A science writer and astronomer and his student, a teen math prodigy, join forces to elucidate fields of math they find weird.

Darling (*Mayday!: A History of Flight Through its Martyrs, Oddballs, and Daredevils*, 2015, etc.) and Banerjee are struck by how some of the most abstruse findings from math turn out to have practical applications in quantum physics or computer science—or lead to concepts like orders of infinity or yield unexpected patterns of numbers or figures. One could argue that these findings are neither weird nor magical but the inexorable results of logic and the permissible rules of operation of mathematical systems by imaginative thinkers. As subjects, the authors examine selected fields of pure, as opposed to applied, math. The first chapter takes on the idea of seeing in the fourth dimension, with descriptions of the 4-D extension of the cube called a tesseract. There follows a chapter on probability emphasizing non-intuitive findings and then one on fractals, a field that deals with curves that have fractional dimensions. This idea grew out of a paper by the field’s inventor, Benoit Mandelbrot, that asked, “how long is the coast of Britain?” Thereafter, the authors’ choices are more self-indulgent, with chapters on chess and music, which will be lost on readers who are not game players or familiar with harmonics. Other areas concern computer science and number theory emphasizing primes. There is a particularly wearisome chapter on competitions to generate large and larger numbers, a sport favored by Banerjee. The text concludes with chapters on topology, set theory, infinity, and the foundations of mathematics. This is difficult material, and readers should be familiar with logical paradoxes, the meaning of “proof,” and notions of consistency and completeness of axiomatic systems as well as the work Gödel and others in establishing the incompleteness of any mathematical system complex enough to embody arithmetic.

The authors offer some beguiling insights on what math is about and how it has evolved but no royal road to easy understanding.