  Spin liquids are a paradigmatic example of a non-trivial state of matter, and
the search for new spin liquids is a key direction of physics, chemistry, and
materials science. Geometric frustration -- where the geometry of the net that
the spins occupy precludes the generation of a simple ordered state -- is a
particularly fruitful way to generate these intrinsically disordered states. A
particular focus has been on a handful of high symmetry nets. There are,
however, many three-dimensional nets, each of which has the potential to form
unique states. In this paper, we investigate the high symmetry nets -- those
which are both node- and edge-transitive -- for the simplest possible
interaction sets: nearest-neighbor couplings of antiferromagnetic Heisenberg
and Ising spins. While the well-known crs (pyrochlore) net is the only
nearest-neighbor Heisenberg antiferromagnet which does not order, we identify
two new frustrated nets (lcx and thp) that possess finite temperature
Heisenberg spin-liquid states with strongly suppressed magnetic ordering and
non-collinear ground states. With Ising spins, we identify three new classical
spin liquids that do not order down to $T/J = 0.01$. We highlight materials
that contain these high symmetry nets, and which could, if substituted with
appropriate magnetic ions, potentially host these unusual states. Our
systematic survey will guide searches for novel magnetic phases.
