We're used to moving in three-dimensional space, but our interactions with higher dimensions are more varied. Science fiction tells us about wormholes and time travel, while the era of big data gives us endless arrays of numbers to wade through. We'll explore a sweet spot between these two extremes, one that offers both mathematical rigor and geometric intuition. Expect hypercubes, analogies, and an atlas or two as we explore techniques and tricks for visualisation beyond three dimensions.