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I am looking for fun, interesting mathematics textbooks which would make good studious holiday gifts for advanced mathematics undergraduates or beginning graduate students. They should be serious but also readable.

In particular, I am looking for readable books on more obscure topics not covered in a standard undergraduate curriculum which students may not have previously heard of or thought to study.

Some examples of suggestions I've liked so far:

  • On Numbers and Games, by John Conway.
  • Groups, Graphs and Trees: An Introduction to the Geometry of Infinite Groups, by John Meier.
  • Ramsey Theory on the Integers, by Bruce Landman.

I am not looking for pop math books, Gödel, Escher, Bach, or anything of that nature.

I am also not looking for books on 'core' subjects unless the content is restricted to a subdiscipline which is not commonly studied by undergrads. (E.g. Finite Group Theory by Isaacs would be good, but Abstract Algebra by Dummit and Foote would not.)

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@ZevChonoles I think this should be made CW. –  Alex Becker Dec 24 '12 at 3:12
How about Fourier Analysis, T.W. Korner, Cambridge University Press, 1988? (the "o" in "Korner" needs an umlaut, but I can't seem to get one there...) –  David Mitra Dec 24 '12 at 3:37
@Zev I've asked some time ago on meta: Should questions about book recommendations be CW? –  Martin Sleziak Dec 24 '12 at 8:27
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37 Answers 37

I'd recommend the new Dover edition of Michael Barnsley's Fractals Everywhere. Barnsley is an expert on iterated function systems (IFS) and shows how fractal geometry can be used to model real world objects. Subjects include metric spaces, dynamical systems, fractal dimension, fractal interpolation, the Julia and Mandelbrot sets, and measures on fractals. The style is engaging and the book is well illustrated.

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As someone in your target audience, I recommend A=B, by Marko Petkovsek, Herbert Wilf and Doron Zeilberger (with a foreword by Donald E. Knuth).

It's basically a book on generating combinatorial identities programmatically, inspired by Exercise in Knuth's Art of Computer Programming, Volume 1:

[50] Develop computer programs for simplifying sums that involve binomial coeffcients.

The mathematics is not above an undergraduate's head, and a lot of the results are intuitive and attractive.

Best of all, the book is available in its entirety from the website (although paper copies can also be purchased)

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I enjoyed this one:

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The Book of Numbers by John H. Conway & Richard Guy. Although this does fall into the "popular mathematics" arena, it contains breadth and depth that will be of interest. Whilst I read many formal textbooks on mathematics, The Book of Numbers is a book I often return to when working on a problem. It contains many gems.

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As leery as I am of adding yet another book (co)authored by John Conway to the list, I have to plug the amazing The Symmetries Of Things, a tremendous introduction to the basics of symmetry groups and tilings. There are things it could be better at (the fact that there's no connection with root lattices and Dynkin diagrams is a little odd to me), but it's still a fine introduction that does take some deep dives into questions of group theory.

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