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I would like some suggestions on a nice book on geometry to gift an undergraduate. I'm not searching for something that is common: I need something new and exciting. Suggestions?

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There are better gifts for an undergraduate than a book on mathematics :) –  Lord Soth Apr 1 at 20:52
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I am not exactly sure what an undergraduate is - I don't know the American education system that well, but I like "The symmetries of things" by Conway et al. –  Oliver Braun Apr 1 at 20:59
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@LordSoth: Glen Livet 25? –  Asaf Karagila Apr 1 at 21:16
    
@AsafKaragila I would take it for sure, but an undergrad would waste it. Heineken $10^3$-pack would be a better option. –  Lord Soth Apr 1 at 21:28
    
@LordSoth: I wouldn't touch Heineken with a poking stick, even before my undergrad years. But my Glen Livet 18 made it through my undergrad without worries, with only the occasional drink. –  Asaf Karagila Apr 1 at 21:30

4 Answers 4

up vote 3 down vote accepted

I love this one. It is one of the most stimulating books on geometry that I have ever read: it is strikingly innovative and really enjoyable (also, it does not require a particularly advanced background).

From the book description:

New Horizons in Geometry represents the fruits of 15 years of work in geometry by a remarkable team of prize-winning authors—Tom Apostol and Mamikon Mnatsakanian. It serves as a capstone to an amazing collaboration. Apostol and Mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Using new and intuitively rich methods, they give beautifully illustrated proofs of results, the majority of which are new, and frequently develop extensions of familiar theorems that are often surprising and sometimes astounding. It is mathematical exposition of the highest order.

The hundreds of full color illustrations by Mamikon are visually enticing and provide great motivation to read further and savor the wonderful results. Lengths, areas, and volumes of curves, surfaces, and solids are explored from a visually captivating perspective. It is an understatement to say that Apostol and Mamikon have breathed new life into geometry.

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Wow! Thank you very much! This is really appropriate! :) –  user139589 Apr 1 at 21:04
    
You're welcome. –  Nick Apr 1 at 21:08

Green Lion Press's:

or

The latter is very unique; it's all the propositions and diagrams, but without the detailed proofs.

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Another very nice book is Indra's pearls The vision of Felix Klein with a preview at google.

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Avatamsaka Sutra –  Andrew Apr 2 at 2:29
    
Yeah @Andrew Of course you knew the Hua-yen Sutra with a nice illustration here. Cheers! –  Raymond Manzoni Apr 2 at 19:03
    
Thanks - very cool! –  Andrew Apr 2 at 19:36

I'll recommend Geometry Revisited by H. S. M. Coxeter and S. L. Greitzer.

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