## Monsters in the Moonlight

The ancient Greek word symmetria means balance. In fact, almost everything we find beautiful has symmetries: faces, flowers, buildings, landscapes, cars. It can be found everywhere in nature, from the AIDS virus to snowflakes to volcanoes. According to studies, people find faces the more beautiful the more symmetrical they are.

Determining the mysteries of symmetry mathematically turns out to be amazingly intricate. Marcus du Sautoy, mathematics professor at the University of Oxford, tells the exciting story of the search for symmetry, embedded in his personal biography: The story begins on his 40th birthday in the Sinai desert and ends a year later, on his 41st, when he gave a lecture holds at a conference. In between, duSautoy not only takes the reader to the leading mathematical institutes, but also to the Alhambra in Granada, Spain, to medieval Arabia and to the Paris of the French Revolution.

The classical symmetries are those of the five platonic solids. Other forms of regularity can be found on the walls of medieval Moorish palaces. Starting with paving with square and hexagonal stones, the builders sought sophisticated patterns with interesting repetitions. DuSautoy is traveling to Granada with his nine-year-old son. "For the mathematician, a visit to the Alhambra is something of a pilgrimage," he claims. As an expert, he knows that there are exactly 17 different symmetry classes in two dimensions, which he finally finds alone in the palace after some rummaging around.

The researchers are of course not satisfied with plane and spatial figures. They like to escape into four or even higher dimensional spaces that mere mortals can hardly imagine. And not just out of pure intellectual curiosity. Symmetries in the fifth dimension turned out to be the key to recognizing which equations of the fifth degree – i.e. those in which the unknown occurs at most to the fifth power – can be solved using algebraic means. The book alone is worth reading for the chapter on Évariste Galois (1811-1832), who solved this long-simmering problem in the early 19th century. Du Sautoy describes the ardent revolutionary who died in a duel at the age of 20 as knowledgeably as his scientific breakthrough, which his successors only really understood decades after his death. Unfortunately, non-expert readers will not be able to fully understand Galois' proof.

It becomes even more hopeless, even for trained mathematicians, when you write to Sautoy about the research of the past decades that examines spaces with several million dimensions. The specialists themselves reverently call some of the objects they seek to fathom monsters.

The "moonlight" from the title can also only be guessed at. The mathematicians are thus describing an astonishing connection between two sub-disciplines. In the so-called modular functions from number theory, the same numbers appear as in symmetries: 196884, 21 493760, 864 299 970. To date, nobody has found an explanation for this. But mathematicians are so fascinated that they call the reflection of one theory in another as moonlight.

You Sautoy is an insider. He vividly depicts not only the historical figures with their peculiarities, but also the contemporary scientists. Among them are some special characters. One looks everywhere for patterns and stares at brick walls for hours and only wants to include the author of the book in his research group if he changes his name, so that the order in the list of its members is not lost. Another is always walking around with plastic bags in which he is lugging around the train timetable of the British railway network.

It seems that a special character is required to approach monsters in millions of dimensions. Du Sautoy himself, who since autumn last year has taken on a special professorship at Oxford that aims to bring science closer to the public, seems completely normal.