I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups)

1. in mathematics

2. outside of mathematics,

besides the applications to $2$-dimensional and $3$-dimensional crystallography (or related fields like chemistry, or physics, of crystals).

One possible application of $4$-dimensional space groups is already reported in the wikipedia article I linked to (see "Magnetic groups and time reversal").