  Nematic order is the breaking of rotational symmetry in the presence of
translational invariance. While originally defined in the context of liquid
crystals, the concept of nematic order has arisen in crystalline matter with
discrete rotational symmetry, most prominently in the tetragonal Fe-based
superconductors where the parent state is four-fold symmetric. In this case the
nematic director takes on only two directions, and the order parameter in such
"Ising-nematic" systems is a simple scalar. Here, using a novel
spatially-resolved optical polarimetry technique, we show that a qualitatively
distinct nematic state arises in the triangular lattice antiferromagnet
Fe$_{1/3}$NbS$_2$. The crucial difference is that the nematic order on the
triangular lattice is a Z$_3$, or three-state Potts-nematic order parameter. As
a consequence, the anisotropy axes of response functions such as the
resistivity tensor can be continuously re-oriented by external perturbations.
This discovery provides insight into realizing devices that exploit analogies
with nematic liquid crystals.
