Mathematics differs in one key respect from taking a stock car around tight curves at high speed: You need not be a professional to experience the excitement.
Likewise, while you shouldn't combine sulfuric acid and sugar in a bowl on your kitchen counter, you can safely engage in mathematics wherever you please.
Tadashi Tokieda wants readers of his paper "Roll Models," which appears in the special March 2013 issue of The American Mathematical Monthly, to recognize these facts about mathematics. As he writes in the paper's abstract:
These notes attempt a case study of applied mathematics for beginning students via problems of rolling. They do so by pointing out diverse surprising phenomena, many of which the reader can try at home, modeling them, and testing the limits of these models. One message is that rolling, because it tightly coordinates different modes of motion, tends to be more exactly solvable than meets the eye. Another message is that the thrill of applied mathematics is not in how difficult the mathematics is, but rather in what diversity of surprises in one's own experience one can discover, then understand.
Tokieda's paper serves as an example of how to dig into a topic, how to mathematically interrogate it. He leaves readers with a reality check and a challenge:
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About the March Monthly
Comprised entirely of contributions from distinguished presenters at the 2011 International Mathematics Summer School for Students in Bremen, Germany, the March 2013 special issue—the first in more than 20 years—of The American Mathematical Monthly captures the spirit of the summer school in journal form. A full-color and generously illustrated 100 pages, the issue allows readers—be they high schoolers, college students, mathematics educators, or lifelong learners—to engage with and learn from leading international mathematicians. Papers range over such topics as unprovable arithmetic statements, bike tire tracks, the uncountability of the rationals, and how objects roll.