Non-Abelian Geometry

The book Eifelheim, by Michael Flynn, is on the whole a pretty lousy book. It is simply to hard a slog for too little reward and has a hard time keeping two plot arcs separated by several hundred years both flowing smoothly (this probably explains why you can get a used hard copy edition of it on Amazon for 34 cents). But, it did have one little snippet early on that was worthwhile, a very intuitive description on non-abelian geometry, in which the distance from point A to point B is not the same as the distance from point B to point A.

In the abstract, this is hard concept to understand. But, examples help. It you think of distance as the time it takes to get from one point to another, then there are lots of examples. Distance is shorter going downstream than upstream. One way streets can make it faster to go to a destination than to return from there. The distance up the mountain is longer than the trip down.

It turns out that non-abelian geometry is a feature of almost all fundamental theories of quantum gravity that can be reconciled with reality, so it turns out to be quite important in practice. But, I've never seen anybody explain the concept in a way that make as much sense as it did when Flynn described it in this book.

There are probably other gems in there as well, but I'm not sure that I have the patience to slug through the book to read them.