The Meaning of 'Symmetry'

There are a lot of discussions of 'symmetry' that give examples and a rather vague definition, but don't make clear why some things aren't 'symmetries' but which seem like they ought to be, at least under the vague definition. My aim here is to give a more precise definition, and show how it applies.

Here I'll just talk about symmetry of figures in the Euclidean plane, since places that give more sophisticated examples usually give a more sophisticated definition as well.

So consider a geometric figure in the plane, that is, a subset of 2D Euclidean space. Informally, a 'symmetry' of the figure is a 'motion' of the figure that leaves it unchanged. For example, an equilateral triangle has six symmetries: three rotations about the center of the triangle (rotations of $0\,^{\circ}$, $120\,^{\circ}$, and $240\,^{\circ}$), and three reflections (through the three angle bisectors). But what counts as a 'motion'? Are these 'motions':

  • Only swapping two points of the triangle? (That is, keep everything else the same.)
  • Moving each point of the triangle a fixed distance along the triangle?

So potentially there's a vast set of 'motions' which seem like they might produce a 'symmetry' but don't. It turns out that many discussions leave out a crucial condition: the motion must be an isometry, that is, it must preserve distances. And that property must hold for the entire space, not just the figure alone. This condition excludes the two 'motions' asked about above, for example, from the set of symmetries.

I don't know why isometry is so often omitted (or glossed over) in symmetry discussions. It's not as if it's a difficult concept.

(By the way, there is one subtle point. It might seem possible that two isometries, while identical close to the figure, are different far away. But this can't happen due to the well-known theorem that, in the Euclidean plane, any isometry is completely determined by its effect on three non-colinear points.)

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