Essential basic geometrical discoveries still possible in our era?

Can we imagine that scientific can still discover some basic simple but essential geometric rules such as the Pythagorean theorem in a near of far future:

$a^2 + b^2 = c^2$

Or do we consider that the future of discovery in mathematic, geometry and physic only rely on heavy computational models ?

I mean, is it possible to imagine that a man with some books and a brain (no computer for simluation, or heavy equipement) can still discover essential stuffs that have never been discovered, such as Euclide, Thales or Pythogore did in the past ?

• Are there elementary results in Euclidean geometry, etc., that are still discovered today? Yes, recreational mathematical journals publish such things all the time. Are they "essential stuffs"? I am not sure what that means. Maybe ancient results became "essential" because they were around for a long time. By the way, Thales is a semi-mythical figure, Pythagoras likely did not discover anything mathematical, and Euclid likely systematized what was discovered before him. – Conifold Jan 24 '17 at 23:11
• There are plenty of recent elementary results in Euclidean geometry, and not only in recreational mathematics: here is one example: ihes.fr/~gromov/PDF/16[58].pdf (Gromov is perhaps the most famous living geometer. And this journal does not publish recreational math:-) – Alexandre Eremenko Jan 25 '17 at 7:13
• i.e. by definition it will be impossible for us to "imagine" such things - if we could, they would not be new. – mobileink Jan 25 '17 at 21:31
• "One of the great theorems of elementary plane geometry was essentially only discovered in the twentieth century; namely, the theorem of Morley that states that the trisectors of the angles of a triangle meet at the vertices of an equilateral triangle (the “Morley triangle”)" From a paper in Acta Arithmetica. – Conifold Jan 27 '17 at 23:27

One incredible discovery of the 1800s was that when you speak of geometry, you need to specify "where your geometry is located" and specifically what constitutes points and lines. At the time this was applied to obtain Hyperbolic Geometry and Elliptic Geometry. Recently, the field of Finite Geometry has become the subject of much foundational research. Finite Geometries arise when you define the concept of points and lines in such a way as to give rise to only finitely many points and lines, and has fundamental connections to theoretical computer science, in particular my field of linear algebra and combinatorics. Here you can see a 140 page masters' thesis on the applications of finite geometry to coding theory. There are a number of open questions that are fundamentally important in finite geometry, particularly the classification of finite projective planes. Although finite projective planes are very important in finite geometry, for $n\geq12$ we don't know very much about the finite projective planes of order $n$, and don't even know how to prove that any exist for most $n$! See here for more on finite projective planes.