By Marcus du Sautoy Win a chance to name one of the new mathematical symmetry groups mentioned in this feature. OSLO, May 2008. King Harald of Norway presents mathematicians John Thompson and Jacques Tits with the Abel prize, one of the highest accolades in mathematics. There is a pleasing symmetry at the heart of this year’s award. The winners are being honoured for ground-breaking work that led to the completion of a project started by Niels Abel, the 19th-century Norwegian mathematician after whom the prize is named. Appropriately enough, that project concerns mathematicians’ attempts to answer the question: what is symmetry? Most people’s response is to point to the left-right reflectional symmetry of the human face. Or a flower, or a snowflake. But a snowflake has additional symmetries to that of a human face: as well as looking at its two halves, you can also turn a snowflake 60 degrees to match up its shape again. This begins to get at the essence of what symmetry is – a transformation or move that you can do to a structure which somehow makes it look like it did before you moved it. So how many other types of symmetry are there? Remarkably, we now have a definitive answer. Thompson, of the University of Florida in Gainesville, and Tits, of the Collège de France in Paris, are responsible for ideas that have culminated in what is essentially a “periodic table” of symmetry. It has been as influential in the world of symmetry as the periodic table of elements has been to chemistry,