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Saturday, June 28, 2003
Theorems in Wheat Fields
Retired astronomer Gerald S. 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.
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 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.
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 mth square, then the ratio of the areas of the original circumscribed circle and the innermost circle is 2m:1.
*Theorem IV: For a regular hexagon, the ratio of the areas of the outer circle and the inscribed circle is 4:3.
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.
The (creators of the crop circles) 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 (creators of the crop circles) 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.
Perhaps Euclid's ghost is stalking the English countryside by night, leaving its distinctive mark wherever it happens to alight.
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