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I don't usually like these types of big list questions, but I think this is actually fairly important as far as education, informing the public about what we do, and getting people excited about math goes. There were some pop math books that exposed me to great ideas and essentially made me want to become a mathematician.

For some examples of what I mean by "pop math" see this or this post.

What circle of ideas, biography of a mathematician, conjecture, or subject would you like to see as a popular math book for non-mathematicians?

Answers might be in the form of a theoretical title plus a brief description. For example,

Bezout's Theorem: a history of how the quest to understand how two curves intersect led to the birth of modern algebraic geometry.

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Do you want a book that recruits future mathematicians? Without a goal I'd suggest this is way too broad a question. – Thomas Andrews Aug 3 '13 at 1:04
I'm not sure I see how stating some goal like that would narrow it down. Surely no one sits down to write a popular math book with that narrow of a goal. Surely these authors find the subject matter fascinating and want to share these ideas with a general audience. If they do a good job of getting their passion across, then they'd probably as a side effect recruit future mathematicians. This big list is not more broad than "Which one result in maths has surprised you the most?" which has 116 upvotes, since only a small number of those would probably make good general audience books. – Matt Aug 3 '13 at 3:05
Then what is the purpose of the question? This is not a discussion forum, it is an answers forum, and questions which are too broad really don't belong here. – Thomas Andrews Aug 3 '13 at 3:07
Can you articulate your complaint better? I've pointed out that it is less broad than other successful list questions. Also, what is the purpose of any question on this site? Asking that question is meaningless. I would like to know people's answers. It is math related, and I don't see how it will generate much "discussion" beyond the answers. If you are uninterested in answering, then ignore the question. No one is forcing you to look at it. Other people seem interested, and are not confused as to what the purpose or specifications are as can be seen below. – Matt Aug 3 '13 at 3:26
Quite a few of the books on the MO list are suitable for non-mathematicians sensu stricto. E.g., Havil’s Gamma is a fine book, but one needs at least calculus to appreciate it. Frankly, there are lots of good popular math books out there; doubtless there are many gaps in their coverage, but I see no burning need for anything in particular. – Brian M. Scott Aug 3 '13 at 15:43

There's room for a really well done biography of von Neumann. The one by MacRae is ok, but not great.

I always thought Von Neumann’s brain indicated that he was from another species, an evolution beyond man. --Hans A. Bethe

Peter Lax described von Neumann as possessing the "most scintillating intellect of this century."

Von Neumann was the first to rigorously establish a mathematical framework for quantum mechanics. He played a key role in inventing the computer. He invented game theory, and later used ideas from game theory in advising US officials at the highest level during the cold war. He is credited with the equilibrium strategy of mutually assured destruction, providing the deliberately humorous acronym, MAD.

"If you say why not bomb [the Soviets] tomorrow, I say, why not today. If you say today at five o’clock, I say why not one o’clock?". -- von Neumann

There's ample material for biographies here.

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The Mathematical Gamer: How deep mathematics arises in video games. Many competitive video games can be analyzed mathematically and the mathematics could get pretty deep. Easily some probability problems, calculus, linear programming (real-time strategy games like starcraft), pursuit-evasion (tag-based games), calculus of variations (geodesics in racing games).

Also various puzzles lend themselves to talking about abstract mathematical ideas. For instance, there's group theory in Peg solitaire, and linear algebra in Fiver. It could be a fun way to introduce these abstact concepts.

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A pop math could do more justice to the work of the ancient mathematicians such as Sieve of Eratosthenes. Modern ways of presentation using applets embedded in a web page would really bring their work to life, and it would help explain such artifacts as base 60 time telling.

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