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, thenreflection
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.