Euclidean Geometry of Crop Circles
Gerald Hawkins' work on crop circles and their
relationship
to Euclidean geometry and diatonic ratios.
Several years ago, astronomer Gerald S. Hawkins, former Chairman of the astronomy department at Boston University, noticed that some of the most visually striking of the cropcircle 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: 2/1/92, 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 a diatonic ratio, the simple wholenumber 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 crop circles. He also discovered a fifth, more general theorem, from which he could derive the other four (see diagram, left). "This theorem involves concentric circles which touch the sides of a triangle, and as the [triangle] changes shape, it generates the special cropcircle geometries," he says.
Hawkins' fifth cropcircle 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 cropcircle patterns; three isosceles triangles generate the other cropcircle geometries.
What is most surprising is that all geometries give diatonic (musical) ratios. Never before have geometric theorems been linked with music.
Curiously, Hawkins could find no reference to such a theorem in the works of Euclid or in any other book that he consulted. When he challenged readers of Science News and The Mathematics Teacher to come up with his unpublished theorem, given only the four variations, no one reported success.
In July 1995, however, "the cropcircle makers . . . showed knowledge of this fifth theorem," Hawkins reports. Among the dozens of circles surreptitiously laid down in the wheat fields of England, one pattern fit Hawkins' theorem based on the stringent definitions, on the rules established by the circles over the period 1980 to the present.
The Circlemakers responsible for this oldfashioned type of mathematical ingenuity remain at large and unknown. Their handiwork flaunts a facility with Euclidean geometry and signals an astonishing ability to bend living plants without cracking stalks, and to trace out complex, precise patterns, most under cover of darkness, with a few notable exceptions during daytime.
NOTE: The circlemakers' fifth theorem has been published in the Mathematics Teacher, the magazine of the National Council of Teachers of Mathematics. Get it through your local library, or order it for US dollars by email Attention "Kay Reuter/Customer Service Dept" orders@nctm.org Ask for information on how to order page 441, volume 91, Number 5. the issue for May, 1998.
The Latest Work
Part 1
T 448 (Andrews Catalogue)
O is the center of circles 1,2,3, and the center of equilateral triangle ADE. ABC is also equilateral with height AD. The moon has center D, radius AD. But OB is also a height of triangle ABC, therefore circle 3 with radius OB is the same size as the moon. Circle 2 is tangent to the moon on OD produced, and circle 1 is the exterior circle of the hexagon tangent to circle 2. This construction fits the crop formation to within the limits of measurement, and we can find the areas of the circles exactly. They give diatonic ratios. From 1 to 2 we get a ratio of 4/3, and from 1 to 3 we get closely a ratio of 10/3.
This geometry is repeated three times by rotating 60º and 120º. The terminator or shadowline of the moon is an arc of radius CB centered on C. Point B is exactly at the middle of the terminator, and exactly where circle 3 intersects the terminator. The circle of the disc of the moon also passes through E, that is why it touches circle 2 on OD produced. This makes the tip of the moon in the crop formation curvein slightly from the outer circle. Is this all an artistic accident, or is it clever design? Are we suppoed to discover where the triangles are,and the exact sizes of the three circles, 1,2, and 3? Is it confirmation of our work that when we get the answer the circles give diatonic ratios?
The six outer loops are embellishments giving a hint of the hexagon. The formation gives the rotational geometry, accurate to a few inches on the ground. The music notes are F and A in the second octave.
T367 (Andrews catalog)
Does this represent the sun and planets? The orbits are exact circles with slighly different centers and diameters of 0.5, 0.7, 1.4, and 2.6. Mercury, Venus, Earth, Mars, and the Asteroids have actual values of 0.4, 0.7, 1.5, and 2.8 — a pretty good fit. If so the nearest date indicated is 11 July 1971. One of the next dates is 30 August 2033, years away because we have to match the motion of three planets, Mercury, Venus, and Mars. By computer graphic measurement the Asteroid circle and the edgecircle beyond Mars give a diatonic ratio of 9/4, note D in the second octave. (See CPRI Newsletter, Autumn/Winter 1995/1996, and 1996/1997).
In astronomy, angles are measured counterclockwise from the oversize Sun at the center. The closest approach of Mercury to the Sun (perihelion) is at 75 degrees, and the angles for Mercury, Venus, and Mars are 189 degrees, 76 degrees and 303 degrees. The Asteroid belt is symbolic and does not give the position of any Asteroid in its orbit.
T370 (Andrews catalog)
This shows the equalring branch of theorem five. Eight equallyspaced rings are needed before you get to a nonequilateral diatonic triangle. It fits with the vertices on ring 8, and sides touching rings 7 and 2. It's an 872 triangle, diatonic ratio 16, note C in the fifth octave. Measurements in the field by Dowell and Vigay are in agreement with this photogrammetry.
Is this confirmed by the outer loop of circles? Perhaps so, because tangents to ring 7 exactly intersect at the ring of the outer loop, and the circles on this ring have diameters of 7 units.
Part 2
T4 (Andrews catalogue)
Theorem I is the first crop theorem found, June 8, 1988. By rule 2 (see Definitions), the DIATONIC RATIO of the areas of the outer and inner concentric circles is 16/3, note F in the 3rd octave. By rule 1, the ratio of satellite diameter to center circle diameter is 1:1, note C. Since there are 3 tangents the geometry is repeated by rotation 3 times. Field measurements by Andrews and Delgado agree with this photogrammetry.
T359 (Andrews catalog)
Crop theorem II and then two applications of theorem III make the area of the outer circle 16 times bigger than the area of the inner disc. 4 X 2 X 2 = 16, which is note C in the 5th octave. Are the circle makers confirming these theorems with the embellishment of the 16 small satellites?
Part 3
T287 (Andrews catalog)
This pattern contains crop theorems, but it is embellished with 3 paws, 3 legs and 6 spokes. Because of the fitted square, the ring gives a ratio of 2 by theorem III. The equilateral triangle and central circle is theorem II.
T340 (Andrews catalog)
Because the inner circle is inside a pentacle, which is also inside a pentacle, the outer circle and inner circle give a DIATONIC RATIO within less than 1% for note D in the 7th octave (see rule 2, DEFINITIONS). The exact ratio by Euclid is 4 times the golden mean raised to the 6th power, a value of 71.8. The loops in the Web Pattern are equally spaced concentric circles, starting with #4 and ending with #8. These give a DIATONIC RATIO of 4, note C in the 3rd octave.
Part 4
T482 (Andrews catalog)
This combines the side of a hexagon, OB, with the side of a pentagon, AB, to get the radius of circle 1, OA. From Ptolemy's theorem of chords, with G equal to the golden mean and OF=1, we can prove that: 20A= G+AB (square root 3), or OA= 1,82709. Therefore by Rule 2, circles 1 and 3 give a ratio of 3.338, note A in the second octave. By crop circle theorem 4 the hexagon circles 1 and 2 give a diatonic ratio of 4/3, note F.
Is the raised circle a clue? D is the center of the arc of the crescent E. Angle CFD is 72 degrees, so CD is also the side of the pentagon.
This example of mathematical art gives the same diatonics as T448, notes F and A2, but the design is better. The diatonic circles now go through the tips of the moon, not the center, and the accuracy is 0.1%, not the previous 0.5%. Artistic as it is, the pattern contains math, and no previous artist has used mathematics as a theme. Ptolemy's theorem of 150 AD is a prehistoric landmark, because it is the foundation of trigonometry.
T487 (Andrews Catalog)
By rule 2, the area of the outer circle is four times the area of the inner circle, giving a diatonic ratio of 8/1, note C''', and letter C by the crop circle code. It is a double application of crop theorem 2, one equilateral triangle drawn inside another.
Text © Gerald Hawkins 1997
Photos, photo corrections and diagrams© Freddy Silva 1997
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