Symmetry makes groups

A square has exactly eight symmetries: turns through $0^\circ$, $90^\circ$, $180^\circ$ and $270^\circ$, and flips in its two diagonals and its two mid-lines. These moves have four properties:

• doing one symmetry and then another always gives a symmetry;
• "do nothing" is a symmetry;
• every symmetry can be undone by another symmetry;
• for three moves, it does not matter how you group them: $(a \text{ then } b) \text{ then } c$ equals $a \text{ then } (b \text{ then } c)$.

Any collection of moves with these four properties is called a group. The symmetries of the square form the dihedral group of order $8$. Order matters: a quarter turn followed by a flip is usually not the same as the flip followed by the quarter turn, so this group is not commutative. A regular polygon with $n$ sides has $2n$ symmetries; a circle has infinitely many, one for every angle of turning.

Groups are the mathematics of symmetry everywhere. There are exactly $230$ space groups describing the possible symmetries of crystals; Évariste Galois used groups to explain why there is no formula for the roots of a general polynomial of degree five; and physicists classify elementary particles by the symmetry groups of the laws they obey. Groups of symmetries that vary continuously, like turning by any angle, are Lie groups, named after Sophus Lie.