The twentieth century was a time of unprecedented development in mathematics, as well as in all sciences: more theorems were proved and results found in a hundred years than in all of previous history. In *The Mathematical Century*, Piergiorgio Odifreddi distills this unwieldy mass of knowledge into a fascinating and authoritative overview of the subject. He concentrates on thirty highlights of pure and applied mathematics. Each tells the story of an exciting problem, from its historical origins to its modern solution, in lively prose free of technical details.

Odifreddi opens by discussing the four main philosophical foundations of mathematics of the nineteenth century and ends by describing the four most important open mathematical problems of the twenty-first century. In presenting the thirty problems at the heart of the book he devotes equal attention to pure and applied mathematics, with applications ranging from physics and computer science to biology and economics. Special attention is dedicated to the famous "23 problems" outlined by David Hilbert in his address to the International Congress of Mathematicians in 1900 as a research program for the new century, and to the work of the winners of the Fields Medal, the equivalent of a Nobel prize in mathematics.

This eminently readable book will be treasured not only by students and their teachers but also by all those who seek to make sense of the elusive macrocosm of twentieth-century mathematics.

forget that mathematics, more than any other art or science, is a young man's game." The prize was established in memory of John Charles Fields, the mathematician who came up with the idea and obtained the necessary funds. It consists of a medal bearing an engraving of Archimedes' head and the inscription Transire suum pectus mundoque potiri, "to transcend human limitations and to master the universe" (fig. 1.1). For this reason the prize is nowadays known as the Fields Medal. This award is

only 230 different types of spatial symmetry groups. A P P LIE D MAT HEM AT i c S 103 Fig. 34 Maurits Escher, Ghosts, 1971. pher Daniel Schechtman discovered an aluminum and manganese alloy having a molecular structure whose surface exhibits a symmetry of the same type, which no crystalline structure can have. These type of structures have been called quasicrystals. The discovery of quasicrystals demonstrates that group theory is not the ultimate tool for the description of nature, and

dimension r. Similarly, there are fractal surfaces with dimensions between 2 and 3. One example of these, called the Menger sponge, may be obtained as follows. Start with a cube and divide it into twenty-seven smaller cubes. Now remove the central seven of these smaller cubes (six on the faces and an internal one). Repeat this process infinitely many times (fig. 4.10). The dimension of the resulting surface is (approximately) 2.72, while the volume that it encloses is o. The above examples of

dimension r. Similarly, there are fractal surfaces with dimensions between 2 and 3. One example of these, called the Menger sponge, may be obtained as follows. Start with a cube and divide it into twenty-seven smaller cubes. Now remove the central seven of these smaller cubes (six on the faces and an internal one). Repeat this process infinitely many times (fig. 4.10). The dimension of the resulting surface is (approximately) 2.72, while the volume that it encloses is o. The above examples of

on multiple scale levels, from maritime coastlines to mountain chains, and they are also used in computer graphics to render realistic images of such objects (fig. 4.12). Due precisely to the rich variety of the applications of fractals, Mandelbrot received the Wolf Prize in 1993, not for mathematics but for physics.