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For my best friend's birthday, I am looking for a geometry book. He's currently doing his math PhD and is really fond of geometry, especially hyperbolic or higher-dimensional ones, also interested in (star or not) polytopes. He is no beginner at all, having read Coxeter or Gromov, so I am looking for a real hard-math book for him.

I looked at Thurston, but it seems most of his works are actually freely available online, and I would prefer not to give him a free-printed pdf as a present :/

It may be a theorem or theory-presentation book, maybe with a historical part though not necessary nor prioritary.

What would you advice to me ?

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closed as off-topic by Anton Petrunin, j.c., R. van Dobben de Bruyn, Nate Eldredge, András Bátkai Sep 7 '18 at 9:43

This question appears to be off-topic. The users who voted to close gave this specific reason:

  • "This question does not appear to be about research level mathematics within the scope defined in the help center." – Anton Petrunin, j.c., R. van Dobben de Bruyn
If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ If I were a geometer, I would want to pick out any "state-of-the-art" book myself. For a present I would much rather receive something historically significant, like a copy of Riemann's Habilitation Lecture for example. $\endgroup$ – Willie Wong Sep 5 '18 at 19:15
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    $\begingroup$ Also, there's a high probability that your friend owns a state of the art book that fits within his interests. You could of course ask him if there's a book he wishes he had but seems to be too expensive. $\endgroup$ – Deane Yang Sep 5 '18 at 19:48
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    $\begingroup$ Maybe buy him some hyperbolic or higher-dimensional sculpture instead? Like ones here shapeways.com/shops/henryseg $\endgroup$ – Zeno Rogue Sep 5 '18 at 22:17
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    $\begingroup$ @ZenoRogue: wow! Thanks for pointing that out. There are some really awesome stuff there. $\endgroup$ – Willie Wong Sep 6 '18 at 0:52
  • $\begingroup$ A state-of-the-art book about all of geometry?? That is going to be taugh... $\endgroup$ – Qfwfq Sep 7 '18 at 9:45
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  1. M. Berger, A Panoramic view of Riemannian geometry,

  2. M. Berger, Geometry revealed. A Jacob's Ladder to modern higher geometry.

I promise you that your friend will enjoy this gift if s/he likes geometry, and does not have these books yet.

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