In this talk I will start with some basics of Lie Theory which relate certain lattices in R^n with the dimensions of irreducible modules for various Lie Groups. In particular, this gives a geometric way to determine these dimensions. Using this relationship, I will then proceed to start with orthoplexes (the duals of n-cubes) to obtain the Catalan numbers in two different ways. The first way is relatively easy to do, but the second expression is not so obvious... until you look at it from the perspective of weights and modules of the Symplectic groups.