TITLE: Book ref. request: "...starting from a mathematically amorphous problem and combining ideas from sources to produce new mathematics..."
QUESTION [38 upvotes]: I couldn't find Charles Radin's Miles of Tiles at the local university library or the public library, and cannot afford its Amazon price right now. 
Thus, while sorely disappointed for the moment, I have decided to try and solicit recommendations for other books like Radin's Miles of Tiles, which share these particular features:
Theme: "In this book, we try to display the value (and joy!) of starting from a mathematically amorphous problem and combining ideas from diverse sources to produce new and significant mathematics--mathematics unforeseen from the motivating problem ... "
Style: The common thread throughout this book is <insert topic here>...the presentation uses many different areas of mathematics and physics to analyze features of <insert topic here>...[as] understanding <insert topic here> requires an unusual variety of specialties...this interdisciplinary approach also leads to new mathematics seemingly unrelated to <insert topic here>...
Readership: Advanced undergraduates, graduate students, and research mathematicians.
(P.S. through the generosity of my parents, I was able to order myself a copy of Miles of Tiles!)

REPLY [6 votes]: In this post, I would like to document suggestions I have come across myself. 
I have been reading Counting on Frameworks recently, and it fits the style of my original request. The author builds up a mathematically amorphous problem ("what is a rigid structure?"), progressively developing a theory (along with the participation of the reader through exercises) to describe "rigidity".
Since I had to learn some basic combinatorics, I came across  a book which fits well with the theme at hand: Combinatorics through guided discovery, by Kenneth Bogart. It seems to be quite fun! You learn combinatorics by going through exercises, rather than by being told what it is -- in other words, you make up the subject yourself, in a guided way. The book is completely free, and available online.
I think the textbook for the online course Formal Methods of Software Design, available online for free again, is an excellent introduction to boolean logic, and general theory building -- it has been a fantastic base for learning other math. As a cool side effect (or was that the original goal?), you'll learn how to derive computer programs (yes, computer programs), from boolean specifications: very much like how you would derive any other proof! Fantastic.
I also came across the course Learning how to Learn recently (offered by the UC San Diego, on Coursera), which is actually, hands down, the most honest and effective course (good MOOCs are rare these days) on the subject that I have ever come across. Generally useful ideas to keep in mind while learning new mathematics!