  In a discrete group generated by hyperplane reflections in the
$n$-dimensional hyperbolic space, the reflection length of an element is the
minimal number of hyperplane reflections in the group that suffices to factor
the element. For a Coxeter group that arises in this way and does not split
into a direct product of spherical and affine reflection groups, the reflection
length is unbounded. The action of the Coxeter group induces a tessellation of
the hyperbolic space. After fixing a fundamental domain, there exists a
bijection between the tiles and the group elements. We describe certain points
in the visual boundary of the $n$-dimensional hyperbolic space for which every
neighbourhood contains tiles of every reflection length. To prove this, we show
that two disjoint hyperplanes in the $n$-dimensional hyperbolic space without
common boundary points have a unique common perpendicular.
