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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 ?

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    $\begingroup$ 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. $\endgroup$ – Conifold Jan 24 '17 at 23:11
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    $\begingroup$ 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:-) $\endgroup$ – Alexandre Eremenko Jan 25 '17 at 7:13
  • $\begingroup$ i.e. by definition it will be impossible for us to "imagine" such things - if we could, they would not be new. $\endgroup$ – mobileink Jan 25 '17 at 21:31
  • $\begingroup$ "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. $\endgroup$ – Conifold Jan 27 '17 at 23:27
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This is exactly what is happening in modern mathematics: people with some books and brain discover essential things that have never been discovered, such as Euclid, Thales and Pythagoras did in the past. In Geometry as well as in other areas of mathematics. And most of these people use computers only to write e-mails and to check their bank accounts).

Type "Poincare conjecture", "Perelman" on Google and you find plenty of popular texts explaining one of the recent top achievements in geometry. If you want very recent (2016) type "Marina Vyazovska", "sphere packing". Average well-educated person can understand what has been done (not how was it done), from these popular explanations.

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  • $\begingroup$ What evidence do you have for those claims? $\endgroup$ – Rory Daulton Jan 25 '17 at 1:46
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    $\begingroup$ @Rory Baulton: I am a professional mathematician myself. Dealing with this every day. I don't know what your background is (not much judging by your question) but you may try to read this site for general public of the Amer Math Soc: ams.org/samplings/feature-column/… $\endgroup$ – Alexandre Eremenko Jan 25 '17 at 7:08
  • $\begingroup$ and as a more general principle of science: it's all, always, open-ended. meaning, we can never foreclose the possibility that somebody will eventually come along and "discover" things we never imagined. the obvious example here is non-euclidean geometry. $\endgroup$ – mobileink Jan 25 '17 at 21:29
  • $\begingroup$ Thanks for these answers: Do you think all these recent discoveries can turns out to be as usefull as an old a²+b²=c², which is quite usefull in our everyday life like for instance: *Get a right angle with a simple rope that can help to build a house *. Does things as 'Non-euclidian geometry' can provide such daily use tools ? $\endgroup$ – matt Jan 27 '17 at 12:38
  • $\begingroup$ @matt: One never knows when mathematical discoveries will find applications. People built houses even before Pythagoras. Some math invented by the Greeks found important applications only 1800 years later.No one knows whether we will exist for another 1000 years, so the question in meaningless. $\endgroup$ – Alexandre Eremenko Jan 27 '17 at 14:23
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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.

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