  We explore visual representations of tilings corresponding to Schl\"afli
symbols. In three dimensions, we call these tilings "honeycombs". Schl\"afli
symbols encode, in a very efficient way, regular tilings of spherical,
euclidean and hyperbolic spaces in all dimensions. In three dimensions, there
are only a finite number of spherical and euclidean honeycombs, but infinitely
many hyperbolic honeycombs. Moreover, there are only four hyperbolic honeycombs
with material vertices and material cells (the cells are entirely inside of
hyperbolic space), eleven with ideal vertices or cells (the cells touch the
boundary of hyperbolic space in some way), and all others have either
hyperideal vertices or hyperideal cells (the cells go outside of the boundary
of hyperbolic space in some way). We develop strategies for visualizing
honeycombs in all of these categories, either via rendered images or 3D prints.
High resolution images are available at hyperbolichoneycombs.org.
