  Let ${\mathcal A}$ be a finite real linear hyperplane arrangement in three
dimensions. Suppose further that all the regions of ${\mathcal A}$ are
isometric. We prove that ${\mathcal A}$ is necessarily a Coxeter arrangement.
As it is well known that the regions of a Coxeter arrangement are isometric,
this characterizes three-dimensional Coxeter arrangements precisely as those
arrangements with isometric regions. It is an open question whether this
suffices to characterize Coxeter arrangements in higher dimensions. We also
present the three families of affine arrangements in the plane which are not
reflection arrangements, but in which all the regions are isometric.
