Crop circles: theorems in wheat fields.

Since the late 1970s, farmers in southern England looking out on their wheat fields in the morning have sometimes been startled to find large circles and other geometric patterns neatly flattened into the crops. How these crop circles were created in the dead of night at the height of the summer growing season remains a puzzle, though hoaxers have claimed responsibility for some of them.

Several years ago, astronomer Gerald S. Hawkins, now retired from Boston University, noticed that some of the most visually striking of these crop-circle patterns embodied geometric theorems that express specific numerical relationships among the areas of various circles, triangles, and other shapes making up the patterns (SN: 2/1/92, p. 76). In one case, for example, an equilateral triangle fitted snugly between an outer and an inner circle (see photo). 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, all of them 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 found that he could use the principles of Euclidean geometry to prove four theorems derived from the relationships among the areas depicted in these patterns. He also discovered a fifth, more general theorem, from which he could derive the other four (see diagram). "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.

Curiously, Hawkins could find no reference to such a theorem in the works of Euclid or in any other book that he consulted. When he challenged readers of Science News and The Mathematics Teacher to come up with his unpublished theorem, given only the four variations, no one reported success.

This past summer, however, "the crop-circle makers . . . showed knowledge of this fifth theorem," Hawkins reports. Among the dozens of circles surreptitiously laid down in the wheat fields of England, at least one pattern fit Hawkins' theorem.

The persons responsible for this old-fashioned type of mathematical ingenuity remain at large and unknown. Their handiwork flaunts an uncommon facility with Euclidean geometry and signals an astonishing ability to enter fields undetected, to bend living plants without cracking stalks, and to trace out complex, precise patterns, presumably using little more than pegs and ropes, all under cover of darkness.