Fw: [ufonewswire] Subject: IUFO: Theorems in Wheat Fields
- An article only a mathematician could love.
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From: "andrew hennessey" <scottishatlantis@...>
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Subject: [ufonewswire] Subject: IUFO: Theorems in Wheat Fields
Subject: IUFO: Theorems in Wheat Fields
Theorems in Wheat Fields
Ivars Peterson is the mathematics/computer writer and online editor at
(http://www.sciencenews.org). He is the author of The Mathematical
Tourist, Islands of Truth, Newton's Clock, Fatal Defect, and The
Jungles of Randomness. He also writes for the children's magazine Muse
(http://www.musemag.com) and is working on a book about math and art
Comments are welcome. Please send messages to Ivars Peterson at ip@....
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
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¡ªwhich
began appearing nearly 3 decades ago¡ªrepresent 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
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¡ª
of the type that appear in antique textbooks on Euclidean
geometry¡ªinto 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
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
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¡ªdrawn so that it passes through the centers of the
three original circles¡ªto 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
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.
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
a common thread¡ªthat 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
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.
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