**SCIENCE NEWS**
©1966 Science Services Inc.
(*With thanks to Science News
Online)*
**October 12, 1996**
**Mathematics**

**Crop
circles: Theorems in wheat
fields**

#### Since the late 1970s, farmers in southern
England looking out on their wheat fields in the morning have
sometimes been startled to find large circles and other geometric
patterns neatly flattened into the crops. How these crop circles were
created in the dead of night at the height of the summer growing
season remains a puzzle, though hoaxers have claimed responsibility
for some of them.

#### Several years ago, astronomer Gerald S.
Hawkins, now retired from Boston
University, noticed that some of the most visually striking of these
crop-circle patterns embodied geometric theorems that express
specific numerical relationships among the areas of various circles,
triangles, and other shapes making up the patterns (SCIENCE NEWS:
February 1st, 1992, p. 76). In one
case, for example, an equilateral triangle fitted snugly between an
outer and an inner circle. It turns out that the area of the outer
circle is precisely four times that of the inner circle.

#### Three other patterns also displayed exact
numerical relationships, all of them involving diatonic ratios, the
simple whole-number ratios that determine a scale of musical notes.
"These designs demonstrate the
remarkable mathematical ability of their creators," Hawkins comments.

**Hawkins found that he could
use the principles of Euclidean geometry to prove four theorems
derived from the relationships among the areas depicted in these
patterns.**

**He also discovered a fifth, more general
theorem, from which he could derive the other four (see diagram).
"This theorem involves concentric circles which touch the sides of a
triangle, and as the [triangle] changes shape, it generates the
special crop-circle geometries," he says.**

** **
** **

#### 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 traingle produces one of the observed
crop-circle patterns; three isosceles triangles generate the other
crop-circle geometries. (*Courtesy of
G. Hawkins*)

#### This past summer, however, "the
crop-circle makers . . . showed knowledge of this fifth
theorem," Hawkins reports. Among the
dozens of circles surreptitiously laid down in the wheat fields of
England, at least one pattern fit Hawkins' theorem.

#### The persons responsible for this
old-fashioned type of mathematical ingenuity remain at large and
unknown. 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 out complex, precise patterns, presumably using little more
than pegs and ropes, all under cover of darkness.

Patterns of circles and triangles surreptitiously flattened into the
crops of southern England reveal the perpetrators' sophisticated
knowledge of Euclidean geometry.

*Further Readings:*

#### Anderson, A. 1991. Britain's crop circles:
Reaping by whirlwind? Science 253(Aug. 30):961.

#### Delgado, P., and C. Andrews. 1989. Circular
Evidence: A Detailed Investigation of the Flattened Swirled Crops
Phenomenon. Grand Rapids, Mich.: Phanes Press.

#### Hawkins, G.S. 1992. Probing the mystery of
those eerie crop circles. Cosmos 2(No. 1):23.

#### Jaroff, L. 1991. It happens in the best
circles. Time (Sept. 23):59.

#### Levengood, W.C. 1994. Anatomical anomalies
in crop formation plants. Physiologia Plantarum 92:356.

#### Nickel, J., and J.F. Fischer. 1992. The
crop-circle phenomenon: An investigative report. Skeptical Inquirer
16(Winter):136.

#### Peterson, I. 1992. Euclid's crop circles.
Science News. 141(Feb. 1):76.

#### Puente, M. 1991. British pair's tale called
tall. USA Today (Sept. 10).

#### Schmidt, W.E. 1991. 2 'jovial con men'
demystify those crop circles in Britain. New York Times (Sept.
10).

#### Riese, T.A., and Y.Z. Chen. 1994. Crop
circles and Euclidean geometry. International Journal of Mathematical
Education in Science and Technology. 25(No. 3):343.

#### Tunis, H.B. 1995. Geometry in English wheat
fields. Mathematics Teacher 88(December):802.

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