  We study reduced nematic equilibria on regular two-dimensional polygons with
Dirichlet tangent boundary conditions, in a reduced two-dimensional Landau-de
Gennes framework, discussing their relevance in the full three-dimensional
framework too. We work at a fixed temperature and study the reduced stable
equilibria in terms of the edge length, $\lambda$ of the regular polygon, $E_K$
with $K$ edges. We analytically compute a novel "ring solution" in the $\lambda
\to 0$ limit, with a unique point defect at the centre of the polygon for $K
\neq 4$. The ring solution is unique. For sufficiently large $\lambda$, we
deduce the existence of at least $\left[K/2 \right]$ classes of stable
equilibria and numerically compute bifurcation diagrams for reduced equilibria
on a pentagon and hexagon, as a function of $\lambda^2$, thus illustrating the
effects of geometry on the structure, locations and dimensionality of defects
in this framework.
