der Meulen, Bethel, 2017, acrylic on canvas, 130 × 130 ...Figure 2.5. Esther Stocker, Untitled, 2010, acrylic on canvas, 200 × 300 cm. Cou...Figure 2.6. Esther Stocker, Unlimited Space, 2013, Roudnice, Czech Republic. Cou...

3 Chapter 3Figure 3.1. Examples of Luca Caciagli’s Komorebi project, showing fractals in na...Figure 3.2. Examples of simple mathematical fractals, from left to right: the Ko...Figure 3.3. Examples of Luca Caciagli’s Komorebi project, showing contradictions...Figure 3.4. René Magritte, Le blanc-seing ( The Blank Signature in English), oil ...Figure 3.5. Simon Hantaï, Étude, oil on canvas, 275 × 238 cm, 1969. National Gal...Figure 3.6. An example of where fractals can be found in the stock market

4 Chapter 4Figure 4.1. Drawing by Leonardo da Vinci. Courtesy of the Library of the Institu...Figure 4.2. Remanta microdrone. Courtesy of ONERA Figure 4.3. Drawing by Leonardo da Vinci of bat wings and Clément Ader's Eole in...Figure 4.4. Marey’s Smoke wind tunnel Figure 4.5. Visualizations created by Marey. Courtesy of Cinémathèque françaiseFigure 4.6. ONERA’s water tunnel at Châtillon (Hauts-de-Seine) Figure 4.7. Visualizations of vortex windings on Concorde. Courtesy of ONERA Figure 4.8. Visualization from the water tunnel of a Citroën DS. Courtesy of ONE...Figure 4.9. The Mirror 1994. Marble, 100 × 49 × 49 cm visualization of an ellips...Figure 4.10. Interférences de chocs, 2005, Stuc, 61 × 7 cm, International Year o...Figure 4.11. Schlieren visualizations carried out at ONERA’s R3Ch wind tunnel in...

5 Chapter 5Figure 5.1. Ink, around 1510/1513, Windsor, 15 × 17 cm Figure 5.2. Leda , black stone and ink, around 1505/1510, Windsor, 20 × 16.2 cm Figure 5.3. Ink, around 1515, Venice, Accademia Gallery, 9.6 × 14.9 cm Figure 5.4.