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Although the concept of "symmetry" is quite intuitive, I'm finding it difficult to pin down exactly what one means by the "symmetries of the equilateral triangle".

Here's an example. Suppose I have an equilateral triangle $ABC$.

enter image description here

One of its particular symmetry is rotation around its center by $120$ degrees, which results in the same triangle, but with its vertices interchanged:

enter image description here

But here is where it gets "fuzzy" (for me). If our "reference frame", or "point of view" also rotates by $120$ degrees, then what we'll see is the first triangle, and the rotation would have had no effect.

So in talking about symmetries of geometric figures, aren't we implicitly assuming that we have to stay put, and not shift our viewpoint? I'm hope I'm making sense here.

To formalise the concept of "fixing our viewpoint", we label the vertices of our triangle, and say that "rotation by $120$ degrees" simply means the abstract map $\begin{pmatrix}A&B&C\\B&C&A\end{pmatrix}$.

Or we could argue that it's the map of the plane $f: \mathbf{R}^{2} \to \mathbf{R}^{2}$, which preserves the properties of the equilateral triangle, but again, what we're really doing is converting the vertices to numbers, and composing maps between these numbers.

In my mind, these are more precise statements, because we can write them down, compose the symmetries, construct the Cayley table, etc., but the intuitive processes of "rotate around center", "reflect in the middle", and "translate by a few units" seem like ghosts of our intuition. There's not much we can do with them.

If we don't put labels on the triangle, there's no way to reason about its symmetries. It has no distinguishing features. For example, if we don't label the plane with real numbers to make it $\mathbf{R}^{2}$, we can't talk about the plane rotating, reflecting, shearing, expanding, etc., because it will look the same no matter what we do to it. No point of the plane is prioritised over the other.

So am I right in saying that thinking about symmetries of geometrical objects is purely for motivation, but when it boils down to the rigorous study of groups, we study, say, abstract permutations of a set of letters $\{A,B,C \}$? In this case, the problem I posed vanishes. But my confusion might just stem from a huge, but simple, misunderstanding.

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When you talk about the symmetries of the triangle, it is implicit that the triangle is a triangle which lives in the Euclidean plane and the rules of the game are the rules of Euclidean geometry. This determines what is a triangle, what are the allowed symmetries of the plane and what it means for the triangle to respect a certain symmetry of the plane (this means the triangle stays the same after performing the symmetry on the plane). Once a specific triangle is given, the symmetry group of the triangle should be well-defined irregardless of how you describe it or whether you give the vertices a name or not.

Rotating the plane (together with the triangle) by $120$ degrees around the center of the triangle leaves the triangle invariant so this is considered a symmetry of the triangle. Rotating the plane by $240$ degrees also leaves the triangle invariant so this is considered another symmetry. It is true that the triangle looks the same before and after both symmetries are done but the symmetries are different (they are different transformations of the plane). You introduce the labels $A,B,C$ on the vertices in order to be able to describe the symmetries of the triangle in a succinct way that makes it clear that the two symmetries are different.

Namely, you know from Euclidean geometry that a symmetry of the plane is completely determined by the image of three non-collinear points and that the symmetry must send lines to lines and so the vertices of the triangle to themselves. Thus, in order to describe a symmetry of the triangle you might as well describe the images of the three vertices of the triangle under the symmetry and by what we said above, this will mean it will act as a permutation on the vertices. In other words, the act of labeling and describing the symmetry group as a subgroup of $S_3$ is just one possible (but very natural) description of the symmetry group.

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