Symmetry, Beauty, Truth

"A group is a set of objects/values with a single binary operator that has a certain set of basic properties: associativity, existence of inverse, existence of identity. It's one of the simplest constructions of abstract algebra. What's really fascinating about it is that that simple construction - the set plus one operation will a simple set of properties - defines the entire concept of symmetry.

Groups don't normally require any structure on their members beyond what's required to make the group operator work properly. You can define a group whose values are a set of points, a set of numbers, a set of coins - very nearly anything you want.

But there are certain structured sets of values that we care about, which you can use as the objects for a group. One of those is a topological space. A topological space is just a collection of objects which have a kind of nearness/adjacency relationship between the objects in the collection. So a group on a topological space is interesting, because what it does is define symmetry on a set of values that preserves the nearness/adjacency relationships of the objects in the space.

Even more interesting, we can define a particular kind of topological space: a manifold, which is a sort of "smooth" topological space: a manifold is a topological space where the structure of the nearness/adjacency relations makes every small finite region of the space appear to be Euclidean.

So a Lie group is a group whose objects form a manifold, and whose group operations preserve the manifold structure of the nearness/adjacency relations."

http://scienceblogs.com/goodmath/2007/03/the_mapping_of_the_e8_lie_grou.php

"Than ours, a friend to man, to whom thou say'st,
"Beauty is truth, truth beauty"---that is all
Ye know on earth, and all ye need to know. "

http://www.eecs.harvard.edu/~keith/poems/urn.html