A short introduction to symmetry and groups

two tables showing the structure of
two different groups

This is a part of maths that I've always liked a lot. It is beautiful and satisfying in itself and it also has very many important applications in areas such as crystallography and particle physics.
I'm going just far enough for you to see some examples of how symmetry can be described by groups. You won't need any previous maths for this, but I will introduce some mathematical terms to make describing things simpler. This will then make a good jumping-off point for anyone who wants to dig deeper in the mathematical theory.

I've divided up what I've written into the following parts:

  1. Some thoughts on symmetry
  2. Looking at some particular symmetries
  3. The symmetry transformations of a square
  4. How do its reflections fit together?
  5. How do all the symmetries of a square fit together? . . . the rules for a group.
  6. Fitting together the symmetry transformations for some other shapes
  7. A new group
  8. Finding this new group in more places
  9. Some examples of infinite groups
  10. When does multiplying give groups?
  11. And finally ...

I've written these parts as separate linked webpages to make it easier for you to think how things will work out before looking at the answer. You just need to click on the potato-head to move to the next section.


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