Symmetry vs isometry

Question

In context of geometry and points in a plane Wikipedia describes symmetry as a type of invariance - the property that something does not change under a set of transformations.

Isn't isometry the exact same thing? A type of invariance that preserves relative distances between points.

This definition from wikipedia adds to my confusion:

If the object X is a set of points in the plane with its metric structure or any other metric space, a symmetry is a bijection of the set to itself which preserves the distance between each pair of points (an isometry).

So from this I can conclude that every symmetry is an isometry, but not every isometry is a symmetry. And which type of invariance, in addition to ones it already has, does an isometry need to have to be considered a symmetry? Coud I apply a rigid motion to any figure in a plane and call that symmetry as well? I have not seen this explicitly stated anywhere.

Answer 1

An isometry is a set bijection $\Phi : (X, d) \to (X', d')$ between metric spaces that identifies $d, d'$, that is, that satisfies $$d(x, y) = d'(\Phi(x), \Phi(y)) \qquad \textrm{for all $x, y \in X$} .$$ A symmetry (as defined in the excerpt), then, is just an isometry from a metric space $(X, d)$ to itself.

Note every metric space admits at least one symmetry, namely the identity map. Checking the axioms directly that the set of symmetries of a fixed metric space $(X, d)$ form a group under composition, which is called the isometry group and is sometimes denoted $\operatorname{Iso}(X, d)$.

As Clara points out in the comments, the term symmetry does not usually have this restricted meaning. More generally, given a set $X$ equipped with some structure $\mathcal S$, we can consider the bijections $X \to X$ that preserve $\mathcal S$ in an appropriate sense. Generically these maps are called automorphisms rather than symmetries, and again for any $(X, \mathcal S)$ the set of such maps form a group under composition called the automorphism group of $(X, \mathcal S)$, and it is often denoted $\operatorname{Aut}(X, \mathcal S)$. In some contexts, often when $(X, \mathcal S)$ has some geometric interpretation, this group is sometimes called the symmetry group of $(X, \mathcal S)$.

Answer 2

The answer Travis provided is right. I just want to emphasize that in your particular case not all isometries of the plane are symmetries of $X$. Suppose $I$ is an isometry of the plane. We say that $I$ is a symmetry of the set of points $X$ if $I(X) = X$.

Just as Travis says, generically, a symmetry of a set $A$ with structure $\mathcal{S}$ is a bijection $A \to A$ such that $\mathcal{S}$ is preserved. In this case, the set $A$ is the plane, and the structure $\mathcal{S}$ is both the metric structure on the plane and the set of points $X$. Thus, $I : A \to A$ is a symmetry of $(A, \mathcal{S})$ if $d(x,y) = d(I(x), I(y))$ (where $d : A \times A \to \mathbb{R}$ is the distance function) and $I(X) = X$, i.e. $I|_X : X \to X$ is a bijection.