Symmetry Corner

The mathematical study of symmetry is called group theory because it studies ‘groups’ of symmetry operations. Most of these can be deconstructed into simpler groups, and those that can be deconstructed no further I have called ‘atoms of symmetry’.

The use of symmetry in mathematics made a huge step forward when Joseph Louis Lagrange introduced it in his work on algebraic equations in 1770. Other mathematicians later took his ideas further, in particular Évariste Galois, who died in 1832 at the age of twenty. He introduced the concept of a group of symmetries, now known simply as a ‘group’ in modern mathematics. Galois used groups in creating a new theory of algebraic equations, and they have since led to important advances in many areas of mathematics, sometimes in cases where there is no obvious symmetry.

In Symmetry and the Monster I describe the quest to find all atoms of symmetry, known technically as ‘simple groups’. These are not simple in the usual sense, but can be very complex and interesting, and most fall into one of several well-understood families. Outside these families lie 26 exceptions — called sporadic groups — that do not fit into the general pattern. Being outsiders makes them particularly interesting.

Among these exceptions the largest is called the Monster, and contains all but six of the others. After its existence was predicted in 1973 — it was constructed seven years later — the Monster began to show strange connections with other parts of mathematics, initially through a coincidence between two numbers: 196,883 and 196,884. These so-called ‘moonshine’ connections were subsequently linked with the mathematics of string theory, a proposed way of combining gravity with quantum theory. However there are coincidences as yet unexplained, one of which concerns the number 163.