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Review: Mythematics

From time to time a publisher offers me a review copy of a book. I always say yes unless I don't think I'll find something positive to say about the book. If, when I receive the book, I can't find something good to say then I won't review it.

Princeton University Press offered me a copy of a new book, Mythematics: Solving the 12 Labors of Hercules by Michael Huber. Before getting into the Math of the book I should give a little bit of background on the mythology. What are the 12 labors of Hercules? Wikipedia comes to the rescue:

The Twelve Labours of Hercules (Greek: Δωδεκαθλος, dodekathlos) are a series of archaic episodes connected by a later continuous narrative, concerning a penance carried out by the greatest of the Greek heroes, Heracles, romanised as Hercules. The establishment of a fixed cycle of twelve labours was attributed by the Greeks to an epic poem, now lost, written by Peisander, dated about 600 BC (Burkert).

As they survive, the Labours of Hercules are not told in any single place, but must be reassembled from many sources. Ruck and Staples assert that there is no one way to interpret the labours, but that six were located in the Peloponnese, culminating with the rededication of Olympia. Six others took the hero farther afield. In each case, the pattern was the same: Hercules was sent to kill or subdue, or to fetch back for Hera's representative Eurystheus a magical animal or plant. "The sites selected were all previously strongholds of Hera or the 'Goddess' and were Entrances to the Netherworld".

The gist of Hercules' mission is in the last paragraph above. "Hercules was sent to kill or subdue, or to fetch back for Hera's representative Eurystheus a magical animal or plant."

So, what does Hercules' adventures have to do with Math? Mr. Huber uses the 12 labors as a backdrop with which to present inventive Math problems. How might Hercules have used mathematics to accomplish each of the labors? Labor 1, for example, involves trapping and killing a lion.

And when the lion took refuge in a cave with two mouths, Hercules built up the one entrance and came in upon the beast through the other, and putting his arm round its neck held it tight till he had choked it; so laying it on his shoulders he carried it to Cleonae.

Here's an inventive mathematical problem derived from the story.

To defeat the lion, Hercules must close up one cave entrance and attack the lion through the other. He finds several stacks of tiles nearby, each of which contains sets of regular polygons. There is one stack of equilateral triangles, one stack of squares, one stack of regular pentagons, one stack of regular hexagons, and one stack of regular octagons. Which stack(s) of polygons will allow Hercules to construct an edge-to-edge tiling in order to close up the mouth of the cave with no two tiles overlapping?

This problem is not too difficult for someone comfortable with the basics of angles within regular polygons.

Not all the problems in the book are this easy. Many are tougher and require a fair amount of engagement and mathematical sophistication to grasp and solve. Also, I wouldn't call this book a recreational Math book because I consider recreational Math to consist of problems which are accessible to students with a high school Math background. Mythematics has problems that span these branches of mathematics:

algebra, combinatorics, difference equations, differential calculus, differential equations, geometry, integral calculus, multivariable calculus, probability, simulations, statistics, and trigonometry

I do think that Mythematics would be enjoyable to college educated folks, or to high school students who learn and enjoy calculus. In particular, I think engineering students in college who like Math would appreciate the problems in this book.