The British mathematician and physicist, Sir Roger Penrose (born 1931), made striking and original contributions to the study of geometry, relativity, quantum mechanics, and the human mind.

Roger Penrose was born in Colchester, England, on August 8, 1931. His father was the geneticist Lionel Penrose, an expert on mental defects, whose interest in geometry was communicated to his son. The Penrose family was illustrious in British intellectual life in the 20th century. Jonathan Penrose won the British chess championship ten times in the 1950s and 1960s. It is not surprising that the intellectual life of the Penrose household was lively.

Penrose received his undergraduate degree from University College, London, and then proceeded to Cambridge for his doctorate. While an undergraduate he discovered a theorem concerning conic sections from which some of the basic theorems of projective geometry follow as special cases. As part of his work for his doctorate he rediscovered some important results in the theory of matrices. From 1964 to 1966 he was a reader in applied mathematics at Birkbeck College at the University of London, advancing to full professor in 1966.

The study of mathematics in Britain has always included a large amount of applied mathematics and even physics, so it is not unexpected that much of Penrose's best-known work looks more like physics than pure mathematics. He and Stephen Hawking studied black holes in collaboration and the two of them identified the basic characteristics of black holes, which result from the collapse of large stars. The mass becomes so concentrated that even photons (light particles) are unable to escape. As a result, even if it is possible to recognize the existence of a black hole from its effects on nearby objects, it would be impossible to observe the interior of the black hole itself.

Starting from his interest in the question of whether space and time are smooth or divided into discrete units, Penrose investigated many aspects of quantum mechanics. While he was at Cambridge, Penrose tried to build mathematical models for quantum mechanics using the basic elements of real numbers. One of the long-standing problems of 20th-century physics has been to combine the apparently conflicting fields of relativity and quantum mechanics. Penrose attempted to find a resolution via twistor geometry, which is based on complex numbers. This ambitious project remains far from completion, but the study of twistors has become an industry within physics in its own right.

Penrose collaborated with his father on the creation of a visual illusion that was incorporated into lithographs by the Dutch artist M. C. Escher, whose work included many mathematical elements. Also within the area of geometry, Penrose made a striking contribution to the study of tilings. A tiling is a method of covering the entire plane with polygons, for example squares or equilateral triangles. Tilings using those figures are called periodic because the pattern repeats regularly in moving about the plane. The question was whether it would be possible to tile (cover) the plane with a nonrepeating pattern.

Before Penrose made his contribution, others had already shown that it was possible to tile the plane in a nonperiodic fashion. The first solution used an immense number of different tiles, and the best solution known in 1974 still used six tiles of different shapes. In that year Penrose found a nonperiodic tiling using only two different shapes. Although this geometric contribution seems far removed from his studies of astrophysics and quantum mechanics, it also reflects the width of his scientific background.

In 1966 Penrose received the Adams Prize from Cambridge University and in 1971 the Dannie Heineman Prize for Physics from the American Physical Society. The next year he was elected to the Royal Society and in 1973 he succeeded to the prestigious Rouse Ball Chair of Mathematics at Oxford University. He shared two awards with his collaborator Stephen Hawking; the 1975 Royal Astronomical Society's Eddington Medal and the 1988 Wolf Prize for physics. Penrose held visiting positions at many leading universities in the United States including Cornell, Texas, California, and Princeton.

Penrose became known to the general public thanks to the best-selling book *The Emperor's New Mind,* which appeared on both sides of the Atlantic in 1989. Hawking had written a book to similar acclaim a couple of years before but had not tried to include any equations other than Einstein's e = mc2. Penrose's book includes that equation and hundreds of others as it ranges over computers, minds, and the laws of physics, to mention just the subjects explicitly named in the subtitle. *The Emperor's New Mind* may have been the best book about modern science yet written. Within 18 months it had run through numerous printings.

During a historic lecture series at the Isaac Newton Institute for Mathematical Sciences at Cambridge University in 1994, Penrose and Hawking recreated the famous Bohr-Einstein debate. In public lectures Penrose and Hawking presented their distinctive views on the universe, its evolution and impact on quantum theory. The same year, Penrose was knighted for his numerous contributions to science. *Shadows of the Mind* (1994) once again demonstrated the ability of Penrose to communicate complex theoretical physics to a general audience.

What distinguished Roger Penrose among the physicists and mathematicians of his time was the breadth and depth in his work. Some of the essays that he wrote illustrate the attention that he gave to his intellectual ancestors, such as Sir Isaac Newton. His influence on his students was profound.

## Further Reading on Roger Penrose

There is an article on Penrose in the McGraw-Hill set on *Modern Scientists and Engineers* (1980). A more personal glimpse is available in Martin Gardner's introduction to Penrose's *The Emperor's New Mind* (1989). A good discussion of tilings and Penrose's work is in B. Grunbaum and G. Shephard's book *Tilings and Patterns* (1986). Articles on Penrose can be found in the popular science journals *Scientific American and Science.* An account of the 1994 Penrose-Hawking debate is presented in *The Nature of Space and Time* (1996).