The concepts of symmetry and isometry are central to the study of geometry. An isometry is a distance preserving map from some space it itself: a rigid motion. For example, f(x)=x+5 is a isometry of the real line; the whole line is shifted by 5 and distances between points remain unchanged. A symmetry of a figure in some space is an isometry of that space which maps the figure to itself. If this seems unclear, do not despair! This whole section is devoted to understanding symmetries and isometries.
Set aside the isometries of a line for a moment and consider the plane. We can still move it along by a translation of the form f(x,y)=(x+a,y+b) for any two numbers a and b. A reflection across a line is also possible, for instance the reflection f(x,y)=(-x,y) across the y-axis. But unlike in the case of a line, we now have an additional type of isometry, a rotation about a point.
Consider a square lying in the plane. We can rotate the plane about the center of the square (denoted p) by 90 degrees. This rotation is an isometry of the plane and at the same time it maps the square to itself. Thus it's a symmetry of the square in the plane.
In the above exercise you have (hopefully) found eight different symmetries: the three rotations illustrated below, the four reflections about the orange lines, and the identity symmetry, which we always count as one of the symmetries of a figure (you can think of it as a 0 degree rotation if you wish).
How do we know these are all the symmetries? By using the theorem that all isometries of the plane are translations, rotations, reflections, and compositions of translations and reflections (that is, a translation followed by a reflection). Since no translation takes the square to itself, only reflections and rotations are possible as symmetries of the square. The square must be mapped to itself by a symmetry, so only the rotations and reflections indicted above actually work.
We can also look at symmetries of more unusual sets. Here by set we just mean some, possibly infinite, collection of points.
As you might suspect, the collection of symmetries of a set depends on the space in which it lies. For instance, the two black points to the right have three different symmetries if placed on a plane, namely the identity symmetry, rotation by 180 degree about point p, and reflection about the orange line. If the space in which the points lie is a line however (so now a symmetry is an isometry of the line which preserves the two points), then there are only two symmetries, the identity and a reflection about the midpoint between them. After all, on a line a 180 degree rotation is the same as a reflection.
Another important concept is that of a fixed point, a point which is mapped to itself by a symmetry or an isometry. In case of a symmetry, there are two types of fixed points involved: the fixed points of the ambient space and the fixed points of the figure itself. For instance, the identity symmetry fixes all points of the space and hence also of the figure. A rotation in a plane fixes only the center point of rotation, while a rotation about a line in 3-spaces fixes only the axis of rotation.
What are the fixed points of the four types of isometries of the plane mentioned before (rotation, reflection, translation, and translation followed by a reflection)?