This is a “Young Readers” science (and math) fiction book — say, ages 10 to 14. And it’s got math galore in it. Some of the many examples are the continued fraction for 4/π (p. 49), chaos theory (“…butterflies can cause storms…” p. 50), and the Mandelbrot set (p. 72 — complete with its definition, though not explaining all the sub-concepts in that definition, such as “complex c-values”, “orbit”, and “bounded”). Hilbert space is also mentioned quite often, though usually in a mystical rather than mathematical way, sometimes almost synonymous with outer space, or some mysterious other-world. On page 3 one of his characters mentions “a Hilbert space of all Hilbert spaces”, and I wonder how mathematically feasible that is. There are, sometimes, references to familiar mathematical apocrypha, such as the one where a mathematician, in a car with our family of protagonists, says, upon passing a field of Angus cows, “There exists a field, containing at least one cow, of which at least one side is black.”

Most of the math in the book is mentioned rather than explained, which is probably for the better; young readers, anticipating an exciting read, might not generally want to be specifically *taught* math in the process. (They can always Google anything that excites their curiosities.) Many aspects of mathematical *life*, however, are described, as ten-year-old Darby and his 17-year-old sister Livey observe the atmosphere in their home with mathematical/theoretical physics parents (later only their father, since the mother leaves the family for a more suave physicist whom she has met at a conference). In particular there are often mathematical guests such as the above-mentioned cow axiomatizer. So readers get to see a slice of life which they might not have otherwise, and for the most part, that slice is depicted in a positive light, although both of the kids — the math prodigy Darby and his less mathematically prodigious sister Livey — seem to feel that mathematicians can be weird. (Kids, though, often feel some version of the idea that *all* adults can be weird.)

One of the premises of the plot is that, once the existence of a mathematical object is proven, that object can take the form of a human being, with powers and privileges. (That seemed a little corny to me, until I remembered that one of my poems asks, about a beautiful, interesting, but badly tattered piece of clothing encountered at a thrift store, “Which mathematical object are you? What’s your Godel number?”) Darby, a math genius from toddler-hood but also a normal playful ten-year-old, proves the existence of “Bob” (named after a Thigamabob Conjecture). Part of the excitement of the book is that this Bob might be a bad guy, though (another of the book’s premises) as a mathematical object he cannot tell a lie.

There are many humorous and lively anecdotes in this book. Suffice it to give the example that Bob, good or bad, has a ball as a new earthling; in particular, he enjoys the food and knock-knock jokes (much, perhaps, as the king in “The King and I” enjoys things like “etc. etc. etc.”, though that doesn’t excuse the evil that he does).

Many things happen in the book — ordinary things like how Darby gets along, or doesn’t, at his new school for genius kids, and science- and math-fiction-y goings on — concerning, for example, some modern version of the Pythagorean Brotherhood, in particular swarms of triangles and the “holy tetraktys”. The plot keeps thickening, the kids’ estranged mother gets involved, along with her new husband, so does a great aunt confined to a mental institution, so does Livey’s math teacher, and so does a new male classmate of Livey’s, with whom (P.S…) she falls increasingly in love but who is, increasingly, enigmatic and magical. Towards the end there’s a chase, or several chases, mostly out of this world. (One world has a blue sun and purple ocean.) There are angels, good and bad, and perhaps predictably, there’s a black hole in the next-to-last chapter, over which Darby is dangled.

To my mind, the plot involves more mysticism than math — which is not to my liking, but perhaps it’s meant to be corniness rather than mysticism. (And I’m okay with corny.) I hope that young readers won’t permanently confuse the mysticism (or corniness) with the math.

What I chuckled most over was the character of Livey, an ordinary teen-age girl “stuck” in a family of math geniuses — both of her parents, her step-father, and her younger brother. Livey’s a cheerleader, slightly boy-crazy but not for just *any* boy, and she’s in danger of failing algebra (and thereby getting kicked off the cheerleading squad); in particular, she can’t remember that, nor understand why, the product of two negatives is a positive.

Of course, one could criticize the choice of the one mathematical “black sheep” in this family being female, but perhaps the mother and great-aunt make up for that. The idea of one non-mathematical family member can be thought of, and was by me, an extended joke, and actually — thought I’m not sure why — the joke takes on a different quality, at least for me, with that member being female; the idea of a male math-dunce would be funny in a different way. That’s probably a reflection of societal attitudes.

At any rate, I do think it a tad surprising (meaning unbelievable) that, throughout the entire book, no one — not her father, not her brother, not her math teacher, not Bob, not her eventual boyfriend — was able or willing to explain to her why a negative times a negative is a positive. (I explain it to pre-Calc students by saying, first, that it has to be that way in order for negative numbers to follow the same laws as positive numbers — in particular the distributive law — and then going into a calculation — involving numbers, not variables — showing why that’s true. I also remind students that, in a sentence or clause, two subsequent not’s can be cancelled out. That last, if I remember correctly, is mentioned is the book, but not to Livey.)

Speaking of whom — somehow she finally does get it and, in the end, that’s what saves the day. This is actually not a “spoiler”, since there’s much more to the ending than that, and in fact, I *couldn’t* become a spoiler even if I wanted to, since I don’t completely understand the ending, especially the very last page! In fact, this is one of those science fiction books that gets so involved that readers (readers like me, at any rate) don’t catch all the details.

Another part of this book that I enjoyed was the author’s Acknowledgements at the beginning. It exudes human warmth and enthusiasm. It’s clear that the author, though not a mathematician, has a special relationship with math. Among the people whom he acknowledges (not always out of personal contact) are Clifford Pickover, Martin Gardner, and Mary Lynn Reed, a writer/mathematician of whom I had not been aware. (When I Googled her, I was glad that I had been introduced to her.) The author also mentions his son and young daughter as well as the cheerleader daughter of a colleague, who provided him with information about Livey’s cheerleading world. And of course he thanks his editors and agent, as well as an encouraging friend, Thea Atkinson.

All in all, this book held my interest, is well written (for the most part; I found one transition on page 4 a little abrupt), and introduces young readers to the world of *adult* math in an engaging way.

Marion Deutsche Cohen is the author of “Crossing the Equal Sign” (Plain View Press, TX), poetry about the experience of math. Her books total nineteen, including the forthcoming “Chronic Progressive”, not specifically about math but containing many math references. She teaches at Arcadia University, in Glenside PA, and would welcome email at: mathwoman199436@aol.com.