If

*X* denotes a subset of

**R**^{n} then its

symmetry group
*Symm(X)* is the group of

isometries of

**R**^{n}
that leave the subset

*X* stable. For examples of isometries
see the write-ups on

isometries of the plane and the

orthogonal group.

It may be that every symmetry of *X* is forced to fix the origin
and so be a linear transformation. This is the case for a regular polygon
centred at the origin in **R**^{2}, for example, and for
also for a Platonic solid centred at the origin in 3-space.

In that case *Symm(X)* will be a subgroup of the orthogonal group
*O(n)* and we can also consider its direct (or rotational) symmetry
group *Rot(X)* which is the intersection of *Symm(X)*
with the special orthogonal group *SO(n)*.

We are mostly interested in the case of *n=2* and *n=3*. In that
case *Rot(X)* consists of rotations, as its name suggests.

__Examples__