What Is Symmetry? The Mathematics of What Stays the Same

To say a shape has a symmetry is to name an action you could perform on it that would leave it indistinguishable from how it started. A square can be rotated by ninety degrees, or reflected across either diagonal or either midline, and after any of those it occupies exactly the same space. Counting the do-nothing action, the square has eight symmetries. The standard operations in the plane are rotation about a point, reflection across a line, translation by a fixed distance and direction, and glide reflection, which is a reflection combined with a slide along the mirror line and is easy to miss because it has no fixed point at all. A shape with no symmetry other than doing nothing is called asymmetric, and a shape that cannot be superimposed on its own mirror image, like a hand or a screw thread, is chiral, which matters enormously in chemistry because two mirror-image molecules can behave completely differently in the body.

The reason symmetry became a branch of mathematics rather than a topic in geometry is that the set of symmetries of any object has a structure. Perform one symmetry and then another and the result is again a symmetry, every symmetry can be undone by another one, and doing nothing is itself a symmetry. Those conditions define a group, and studying symmetry therefore becomes studying groups, an abstraction that applies equally to the rotations of a cube, the rearrangements of a set of objects and the solutions of an equation. The historical breakthrough came from Evariste Galois, who showed that whether a polynomial equation can be solved by a formula depends on the structure of the group of symmetries among its roots, which is why there is a formula for the quadratic, cubic and quartic and provably none for the general fifth degree equation. He worked this out in his teens and died in a duel at twenty.

One of the most satisfying results in the subject is that repeating patterns can be completely catalogued, and the catalogue is short:

• •Rosette groups, patterns with a single centre, which come in exactly two infinite families: those with rotations only, and those with rotations and reflections
• •Frieze groups, patterns repeating along a strip, of which there are exactly seven, distinguished by which combinations of translation, reflection, half turn and glide reflection they contain
• •Wallpaper groups, patterns repeating in two directions across the plane, of which there are exactly seventeen, a result proved in 1891 and satisfied in full by the tilework of the Alhambra centuries earlier
• •Crystallographic space groups in three dimensions, of which there are exactly 230, which is the framework the whole of crystallography is built on
• •A striking consequence of the same reasoning is that a repeating pattern can have two, three, four or six fold rotational symmetry and never five fold, which made the 1982 discovery of quasicrystals with fivefold symmetry so disruptive that it took years to be accepted and eventually earned a Nobel prize

The deepest application is a theorem proved by Emmy Noether in 1918, which states that every continuous symmetry of a physical system corresponds to a conserved quantity. If the laws of physics work the same way tomorrow as today, meaning they are symmetric under shifts in time, then energy is conserved. If they work the same here as a metre to the left, momentum is conserved. If they work the same whichever direction you face, angular momentum is conserved. This reframes conservation laws from a collection of empirical facts into consequences of the structure of the laws themselves, and it is the organising principle of modern theoretical physics, where the fundamental forces are described by the symmetries their equations obey. Equally revealing are the places symmetry breaks: the universe contains far more matter than antimatter, which requires a violated symmetry, and much of particle physics concerns symmetries that hold at high energy and break at low energy, which is how particles acquire mass.

A symmetry is any transformation that leaves an object indistinguishable from how it started, which in the plane means rotation, reflection, translation or glide reflection. The symmetries of anything form a group, and that abstraction connects geometry to the solvability of equations, which is how Galois proved no general formula exists for the fifth degree. Repeating patterns are fully classified: seven frieze groups, seventeen wallpaper groups, 230 crystal space groups. Noether showed every continuous symmetry yields a conservation law.