Excerpts from a Science News article titled **Math in the Fields**, from the *New Age Journal*, June '97

Whoever or whatever is creating crop circles in southern England is apparently a whiz at math. Several years ago, mathematician/astronomer Gerald S. Hawkins, author of "

Stonehenge Decoded" and now retired fromBoston University, noticed that some of the most visually striking of these crop-circle patterns expressed specific numerical relationships among the areas of various circles, triangles, and other shapes that make up the patterns.Hawkins found that he could use the principles of Euclidean geometry to prove four theorems derived from the relationships depicted in these patterns. In one case, for example, an equilateral triangle fitted snugly between an outer and an inner circle. It turns out that the area of the outer circle is precisely four times that of the inner circle. Three other patterns also displayed exact numerical relationships involving diatonic ratios, the simple whole-number ratios that determine a scale of musical notes. "These designs demonstrate the remarkable mathematical ability of their creators," Hawkins comments.

Hawkins also discovered a fifth, more general theorem, from which he could derive the other four. "This theorem involves concentric circles which touch the sides of a triangle, and as the [triangle] changes shape, it generates the special crop-circle geometries," he says. Different triangles give different sets of circles. An equilateral triangle produces one of the observed crop-circle patterns; three isosceles triangles generate the other crop-circle geometries.

Curiously, Hawkins could find no reference to such a theorem in the works of Euclid or in any other book that he consulted. He challenged readers of

Science NewsandThe Mathematics Teacherto come up with his unpublished theorem, given only the four variations. No one reported success. And then, during the crop circle season of 1996, among the dozens of circles surreptitiously laid down in the wheat fields of England, at least one pattern fit Hawkins' theorem. "The crop-circle makers showed knowledge of this fifth theorem," Hawkins reports.Those responsible for this old-fashioned type of mathematical ingenuity remain unknown. Their handiwork reveals an astonishing ability to enter fields undetected, bend living plants without cracking stalks, and trace out complex, precise patterns demonstrating an uncommon facility with Euclidean geometry. (JG)