What exactly is meant by symmetry? This is something that has troubled me for long.
M. Artin, in his book Algebra (pp.155-156), describes four types of symmetry (of plane figures):

*

*bilateral (i.e. reflective),

*rotational,

*translational, and

*glide.

I somehow understand symmetry to be a "condition" in which a figure, under a specific set of operations, remains "essentially the same" or congruent.
How do we prove that the figure before the operation and after the operation are the same? By superimposing the two. For example, if I take an equilateral triangle and rotate it by $60^\circ$, the figure before rotation and after rotation will superimpose perfectly. I am not allowed to rotate the resultant figure by $-60^\circ$ to superimpose the two figures. I have to superimpose the two figures as they are.
Hence, symmetry seems to be a marriage of certain figures and specific operations. If, after a certain operation has acted upon a figure, the two figures can be imposed without performing any further operation, we have symmetry.
Is this correct?
If it is, why is translation a "symmetry-related" operation then? If I have a triangle, and I translate it by vector $\overline{a}$, I will have to translate it by $-\overline{a}$ to superimpose the two figures! In other words, I will have to perform an additional operation for super-imposition! Same is the case for glide.
Sorry for the ambiguous wording. I have tried to elucidate my doubts to the best of my abilities. Thanks
 A: Translation would not preserve a triangle. But if it were translation in just the right direction, it could preserve an entire plane tesselated by triangles in the way that you understand symmetry, without having to translate back. There are similar infinite tesselations of a plane that use glide reflections.
Technically, a symmetry of an object (whether it is a single triangle or an infinite plane) is a member of a group (or possibly some other algebraic structure like a monoid) that acts on that object without changing something special about it. The "something special" can vary, but what you have in mind is the distance between two points.
So for example, the symmetry of an equilateral triangle that you imagine is not really about rotating it through the surrounding plane. It's more about the net effect of where a point on that triangle would end up once the rotation was done. And the distance between any two points before the rotation will equal the distance between them after. So that action (moving a point from its preimage to its postimage) is the symmetry.
And the examples of the infinite tesselations of the plane is more about moving points on the wires that tesselate to other points on the wires than it is about moving the entire plane. (Although you can treat the complement equally well).
