Geometry and the Imagination

The princeton companion is nice to have around, but you do not really read it end to end. Spivak's calculus you bring to the beach and read it between swim and swim. EDIT: Also, some books by Hilbert are breathtakingly beautiful: Geometry and the Imagination (just the chapter on synthetic differential geometry is worth more than 10 other great books), and the Methods of Mathematical Physics is also great. It begins by giving three proofs of cauchy-schwartz inequality, and then goes on to give several different definitions of the eigenvectors of a matrix. Both of those make great beach readings for this summer.

All the books cited here by Arnold, Spivak, and Lanczos are extraordinarily good. Nobody has mentioned yet "Geometry and the Imagination" by Hilbert and Cohn-Vossen. If there is a Feynman equivalent in math it is certainly this book. For elementary geometry, the Feynman equivalent is probably "Introduction to Geometry", by H.S.M.Coxeter. Beautifully written, figures on every page, covers all geometric topics (affine, projective, ordered, differential, ...) For differential geometry, nothing beats "A Panoramic view of Differential Geometry" by Berger. It is a stunning comprehensive overview of the whole field, focused on the meaning and the applications of each part and, strangely for a math book, with no formal proofs. Only the main ideas of the proof and the relationships between them are given, but this allows to fit the whole subject into a single, manageable whole.