I tried using, for example, the HypercubeGraph function:
As you can tell from the lack of symmetry (notice the shape of the innermost ring, for example), this is not the right projection. I also tried looking at some GraphPlot options like
GraphPlot[HypercubeGraph, Method -> "SpringElectricalEmbedding"]
but to no avail. How can I generate the Petrie polygon projection of an 8-cube in Mathematica?