Is Euclidean Geometry studied at all?

Question

Is there a place for Euclidean geometry in the hearts or minds of any mathematicians? I personally find it to be the most beautiful mathematics I have yet encountered but I see little of it on sites such as these, leading me to believe that it somehow disappears from the minds of all college mathematics students unless it is to calculate a distance or angle, which is not at all what appeals to me.

The only modern day type of Euclidean geometry research, etc. that I see being done at a college level would be the Forum Geometricorum. (Highly recommend it)

In short, will anyone see or study Euclidean geometry after high school?

Answer 1

This "comment" is a little too long for the comments.

I think, without diving too deep in to the link that you posted, that what you're lamenting is the lack of what I would call synthetic geometry, in particular as opposed to analytic geometry or other approaches to geometry ( eg algebraic geometry ). This would be things like ruler/compass constructions. To this complaint, I think many many mathematicians consider synthetic geometry very beautiful and it has a special place in their hearts, but it's kind of like owning a horse that you ride for pleasure and exercise, but you still drive a car to get places; we typically have better tools now. Synthetic geometric intuitions still play a big part and occasionally I've seen proofs that were more enlightening when done synthetically, but for the most part, practicing mathematicians use the more modern tools.

If it's synthetic geometry you're actually interested in, you shouldn't restrict yourself to Euclidean geometry! There's actually a lot of fun to be had trying to do some of the same things in non-euclidean geometry that were done for centuries in Euclidean geometry. I recommend the book "Euclidean and non-Euclidean Geometry". For more of a similar flavor, you could study projective geometry. I've never read it, but Robin Hartshorne has a very popular book that is of a synthetic flavor ( Note: His more famous algebraic geometry books are decidedly not of this flavor ).

As far as "Euclidean" geometry goes, though, that is still quite actively studied. It may not be the hottest topic, and the tools used are often pretty far removed the days of the Greeks or even Descartes, but there is plenty of geometry still done that studies good old Euclidean space.

Answer 2

The area of circle packings is currently pretty wide open. Here's a specific easy example: take a circle of radius $r$ and place $n$ circles around it so that they are tangent to the center circle and to each other. The question is, what is the ratio of radii of the center circle to the smallest outer circle? So if $r_c$ is the radius of the center circle, and $r_1$ is the radius of the smallest circle, then how big can $r_c/r_1$ be? Here's an image:

Center circle, with 7 tangent outer circles. The left/right images are not allowed:

The answer is surprising, and is known as the Ring Lemma: $r_c/r_1$ is bounded above by a constant that only depends on $n$: the number of outer circles. In other words, the outer tangent circles can be arbitrarily big (this is obvious, just make one big) but they cannot be arbitrarily small for a given $n$. The proof of this is not trivial! Also surprisingly, this idea was used in proving distributional limits of random graphs.

As far as I'm aware, these results have not been proven in dimensions 4 and higher. These questions become very difficult in higher dimensions!

These kinds of things also come up in complex analysis (Schramm Loewner theory in particular), where one is trying to prove regularity results about complicated limits of stochastic processes. It helps to think of your domain as a packing of circles, and then look at conformal maps of this.