Exploring Classical Greek Construction Problems with Interactive Geometry Software

Double a cube? Trisect an angle? Square a circle? Yes, you! In their undergraduate text Exploring Classical Greek Construction Problems with Interactive Geometry Software, Ad Meskens and Paul Tytgat guide the reader on a historical journey to perform three “impossible” feats, treating the reader to myriad constructions, not purely by compass and straightedge, of course, but by incorporating other means including conic sections, loci, and mechanical devices.

Proceeding through a sequence of 150 purposefully chosen problems, woven together by historical motivation and stories, the reader encounters pirate’s treasure and tomahawks, spirals and conchoids, and last but not least, a heptagon named Cinderella. Some problems call for pencil and paper, while others call for interactive geometry software (IGS). Most are revisited from several perspectives. Given the heavy emphasis on neusis constructions and loci, IGS is an invaluable visualization tool. While the problem sequence structure and incorporation of IGS are similar to Venema's Exploring Advanced Euclidean Geometry with GeoGebra, the choice of topics and stronger emphasis on history set the book under review apart.

The journey begins with an ample introduction to three construction modes in “Plato’s hierarchy”: by conics, by mechanical devices, and by compass and straightedge. Highlights here include a fun and elaborate IGS exploration to solve Apollonius’s three circles tangency problem via hyperbolae as well as several IGS exercises to create digital versions of an assortment of mechanical devices (e.g., Archimedes’s trammel) to draw an ellipse.

Continuing, we reach the core chapters on the three main problems: doubling a cube, trisecting an angle, and squaring a circle. With the story of the Delian altar setting the stage, the reader sets out to double a cube, i.e., to construct the side of a cube with double the volume of a given cube. Whether by rotating and sliding configurations of rods or by intersecting conics or conchoids, without fail, the cube root of two appears. For the second feat, trisecting an angle, mechanical constructions again abound, along with constructions using special curves, such as conchoids, Archimedean spirals, and the quadratrix. Come time to find a square with the same area as a circle, the mathematics does get a bit sketchy, as the reader gets a taste of some of the missteps of mathematicians past. A personal favorite is Franco de Liège’s attempt to square a circle of radius seven, which leads to some dubious mathematics, not too much unlike the infinite chocolate bar trick. In the final chapters, where IGS exercises become sparse and trig equations spawn, the reader at last faces the fact that, in a finite number of steps, compass and straightedge alone cannot produce solutions to any of the three classical problems at hand.

Throughout, the authors' sense of humor and enthusiasm shine through as they remind the reader to take precautions in cutting the cardboard (yes, there are instructions for making your own tools) and, in the penultimate chapter, plant a gentle invite to discover the nature of \(\pi^{\pi}\). Figures and images of historical texts dot nearly every page, and numerous footnotes lead to references (including some primary sources) for the reader who wants to learn more.

As a whole, the book captures a great story that would be fitting for a semester-long inquiry-based course or independent study. Yet teachers will also be able to use this collection of gems as a source of inspiration for problems and projects to enrich a variety of courses at the undergraduate or high school level. In the reviewer’s opinion, it is powerfully motivating to see polar curves and conic sections emerge naturally in pursuit of solutions to what at first seem to be unrelated problems. Even if a construction cannot itself be performed using only compass and straightedge, its validity often follows pleasingly from right triangle trig and basic Euclidean geometry. An occasional limit or series sneaks in, but calculus is not needed for the majority of the book.

As for quibbles, there are a few instances of inconsistencies in the labels in figures and the accompanying prose, but they are nothing a patient reader cannot work through, and indeed, may lead the reader to a deeper understanding or the discovery of an additional truth.

Overall, Meskens and Tytgat have admirably accomplished their proclaimed goals for the text. The historical context piques the reader's interest and highlights the human element of mathematics, while the explorations with IGS allow the reader to feel a sense of discovery for themselves. By the end of it all, the reader will no longer joke about impossible tasks but instead seriously inquire, “With what tools, may I ask?”

Briana Foster-Greenwood is an Assistant Professor of Mathematics at Cal Poly Pomona. In her spare time, she plays the flute and folds too much modular origami.