I learned this when I took a graduate-level mathematics course from Brian Greene, the Columbia University physicist who has done a very nice job popularizing string theory, the theory that requires our universe to be made of 10 dimensions. (Actually, recent developments in string theory suggest
that there may be yet another dimension, for a total of 11, and that this new extra dimension is invisible because it is "curled up" into an infinite number of tiny loops. But to avoid further brain pain, let's stick with 10.) At the time I took the class, I was working toward a master's degree in the philosophical foundations of physics. This course was by far the most difficult one I have ever taken. After Week 3, I understood very little of what was going on. Yet the dimension stuff, that was Week 2, I think. Anyway, the course was this year-long journey through the world of differential geometry, which, as far as I could tell, should have been named abstract geometry, concerned as it was with the properties of surfaces and spaces of things in n dimensions. (Remember, n here can be any whole number you wish--2, 5 or 12,497.) The culmination of this course, which paused only once for a quiet moment of repose somewhere around Week 8 when Professor Greene revealed to us that we had just derived the fundamental equation of general relativity (who knew?!), was an introduction to the basics of string theory.