by pat » Tue Mar 28, 2006 5:31 pm
As usually defined, a symmetry is a continuous, distance-preserving function that maps a shape onto itself. So, if your shape is S, then for any point s in S, your function f maps s onto a point in S.
By virtue of being distance-preserving, the function must be one-to-one. If a and b are two points in S, then |a-b| = |f(a) - f(b)|. The distance is only zero if a = b.
What you are calling a symmetry are the fixed points of what I'm calling a symmetry. The fixed points of a symmetry f are all points s in S such that f(s) = s.
Your concern is with the dimension of the set of fixed points or with the number of unique sets of fixed points at each dimension.
It is clear from the above that the fixed points are a subset of the shape. So, automatically, we cannot have that m bigger than n.
In the above, the trivial symmetry is also possible where f(s) = s for all S. This is the identity map on the shape. For the identity map m = n.
For some shapes, there are symmetries where m = 0. For example, rotating an annulus in its plane around its center leaves no fixed points.
The annulus has an infinite number of symmetries for m = 1 (where you flip the annulus over a line through its center).
Maybe more later... it's lunchtime now...