Geometry revisited

What Hacker News says

1. generationPJul 2020
   Concrete Mathematics by Knuth and Patashnik (already mentioned for u/pmiller2) if the kid likes numbers. That's perhaps the guiding thread of the book -- it's about the beautiful (yet usually very elementary and natural) things you can do with numbers.

   Geometry Revisited by Coxeter and Greitzer and/or Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Honsberger if the kid is into plane geometry. It's an idyllic subject, great for independent exploration, and the books shouldn't take long to read. Not very deep, though (at least Honsberger).

   Anything by Tom Körner, just because of the writing. Seriously, he can make the axiomatic construction of the real number system read like a novel; open https://web.archive.org/web/20190813160507/https://www.dpmms... on any page and you will see.

   Proofs from the BOOK by Aigner and Ziegler is a cross-section of some of the nicest proofs in reasonably elementary (read: undergrad-comprehensible) maths. Might be a bit too advanced, though (the writing is terse and a lot of ground is covered).

   Problems from the BOOK by Andreescu and Dospinescu (a play on the previous title, which itself is a play on an Erdös quote) is an olympiad problem book; it might be one of the best in its genre.

   Oystein Ore has some nice introductory books on number theory (Number Theory and its History) and on graphs (Graphs and their uses); they should be cheap now due to their age, but haven't gotten any less readable.

   Kvant Selecta by Serge Tabachnikov is a 3(?)-volume series of articles from the Kvant journal translated into English. These are short expositions of elementary mathematical topics written for talented (and experienced) high-schoolers.

   I wouldn't do Princeton Companion; it's a panorama shot from high orbit, not a book you can really read and learn from.

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2. nextosNov 2013
   There are some great books to build up some intuition and rigor in all branches of math, with no prior knowledge:

   * How to Prove It (Velleman)

   * Algebra; Trigonometry; F&G; The Method of Coordinates (Gelfand)

   * Geometry (Kiselev)

   * Calculus Made Easy (Thompson)

   * How to Count without Counting (Niven)

   * Introduction to Probability Theory (Hoel)

   * The Little Schemer (Friedman)

   The proceed to more advanced texts like:

   * Naive Set Theory (Halmos)

   * Linear Algebra Done Right (Axler)

   * Geometry Revisited (Coxeter)

   * Infinitesimal Calculus (Keisler)

   * Concrete Mathematics (Graham)

   * Information Theory, Inference and Learning (MacKay)

   * SICP (Abelson)

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