A few weeks ago New Scientist wrote Beyond space and time: Fractals, hyperspace and more.
"We don't have any trouble coping with three dimensions – or four at a pinch. The 3D world of solid objects and limitless space is something we accept with scarcely a second thought. Time, the fourth dimension, gets a little trickier. But it's when we start to explore worlds that embody more – or indeed fewer – dimensions that things get really tough. These exotic worlds might be daunting, but they matter. String theory, our best guess yet at a theory of everything, doesn't seem to work with fewer than 10 dimensions. Some strange and useful properties of solids, such as superconductivity, are best explained using theories in two, one or even no dimensions at all. Prepare your mind for boggling as we explore the how, why and where of dimensions."
It's several short articles that don't go into much depth but I did learn a few things about what physicists do with higher dimensions.
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