A circle has a much higher form of rotational symmetry. We can express it in terms of the Lie group of rotations that we have now encountered – with the whole lot of them leaving the shape unaltered. And we can go on to look at the symmetry of shapes like a sphere or a cylinder and in each case find the appropriate Lie group.
Thus the language of the mathematical group applies to geometrical shapes – and it also can be applied to any form of data that looks to have a pattern. The concept of a group involves the categorising of the form of symmetry found in data.
And this leads to the power of this thinking when applied to the world of elementary particles. When people first looked for elementary particles, they found just a very few – ones familiar to us today, such as the proton, the neutron, which between them form the core of an atom, its nucleus, and the electron, which swirls in a cloud around the nucleus to form an atom. But as time went on and particle-detectors became more sophisticated, the number grew of particles, and now there are several hundred.
Quite a lot of them might hardly seem to deserve the name ‘particle’, since they twinkle in and out of existence in a micro-fraction of a second. But to the researchers they still count, and they have been carefully analysed and recorded.
The particles are categorised by numbers, specifying aspects such as their electric charge and various other features. And when the particles are all set out in array, according to the various numbers, patterns start to emerge.