General audience books that teach deep mathematics

Solution 1:

• I recommended Conway et al.'s Symmetries of Things in a recent answer, and I'll do it again!

Essentially, the first third of the book is devoted to two-dimensional symmetry, and is generally very layperson-friendly, with lots of good pictures and things that utilize intuition. Generally the first 100 pages should be doable for anybody with a genuine interest. The next 100 pages are a bit more difficult, and then abstract groups are encountered, at which point all bets are off (but to be fair, it's still a pretty gentle treatment!). So, maybe it's not a very good fit for young children, but I'd say ages 10-12+ should be able to get plenty out of it. It doesn't hide the fact that it's a book about math, whatever that means.

• And while I realize this isn't a book, Satyan Devadoss has a series of video lectures about quality mathematics for a general audience, called The Shape of Nature. It's not cheap, but I managed to borrow a copy through an interlibrary loan and it was well worth it.

Solution 2:

In Russia, those 2 books were very popular:

• http://www.amazon.com/Algebra-Fun-Yakov-Perelman/dp/2917260262/ref=sr_1_1?ie=UTF8&qid=1450470498&sr=8-1&keywords=perelman
• http://www.amazon.com/Physics-Entertainment-Yakov-Perelman-ebook/dp/B00MVB5PRY/ref=sr_1_2?ie=UTF8&qid=1450470498&sr=8-2&keywords=perelman

not sure how their english translation compares to the original Russian though