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Crop Circle Geometry Emily Yeung 24243054 Math 308 Winter 2008 Term 2 Prof. Dale Rolfsen
In the history of crop circles, they first appeared over 300 years ago. The first completedrecord of crop circle appeared in England at 1972 summer, its shape was assembled as a simplecircle. Since the first documented crop circle, they have become bigger and more complex similarto some pictorial designs. Complex geometry style designs begin to appear in 1991 and binarynumber designs appeared in 2001, these are the important time interval of crop circle history. At1970’s, crop circles diameter measured around nine meters, they grew over time and the biggestcrop circles was recorded over 300 meters in diameter. The compositions of crop circles have alsochanged from a simple circle to some complicate geometry patterns. In 1990, crop circles startedto appear as non-circular and in a more elegant pattern. However, most of the crop circles are inprecise geometry patterns, these patterns includes circles, ellipses, rectangles, triangles andsometime as crosses. Crop circles divided into different sub categories: clockwise,counterclockwise, actinomorphous, radioactivity, lines, or mixed patterns. Ever since the firstdiscovery, to as late as the year 2006, over 10,000 crop circles have been discovered! Crop circlesare all over in the world, in 29 different countries. All the crop circles mysteriously appear inperfect patterns. Some people argue these perfect patterns seem made up by a mathematicalformula. Some of them appeared similar to the graph pattern as the complex number which ourprofessor was mentioned in our class earlier. As of today, the formation of crop circle is still a mystery. There are many differentexplanations to this phenomenon such as cyclonic storm, magnetic field, extra-terrestrial (ET)...etc. One of the theories is that these crop circles are being created by cyclonic storm. Howeverthis theory cannot explain those with a complex or non-circle patterns, since cyclonic storm canonly make simple circles! There are many different scientist from different fields tries to explain
the formation of these mysterious crop circles, however none of them seems to able to explain allthe mysteries about crop circles. Hence, making crop circles more mystery and were popular. In the experiment of making crop circles, some people had claimed that these crop circleswere made by them. They mentioned the circles were created by simple tools like a rope and awooden board. I personally believe that would be possible if the crop circle is in a geometricdesign. Many people might have a feeling that is not easy to make a geometry graph, however theycan be composed easily on a piece of paper using simple tools such as pens and rulers. As we havetaught in our math class, to draw a geometry graph, the only information we will need is the anglesof the pattern. While we are making a crop circles in a large area we would need to be very precisefor the angles and degree measurements. The reason for a precise angles is that the errors will beexaggerated when the pattern grow in sizes while compares to draw a geometry graph on piece ofpaper, the error is un-noticeable. Hence, making a geometric pattern on a crop field might not beso trustable with only rope and wooden board, but with some scientific tools. Some mathematicians came together and form themselves into small groups, each groupwas in charge of different parts of the crop circles, like the external shape of geometry, the internalgeometry, size, placing, ratios, construction points, and construction lines. Crop circles not onlyappear in 2 dimensional forms, they also appear in 3 dimensional patterns. Therefore one of thegroup was in charge of the 3 dimensional diagrams. For example, one of the pattern was anunfolded tetrahedron. As a result, the group of research in the internal geometry discovering moredetails than the group research in external shape of geometry. First, their size and placing of thedifferences in crop formation followed by the geometrical construction rules. Second, some of the
elements from the crop internal geometry have special ratios to the others. Thirdly, theconstruction points found in the studied formation which never been found in the standing crops.Fourth, some formation has the elements that are strictly necessary to avoid construction points instanding crops. Base on the above points, this group of mathematicians argues that theconstruction lines are not the ultimate proof of human activity meaning that is an indication ofnon-human involvement. On the other hand, some mathematicians manage to use computers to calculate the area,length, and circumference of the crop patterns. By using computers, they found out every circle inthe crop patterns fit in within the theorem of geometry, also their errors were less than 0.25 inches.Mathematicians believe these crop circles hide a lot secret methods of mathematics which studiesdo not have knowledge of. The most popular theorem was found in the crop circles and that wasthe Fifth Theorem. In 1990, there was a mathematician called Prof. Gerald S. Hawkins who founda new theorem from the crop circles, which theorem is totally new in mathematics. Prof. Gerald S.Hawkins who published this theorem to public and the theory which people called the fifththeorem. The fifth theorem is a geometrical theorem which he found by his mathematical andgeometrical analyses on crop circles. Prof. Hawkins geometrical theorem is Euclidean geometryin nature but expressed in diatonic ratios which are the ratios used in the diatonic scale of music.By analyzing the relationship between the areas and diameters of the crop circles within theformations, he discovered that those regular polygon patterns or circumscribed circles conveyingdiatonic ratios are the triangle, square and hexagram. Circumscribed circle is a circle that draws bythe vertex of polygon. By using the same center as circumscribed circle, decreasing the radius sothat the circles circumference touches each of the inner-aspects of the triangles sides, one draws
its inscribed circle. This is a new theorem which neither Euclid had discovered nor appeared in theworld of mathematics. Here are the four theorems which Euclid has discovered. 1st Theorem (discovered on June 4, 1988),The ratio of the diameter of the triangles circumscribed circle to the diameter of the circles at eachcorner is 4:3. Let say there are three equal circles which share a common tangent and thosetangents create an equilateral triangle. The circumscribed circle is on the vertex of the triangle. 2nd TheoremFor an equilateral triangle, the ratio of areas of the circumscribed and the inscribed circles is 4:1.The area between the annulus is 3 times the area of the inscribed circle.
3rd TheoremFor a square, the ratio of the areas of circumscribed and inscribed circles is 2:1 4th TheoremFor a regular hexagon, the ratio of areas of circumscribed and inscribed circles is 4:3. 5th Theorem (a general theorem derived from 1-4 theorem)
The 2nd, 3rd, 4th theorems are the special of regular polygons. Only triangle, square, and hexagonwill give a diatonic ratio from the circumscribed and inscribed circles. The 5th theory involves atriangle and various concentric circles touching the triangles sides and corners. As therelationships between circles (rings) and triangles, different circles or rings are explored usingvarious kinds of triangles (i.e., equilateral, isosceles, right). Each triangle discloses differentgeometrical relationships. Some of the crop circles also follow other geometry rule in Mathematics, for example, cropMandela pattern which have rotational symmetry and give diatonic ratios, which satisfy the rulesof Satellite Circles (1981) and Concentric Circles (1986). The geometry crop circles have a good influence in teaching and learning area. Cropcircles were used by the teacher in class. It led to students developing more of a sense of theMathematical area of thinking, for example you might ask: Can you describe the patterns of thecrop circles? How could you compare them? Which features are the same and which are different?Can you make the pattern on paper? Therefore there are many websites published. For example,(http://www.hypermaths.org/cropcircles/chapter3/index.html) would be a nice website which
have named crop circles and contain both crop circles in photos and in a 2D graphical diagram.Also there are many Mathematics related books published after the crop circles which appeared,for example, Crop Circles: the Hidden Form by Nick Kollerstrom. Nowadays we can even findinformation on YouTube about geometric crop circles. Although the mystery of crop circles has never been explained fully, we had managed tolearn from them. Scientists get the opportunity to get together and work with other experts. InMathematics, people find new theory in that, and teaching geometry using crop circles. As today,crop circles remain as an interesting topic and attract many students to its craft. Many books arepublished base on this topic which shows there are many more to learn beside from mathematics.As the conclusion for this essay, I would like to say that I have learn many new theories from thecrop circle and was inspired by them and we can expected more founding while many scientist arehaving their attention and interest to the crop circles.