Mathematics and Modern Art : Proceedings of the First ESMA Conference held in Paris July 19 22 2010 Preface A Mathematician and an Artist. The Story of a Collaboration by R.Palais Dimensions a Math Movie by A.Alvarez J.Leys Old and new Mathematical Models: saving the Heritage of the Institute Henri Poincaré by F.Apéry An Introduction to the Construction of some Mathematical Objects by C.P.Bruter Computer Mathematics and Art by J. F.Colonna Structure of Visualization and Symmetry in iterated Function Systems by J.Constant Polyhedral eversions of the sphere; gastrulation by R.Denner M.C. Escher’s Use of the Poincaré Models of Hyperbolic Geometry by D.Dunham Mathematics and Music Boxes by V.Hart Mes Gravures Mathématiques by P.Jeener Knots and Links As Form Generating Structures by D.Kozlov Geometry and Art from the Cordovan Proportion by A.Redondo Buitrago E.Reyes Dynamic Surfaces by S. Salamon Pleasing Shapes for Topological Objects by J.Sullivan Rhombopolyclonic Polygonal Rosettes Theory by F.Tard . The link between mathematics and art remains as strong today as it was in the earliest instances of decorative and ritual art. Arts architecture music and painting have for a long time been sources of new developments in mathematics and vice versa. Many great painters have seen no contradiction between artistic and mathematical endeavors contributing to the progress of both using mathematical principles to guide their visual creativity enriching their visual environment with the new objects created by the mathematical science. Owing to the recent development of the so nice techniques for visualization while mathematicians can better explore these new mathematical objects artists can use them to emphasize their intrinsic beauty and create quite new sceneries. This volume the content of the first conference of the European Society for Mathematics and the Arts (ESMA) held in Paris in 2010 gives an overview on some significant and beautiful recent works where maths and art including architecture and music are interwoven. The book includes a wealth of mathematical illustrations from several basic mathematical fields including classical geometry topology differential geometry dynamical systems. Here artists and mathematicians alike elucidate the thought processes and the tools used to create their work.