# Theorems in Wheat Fields

It's no wonder that farmers with fields in the plains surrounding Stonehenge, in southern England, face late-summer mornings with dread. On any given day at the height of the growing season, as many as a dozen farmers are likely to find a field marred by a circle of flattened grain.

Plagued by some enigmatic nocturnal pest, the farmers must contend not only with damage to their crops but also with the intrusions of excitable journalists, gullible tourists, befuddled scientists, and indefatigable investigators of the phenomenon.

Indeed, the study of these mysterious crop circles has itself grown into a thriving cottage industry of sightings, measurements, speculations, and publications. Serious enthusiasts call themselves cereologists, after Ceres, the Roman goddess of agriculture.

Most crop deformations appear as simple, nearly perfect circles of grain flattened in a spiral pattern. But a significant number consist of circles in groups, circles inside rings, or circles with spurs and other appendages. Within these geometric forms, the grain itself may be laid down in various patterns.

Explanations of the phenomenon range from the bizarre and the unnatural to the merely fantastic. To some people, the circles—which began appearing nearly 3 decades ago—represent the handiwork of extraterrestrial visitors. Others attribute the formations to crafty tradesmen bent on mischief after an evening at the pub, pranksters commemorating a recent movie, or even hordes of graduate students driven by a mad professor. To a few, the circles suggest the action of numerate whirlwinds, microwave-generated ball lightning, or some other peculiar atmospheric phenomenon.

These scenarios apparently suffered a severe blow in 1991, when two elderly landscape painters, David Chorley and Douglas Bower, admitted to creating many of the giant, circular wheat-field patterns that had cropped up during the previous decade in southern England. The chuckling hoaxers proudly displayed the wooden planks, ball of string, and primitive sighting device they claimed they had used to construct the circles.

But this newspaper-orchestrated, widely publicized admission didn't settle the whole mystery, and new patterns continued to appear during subsequent summers. Moreover, in the wake of their admission, retired astronomer Gerald S. Hawkins felt compelled to write to Bower and Chorley. He asked how they had managed to discover and incorporate a number of ingenious, previously unknown geometric theorems—of the type that appear in antique textbooks on Euclidean geometry—into what he called their "artwork in the crops." Hawkins concluded his letter as follows: "The media did not give you credit for the unusual cleverness behind the design of the patterns."

Hawkins' first encounter with crop circles had occurred early in 1990. Famous for his investigations of Stonehenge as an early astronomical observatory, he responded to suggestions by colleagues that he look into crop circles, which were defacing fields suspiciously close to Stonehenge.

Of course, there was no connection between crop circles and the stone circles of Stonehenge, but Hawkins found the crop formations sufficiently intriguing to begin a systematic study of their geometry. Using data from published ground surveys and aerial photographs, he painstakingly measured the dimensions and calculated the ratios of the diameters and other key features in 18 patterns that included more than one circle or ring.

In 11 of those structures, Hawkins found ratios of small whole numbers that precisely matched the ratios defining the diatonic scale. These ratios produce the eight notes of an octave in the musical scale corresponding to the white keys on a piano.

The existence of these ratios prompted Hawkins to begin looking for geometric relationships among the circles, rings, and lines of several particularly distinctive patterns that had been recorded in the fields. Their creation had to involve more than blind luck, he concluded.

Hawkins' first crop-circle candidate, which had appeared in a field in 1988, consisted of a pattern of three separate circles arranged so that their centers rested at the corners of an equilateral triangle. Within each circle, the hoaxers had flattened the grain to create 48 spokes.

Hawkins approached the problem experimentally by sketching diagrams and looking for hints of geometric relationships. He found that he could draw three straight lines, or tangents, that each touched all three circles. Measurements revealed that the ratio of the diameter of a large circle—drawn so that it passes through the centers of the three original circles—to the diameter of one of the original circles is close to 4:3.

Was there an underlying geometric theorem proving that a 4:3 ratio had to arise in such a configuration of circles? Armed with his measurements and statistical analyses, Hawkins began pondering the arrangement. After several weeks, he had his proof.

Hawkins' first theorem was suggested by a triplet of crop circles discovered on June 4, 1988, at Cheesefoot Head. Hawkins noticed that he could draw three straight lines, or tangents, that each touched all three circles. By drawing in the equilateral triangle formed by the circles' centers and adding a large circle centered on this triangle, he found and proved Theorem I: The ratio of the diameter of the triangle's circumscribed circle to the diameter of the circles at each corner is 4:3. |

Over the next few months, Hawkins discovered three more geometric theorems, all involving diatonic ratios arising from the ratios of areas of circles, among various crop-circle patterns. In one case, for example, an equilateral triangle fitted snugly between an outer and inner circle, with the area of the outer circle precisely four times that of the inner circle.

Theorem II: For an equilateral triangle, the ratio of the areas of the circumscribed (outer) and inscribed (inner) circles is 4:1. The area of the ring between the circles is 3 times the area of the inscribed circle. | |

Theorem III: For a square, the ratio of the areas of the circumscribed and inscribed circles is 2:1. If a second square is inscribed within the inscribed circle of the first, and so on to the ratio of the areas of the original circumscribed circle and the innermost circle is 2 | |

Theorem IV: For a regular hexagon, the ratio of the areas of the outer circle and the inscribed circle is 4:3. |

For Hawkins, it was a matter of first recognizing a significant geometric relationship, and then proving in a mathematically rigorous fashion precisely what that relationship is. "That was the approach I had taken at Stonehenge," Hawkins remarked. "It wasn't just one alignment here and nothing there. That would have had no significance. It was the whole pattern of alignments with the sun and the moon over a long period that made it ring true to me. Once you get a pattern, you know it probably won't go away."

There was more. Hawkins came to realize that his four original theorems, derived from crop-circle patterns, were really special cases of a single, more general theorem. "I found the underlying principles—a common thread—that applied to everything, which led me to the fifth theorem," he said. The theorem involves concentric circles that touch the sides of a triangle, and as the triangle changes shape, it generates the special crop-circle patterns.

Hawkins' fifth crop-circle theorem involves a triangle and various concentric circles touching the triangle's sides and corners. 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 |

Remarkably, Hawkins could find none of these theorems in the works of Euclid, the ancient Greek geometer who had established the basic techniques and rules for what is known as Euclidean geometry. Hawkins was also surprised at his failure to find the crop-circle theorems in any of the mathematics textbooks and references, ancient and modern, that he consulted.

This suggested to Hawkins that the hoaxer (or hoaxers) had to know a lot of old-fashioned geometry. Hawkins himself had had the kind of British grammar-school education that years ago had instilled a healthy respect for Euclidean geometry. "We started at the age of 12 with this sort of stuff, so it became part of one's life and thinking," Hawkins said. That generally doesn't happen nowadays.

The hoaxers apparently had the requisite knowledge not only to prove a Euclidean theorem but also to conceive of an original theorem in the first place—a far more challenging task. To show how difficult such a task can be, Hawkins often playfully refused to divulge his fifth theorem, inviting anyone interested to come up with the theorem itself before trying to prove it. In an article published in *The Mathematics Teacher*, he challenged readers to come up with his unpublished theorem, given only the four variations. No one reported success.

What Hawkins had obtained was a kind of intellectual fingerprint of the hoaxers involved in creating these particular crop-circle patterns. "One has to admire this sort of mind, let alone how it's done or why it's done," he remarked. Curiously, in 1996, the crop-circle makers showed knowledge of Hawkins' fifth theorem by laying down a new pattern that satisfied its geometric constraints.

Did Chorley and Bower have the mathematical sophistication to depict novel Euclidean theorems in the wheat? Not likely. The persons responsible for this old-fashioned type of mathematical ingenuity remain at large. 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 complex, precise patterns, presumably using little more than pegs and ropes, all under cover of darkness.

Perhaps Euclid's ghost is stalking the English countryside by night, leaving its distinctive mark wherever it happens to alight.