Ivars Peterson's MathTrek
June 30, 2003
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 circleswhich began appearing nearly 3 decades agorepresent 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 theoremsof the type that appear in antique textbooks on Euclidean geometryinto 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 circledrawn so that it passes through the centers of the three original circlesto 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.
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.
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 principlesa common threadthat 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.
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 placea 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 out 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.
Copyright 2003 by Ivars Peterson
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Andrews, C., and S.J. Spignesi. 2003. Crop Circles: Signs of Contact. Franklin Lakes, N.J.: Career Press.
Delgado, P., and C. Andrews. 1989. Circular Evidence: A Detailed Investigation of the Flattened Swirled Crops Phenomenon. Grand Rapids, Mich.: Phanes Press.
Hawkins, G.S. 1992. Probing the mystery of those eerie crop circles. Cosmos 2(No. 1):23-27.
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Peterson, I. 1996. Crop circles: Theorems in wheat fields. Science News 150(Oct. 12):239. Available at http://www.sciencenews.org/sn_arch/10_12_96/note1.htm.
______. 1992. Euclid's crop circles. Science News 141(Feb. 1):76-77.
Pinchbeck, D. 2002. Wheat graffiti. Wired (August):114-117.
Puente, M. 1991. British pair's tale called tall. USA Today (Sept. 10).
Ridley, M. 2002. Crop circle confession. Scientific American 287(August):25.
Riese, T.A., and Y.-Z. Chen. 1994. Crop circles and Euclidean geometry. International Journal of Mathematical Education in Science and Technology 25(No. 3):343-346.
Schmidt, W.E. 1991. 2 'jovial con men' demystify those crop circles in Britain. New York Times (Sept. 10).
Tunis, H.B. 1995. Geometry in English wheat fields. Mathematics Teacher 88(December):802.
Gerald Hawkins died suddenly on May 26, 2003, at the age of 75. See http://www.cropcirclenews.com/modules/news/article.php?item_id=30.
Comments are welcome. Please send messages to Ivars Peterson at firstname.lastname@example.org.
A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the MAA book Mathematical Treks: From Surreal Numbers to Magic Circles. Find it at the MAA Bookstore.