Consider a set $S \subseteq \mathbb{R}^2$. What does it mean for a function $f$ to be a symmetry of $S$? After all, there are usually infinitely many bijections on $\mathbb{R}^2$ that send $S$ to $S$. I would very much like some clarification of this topic.
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2Have you not given the answer yourself? A symmetry (yes, not the symmetry) is a bijection $f : \mathbb R^2 \to \mathbb R^2$ such that $f(S) = S$. – azif00 Jul 11 '21 at 05:18
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How many symmetries are there for an equilateral triangle? How about a square or some other n-gon? There may be infinitely many bijections but the symmetries have invariants. I recommend you do a Google search for symmetries of the plane. They have been studied extensively and classified. – John Douma Jul 11 '21 at 05:23
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The missing ingredient is that you want $f$ to preserve distances: if $d(x,y)$ is the distance between point $x$ and $y$, then you want $$d(x,y) =d(f(x),f(y))$$ for all points $x$ and $y$.
A function with this property is called an isometry. When we speak of geometric symmetries, we always require them to be isometries. It can be shown that any isometry of the plane is a reflection, a rotation, a translation, or a composition of these.
MJD
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