Misha's answer is almost always the right one, but you technically can project points into a higher-dimensional space using a kernel function. This comes up in Support Vector Machines where you're trying to separate two classes of data points by drawing a hyperplane between them. If your data isn't linearly separable, projecting it into a higher-dimensional space can sometimes help.

But most of the time, what you want to so is just measure everything you can think of, and let those measurements be your dimensions. When looking at rubes and bleggs, measure things like redness, blueness, roundedness, furryness, whatever you can think of. Each of those is one dimension. Before you know it, you've got a high-dimensional featurespace. Good luck dealing with it.