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From: "andrew hennessey" <scottishatlantis@...>

To: <ufonewswire@yahoogroups.com>

Sent: Thursday, July 27, 2006 7:49 AM

Subject: [ufonewswire] Subject: IUFO: Theorems in Wheat Fields

Subject: IUFO: Theorems in Wheat Fields

Theorems in Wheat Fields

Ivars Peterson

Ivars Peterson is the mathematics/computer writer and online editor at

Science News

(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

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¡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

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¡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

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¡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

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.

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

Hawkins

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

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.

References:

More- http://www.combat-diaries.co.uk/diary24/diary24chapter9.htm

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