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

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  • Linette Sukup
    An article only a mathematician could love. Peace. Linette ... From: andrew hennessey To: Sent:
    Message 1 of 1 , Aug 7 12:06 AM
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      An article only a mathematician could love.


      ----- Original Message -----
      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

      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
      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.


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

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