I was doing good at school in plane geometry and trigonometry – especially in geometric proofs like proving the equality of two line segments or two angles – more than I was doing in analytic geometry.
I am considering doing research in mathematics to be my career (and my life) someday. and I am wondering about the most interesting research area for me.
What I am asking about is :
Is there still open research fields in plane geometry or Euclidean geometry? Or this area is considered to be fundamentals that are already investigated enough?
Thank you!
Best Answer
Actually there is a theorem of Tarski that elementary Euclidean geometry is decidable. Roughly speaking this means that there is a computer program that can decide if a given statement of elementary Euclidean geometry is true (given enough time). Generally mathematicians do not like to do things that can be done on a computer. And so there is no active research in elementary geometry. There is some information on this in wikipedia http://en.wikipedia.org/wiki/Euclidean_geometry#Logical_basis. You can also check this question out Is it possible to solve any Euclidean geometry problem using a computer?.