### Classroom Application #3: Art and Geometrical Constructibility

Crockett Johnson's mathematical paintings can be an excellent way to show students intersections between mathematics and art while exploring the principles of geometry. Stephanie Cawthorne of Trevecca Nazarene University used a version of her article, “Cubes, Conic Sections, and Crockett Johnson," to introduce constructibility to undergraduates studying Euclidean and non-Euclidean Geometry.

A study of the constructibility constraints of a compass and unmarked straightedge, discussed in “Three Greek Problems” and “Constructions in Euclid’s Elements,” can be enhanced by having students construct various segments of length a + b, a – b, a·b, a/b and \(\sqrt{a}\). Crockett Johnson’s painting *Square Roots to Sixteen* provides a nice visualization of the constructibility of \(\sqrt{a}\). When considering the inconstructibility of some of the ancient problems, Crockett Johnson's paintings suggest examples of other means of construction. For instance, *Problem of Delos (Menaechmus)* shows how Johnson portrayed a conic section solution to doubling a cube, and *Construction of the Heptagon* gives an original Neusis construction of a regular heptagon. Students are often drawn in by Crockett Johnson's creative interpretations of ancient constructions, along with his fascinating original constructions for well-known problems.

At the other end of the educational spectrum, Amy Ackerberg-Hastings has used *Square Roots to Sixteen* and *Squares of a 3–4–5 Triangle in Scalene Perspective* with pre-K, kindergarten, and 1st-grade children to discuss shapes, colors, and patterns. After the kids in each class identified the squares and triangles in the second painting's representation of the Pythagorean Theorem, they worked together with the adults to count the squares and figure out that the 9 + 16 squares on the two legs of the right triangle were the same as the 25 squares along the hypotenuse. While it is unlikely the children will remember that relationship when they next encounter the preliminaries of analytic geometry, they did get an initial exposure to these concepts in a fun context.

They really loved how the triangles curved around from small to large in *Square Roots of Sixteen*. Although they did not know that the color pattern had a specific meaning—the 3 dark gray triangles have hypotenuses whose lengths are whole numbers (the square roots of 4, 9, and 16); the 6 white triangles have hypotenuse lengths that are irrational and are square roots of even integers (2, 6, 8, 10, 12, 14); the 6 tan triangles have hypotenuses whose lengths are irrational and are square roots of odd integers (3, 5, 7, 11, 13, 15)—they much enjoyed describing the pattern and making up their own rules for how to extend the pattern. Sessions with this age group should be kept short, around 20 minutes, to be meaningful, and it is very helpful to end with a reading of Johnson's most famous children's book, *Harold and the Purple Crayon*.