Is there still active research in Euclidean geometry?

Question

Is there still ongoing and active research in Euclidean geometry? I have heard somewhere that Euclidean geometry is a solved subject with no open problems. But perhaps I heard wrong. Is there still mathematical research literature on Euclidean geometry? If so, can I see some recent or relatively recent research texts on Euclidean geometry? I would be very interested in such references.

Answer 1

I'm not aware of it being active in the traditional sense, but it's still valuable and active in different ways.

Despite both being the roots of mathematics, Euclidean geometry is fundamentally different from (Elementary) number theory: While number theory suffers (celebrates?) the Godel's incompleteness theorem, first order Euclidean geometry is decidable (though this is a subtle issue). That is, there exists a computer program that can determine any given statement in Euclidean geometry is true or false. This doesn't necessarily kill the field, as Euclidean geometry can be easily shown to be NP-complete, so we would not expect to find an efficient algorithm. And one may still be interested in higher order Euclidean geometry that is not decidable.

Still, Euclidean geometry not only has special places in education (for being practical as well as intellectually challenging) and mathematical competition, but also attracts researchers' attention in developing more efficient algorithms to solve larger class of problems, thus as a test field of computational algebraic geometry (Grobner's basis and Wu's method) and more recently deep learning (or large language models or AI, whatever you want to call it). In particular, DeepMind's AlphaGeometry has grabbed a lot of attention.