Conspiracy for the Day -- December 9, 1993
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("Quid coniuratio est?")
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Euclid's Crop Circles
by Ivars Peterson
[From *Science News*, Feb. 1, 1992.]
[Excerpts]
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 are likely to find a field marred by a circle of
flattened grain.
The study of these mysterious crop circles has 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 to the
unnatural. To some people, the circles -- which began appearing
about a decade ago -- represent the handiwork of extraterrestrial
invaders, or crafty tradesmen bent on mischief after an evening
at the pub.
[The paranormal-type scenarios] suffered a severe blow late last
summer, when two elderly landscape painters, David Chorley and
Douglas Bower, admitted to creating many of the giant, circular
wheat-field patterns that cropped up over the last decade in
southern England.
But this newspaper-orchestrated, widely publicized admission
didn't settle the whole mystery. Gerald S. Hawkins, a retired
astronomer, felt compelled to write to Bower and Chorley last
September, asking how they managed to discover and incorporate a
number of ingenious, previously unknown geometric theorems -- of
the type that appear in antique textbooks -- into what he called
their "artwork in the crops."
[Hawkins had begun his investigations in 1990. He] 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 these structures, Hawkins found ratios of small whole
numbers that precisely matched the ratios defining the diatonic
scale. These ratios produce the eight tones 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 geometrical 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 insists.
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.
Hawkins came to realize that his four original theorems, derived
from crop-circle patterns, were really special cases of a single,
more general theorem.
Remarkably, he could find none of these theorems in the works of
Euclid, the ancient Greek geometer who established the basic
techniques and rules of what is known as Euclidean geometry. He
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 suggests that the hoaxer or hoaxers "had to know a
tremendous lot of old-fashioned geometry," he argues.
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
refuses to divulge his fifth theorem, inviting anyone interested
to come up with the theorem itself before trying to prove it.
"It's a good test," he says. "It's easy to prove the theorem but
so difficult to conceive it."
What Hawkins now has is a kind of intellectual fingerprint of the
hoaxers involved. "One has to admire this sort of mind, let alone
how it's done or why it's done," he says.
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CfD Recommends: The video documentary entitled
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(Watch it with someone you love.)
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Today's conspiracy brought to you by.......
Brian Francis Redman
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: Aperi os tuum muto, :
: et causis omnium filiorum qui pertranseunt. :
: Aperi os tuum, decerne quod justum est, :
: et judica inopem et pauperem. :
: -- Liber Proverbiorum XXXI: 8-9 :
:.................................................:
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