During my fellowship, I studied crossing numbers of graphs, which are the number of times edges in a graph cross in a drawing of that graph. This idea was first worked on in 1944 by Paul Turán and one way it is applied is to plan roads in a city or rails in a storage yard. Most crossing number work done has been with complete (Anthony Hill) or complete bipartite (Kazimierz Zarankiewitz) graphs. I mainly worked with the crossing number of the generalized hypercube graph, focusing on the rectilinear crossing number. The crossing number of the generalized hypercube graph has been worked on very minimally and in this paper, I took several approaches at finding an upper bound for the rectilinear crossing number.