This talk gives an introduction to Leyton's new book The Structure of Paintings in Springer-Verlag. In a series of 4 books, Leyton has developed new foundations for geometry in which shape is equivalent to memory storage. A principal argument of these foundations is that artworks are maximal memory stores. At the basis of this geometry are Leyton's fundamental laws of memory storage, and these laws are shown to determine the structure of artworks. That is, the central argument is that artworks are structured so that they allow the maximal extraction of stored memory. Furthermore, the book demonstrates that the emotion expressed by an artwork is actually the memory extracted by the laws. Therefore, the laws of memory storage allow the systematic and rigorous mapping not only of the compositional structure of an artwork, but also of its emotional expression. This fundamentally opposes the view that the emotional expression of an artwork is undefinable. Leyton's methodology makes the structure and emotional content of an artwork fully definable, rich, systematic and complete. The argument is supported with detailed analyses of paintings by Picasso, Raphael, Cézanne, Gauguin, Modigliani, Ingres, De Kooning, Memling, Balthus and Holbein.