Mathematics in the Visual Arts

Ruth Scheps holds a doctorate in biochemistry from the Weizmann Institute of Science, Israel. She is also a radio producer and editor of the magazine Mikhtav Hadash.Marie-Christine Maurel is a biologist and Professor at Sorbonne University, France. She is also a researcher at the Institut de Systématique, Évolution, Biodiversité (CNRS, MNHN, SU and EPHE).

• Introduction ixRuth SCHEPS and Marie-Christine MAURELChapter 1 Infinity of God and Space of Men in Painting, Conditions of Possibility for the Scientific Revolution 1Giuseppe LONGO and Sara LONGO1.1 A brief introduction to infinity 11.2 Infinity in painting and the invention of mathematical space 21.3 Geometrical optics and the subject in projective space 81.4 The limit of time, calculus and algebra 161.5 Rational spaces: from trade to physics 181.6 Setting a priori conditions of representation and knowledge 201.7 Spaces of possibilities for the evolution of life? 231.8 Conclusion and opening: heterogeneous spaces of biological evolution 25Chapter 2 Geometry and the Life of Forms 29Ruth SCHEPS2.1 Introduction 292.2 Taking form 302.2.1 Early geometries 302.2.2 Geometrizing complexity 312.2.3 Morphogeneses 352.3 Art and geometry 372.3.1 Geometric art before its time 372.3.2 From geometric abstraction to digital art 382.3.3 Three legatees of geometric art 422.4 Beyond geometry 462.4.1 Quantic and cosmic 472.4.2 Outline and content 492.4.3 From form to the sublime 50Chapter 3 Among the Trees: Iterating Geneses of Forms, in Art and Nature 53Giuseppe LONGO and Sara LONGOChapter 4 The Passion of Flight: From Leonardo da Vinci to Jean Letourneur 63Bruno CHANETZ4.1 Introduction: from legend to reality 634.2 Leonardo da Vinci and the basis of the theory of flight 644.2.1 Chief engineer to Francis I of France 644.2.2 The flying propeller 654.2.3 Flapping-wing flight 674.2.4 Why can’t man fly like a bird? 684.2.5 The basis of Leonardo da Vinci’s theory of flight 694.3 Pioneers of the air and the first fluid movement visualizations 704.3.1 Clément Ader (1841 - 1925), a distant successor of Leonardo da Vinci, invents the aeroplane 704.3.2 The oil king presides over the surge in flight 714.3.3 From Magnus to Lanchester: the difficult gestation of the theory of flight 714.3.4 Gustave Eiffel highlights the suction component of lift force 724.3.5 Étienne-Jules Marey takes the first images of fluid movement 724.4 From Henri Werlé to Jean Letourneur, the sculptor of fluid movement 754.4.1 Henri Werlé or "the Master" of ONERA’s water tunnel 754.4.2 Jean Letourneur, interpreter of snapshots 774.4.3 As the 21st Century dawns, Jean Letourneur gathers momentum 794.5 Conclusion 814.6 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) 81Chapter 5 Sculptor of Fluid Movement 83Jean LETOURNEUR5.1 References 101Chapter 6 Internal Geometry of "Salvator Mundi" (The "Cook Version", Attributed to Leonardo da Vinci) 103Jean-Pierre CRETTEZ6.1 Introduction 1036.2 Distinctive features of the works of Leonardo da Vinci 1046.3 Presentation of the Salvator Mundi, Cook version 1066.4 Investigating the compositional mesh 1086.5 Compositional format 1126.6 Elements of the internal geometry of the Salvator Mundi, Cook version 1126.7 A detailed look at the ellipses of the head of the Salvator Mundi 1146.8 Visual consonance 1156.9 Properties of the type 1 ellipse 1166.10 Other applications of the type 1 ellipse 1176.11 The decoration of two intersecting bands of the stole 1186.12 The internal geometry of the Salvator Mundi (Ganay version) 1236.13 Conclusion 1256.14 References 126Chapter 7 Internal Geometry of a Night Scene by Georges de La Tour: "The Apparition of the Angel to St Joseph" 127Jean-Pierre CRETTEZ7.1 Introduction 1277.2 Methodology 1287.3 Distinctive features of the work of Georges de La Tour 1297.4 Internal geometry of The Appearance of the Angel to St Joseph 1307.5 The search for the compositional mesh 1327.6 Compositional format 1347.6.1 Position of the two protagonists’ heads 1347.7 The compositional architecture 1357.8 The ellipse of light 1367.9 Curved or elliptical forms 1377.10 Internal geometry of the two protagonists’ heads 1397.11 Discussion 1407.12 Compositional construction 1417.13 Conclusion 1417.14 References 142Chapter 8 Emergilience, an Art Research Project 143Sophie LAVAUD8.1 Background of the project Emergilience 1438.1.1 Preamble 1438.1.2 The artistic challenges 1468.1.3 A little bit of art history 1468.2 Description of the Emergilience project 1508.2.1 A continually developing research in progress 1508.2.2 What has already been achieved? 1518.2.3 Future developments 1688.3 Let us finish with a conclusion that looks to the future 1718.4 References 173List of Authors 175Index 177