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About 2000 years ago Euclid wrote a book that contains (almost) all the geometry that was known at his time. Today, in the 21st century, our knowledge of geometry increased drastically: our knowledge of euclidean geometry is better and we have better foundations. Also we have so many (interlinked) branches of geometry:

  • Euclidean geometry
  • Neutral geometry
  • Affine geometry
  • Vector geometry
  • Analytic geometry
  • Non euclidean geometry
  • Projective geometry
  • Discrete geometry
  • Differential geometry
  • Integral geometry
  • Algebraic geometry
  • Discrete differential geometry
  • Combinatorial geometry
  • Computational geometry
  • Symplectic geometry
  • Kahlerian geometry
  • Complex geometry
  • Descriptive geometry
  • Diophantine geometry
  • Metric geometry
  • Convex geometry
  • Noncommutative geometry
  • Nonriemanniann geometry
  • Arithmetic geometry
  • Topology

Did anyone try to do today what Euclid did long ago. I understand this is impossible for one person, but a group of specialists can do it. I'm not asking for an encyclopedic work but for a treatment of geometries from the most primitive to the most advanced. It will span thousands of pages but perhaps it will be the best work on geometry for the centuries to come. Does anyone have this idea? If Dieudonne alone could do a treatment of analysis in 10+ volumes, a group of mathematicians can do it.

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    $\begingroup$ Bourbaki, perhaps? $\endgroup$ – kimchi lover Apr 21 at 15:31
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    $\begingroup$ Actually, yes, you are asking for an encyclopedic treatment. As far as I know, there is no such thing. The most comprehensive books will usually only cover euclidean, affine, projective and some non-euclidean geometry. Everything else is a highly specialized field. You can't hope to cover all these topics at once. $\endgroup$ – Jean-Claude Arbaut Apr 21 at 15:31
  • $\begingroup$ @Jean-ClaudeArbaut Spivak's book span about 1900 pages so a complete treatment of geometry may need some 80 000 pages, that's a scary 80 volumes of 1000 pages each. You'll need a truck to move them. $\endgroup$ – Euclidos Apr 21 at 15:50
  • $\begingroup$ Analysis, algebra, geometry -- you are asking 1/3 of the whole repetoire of the modern mathematics, and I am afraid it is too vast to be treated by a single book done by a single human being, and is still impossibly difficult by a series of books done by a group of people. $\endgroup$ – Aminopterin Apr 21 at 18:41
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    $\begingroup$ Euclid's Elements was far from comprehensive--it was an introductory textbook, even in his time. $\endgroup$ – Eric Wofsey Apr 21 at 19:57
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I can't give a universal answer but if you are interested in the unification of several areas of geometry and group theory I would highly recommend "Metric Spaces of Non-Positive Curvature" by Bridson and Haefliger.

The book covers a vast number of topics in metric and Riemannian geometry as well as their connections to geometric group theory.

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Marcel Berger's two volume Geometry (https://books.google.com.mt/books/about/Geometry_I.html?id=5W6cnfQegYcC&redir_esc=y ) might be close to what you're looking for. As you note, the subject is now vast and so the books are not comprehensive, but they definitely give an introduction to many of the areas.

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  • $\begingroup$ I actually have these books, haven't read them yet. They treat geometry abstractly. I was searching for a treatment that starts from elementary (axioms, angles, congruent triangles...) to reach the more advanced theorems and the abstract treatment a la Marcel Berger. $\endgroup$ – Euclidos Apr 21 at 15:48
  • $\begingroup$ Euclid himself wrote other books on geometry and he left out much that was already known. Read the Wikipedia ariticle Euclid for more details. You should ask a more modest and revised question because what you asked for does not exist. $\endgroup$ – Somos Apr 21 at 17:49

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