Crop circles first appeared over 300 years ago in England. Over time, they grew in size and complexity, evolving from simple circles to intricate geometric patterns. Mathematicians have studied crop circles and discovered that many follow precise geometric rules and relationships. Some mathematicians even proposed new theorems based on analyses of crop circle patterns. While the origins of crop circles remain mysterious, they have inspired new mathematical discoveries and interest in geometry.

1. The Best Selling Ufology Books Collection
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2. Crop Circle Geometry
Emily Yeung
24243054
Math 308
Winter 2008
Term 2
Prof. Dale Rolfsen

3. In the history of crop circles, they first appeared over 300 years ago. The first completed
record of crop circle appeared in England at 1972 summer, its shape was assembled as a simple
circle. Since the first documented crop circle, they have become bigger and more complex similar
to some pictorial designs. Complex geometry style designs begin to appear in 1991 and binary
number designs appeared in 2001, these are the important time interval of crop circle history. At
1970’s, crop circles diameter measured around nine meters, they grew over time and the biggest
crop circles was recorded over 300 meters in diameter. The compositions of crop circles have also
changed from a simple circle to some complicate geometry patterns. In 1990, crop circles started
to appear as non-circular and in a more elegant pattern. However, most of the crop circles are in
precise geometry patterns, these patterns includes circles, ellipses, rectangles, triangles and
sometime as crosses. Crop circles divided into different sub categories: clockwise,
counterclockwise, actinomorphous, radioactivity, lines, or mixed patterns. Ever since the first
discovery, to as late as the year 2006, over 10,000 crop circles have been discovered! Crop circles
are all over in the world, in 29 different countries. All the crop circles mysteriously appear in
perfect patterns. Some people argue these perfect patterns seem made up by a mathematical
formula. Some of them appeared similar to the graph pattern as the complex number which our
professor was mentioned in our class earlier.
As of today, the formation of crop circle is still a mystery. There are many different
explanations 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. However
this theory cannot explain those with a complex or non-circle patterns, since cyclonic storm can
only make simple circles! There are many different scientist from different fields tries to explain

4. the formation of these mysterious crop circles, however none of them seems to able to explain all
the 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 circles
were made by them. They mentioned the circles were created by simple tools like a rope and a
wooden board. I personally believe that would be possible if the crop circle is in a geometric
design. Many people might have a feeling that is not easy to make a geometry graph, however they
can be composed easily on a piece of paper using simple tools such as pens and rulers. As we have
taught in our math class, to draw a geometry graph, the only information we will need is the angles
of the pattern. While we are making a crop circles in a large area we would need to be very precise
for the angles and degree measurements. The reason for a precise angles is that the errors will be
exaggerated when the pattern grow in sizes while compares to draw a geometry graph on piece of
paper, the error is un-noticeable. Hence, making a geometric pattern on a crop field might not be
so trustable with only rope and wooden board, but with some scientific tools.
Some mathematicians came together and form themselves into small groups, each group
was in charge of different parts of the crop circles, like the external shape of geometry, the internal
geometry, size, placing, ratios, construction points, and construction lines. Crop circles not only
appear in 2 dimensional forms, they also appear in 3 dimensional patterns. Therefore one of the
group was in charge of the 3 dimensional diagrams. For example, one of the pattern was an
unfolded tetrahedron. As a result, the group of research in the internal geometry discovering more
details than the group research in external shape of geometry. First, their size and placing of the
differences in crop formation followed by the geometrical construction rules. Second, some of the

5. elements from the crop internal geometry have special ratios to the others. Thirdly, the
construction 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 in
standing crops. Base on the above points, this group of mathematicians argues that the
construction lines are not the ultimate proof of human activity meaning that is an indication of
non-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 in
the 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 studies
do not have knowledge of. The most popular theorem was found in the crop circles and that was
the Fifth Theorem. In 1990, there was a mathematician called Prof. Gerald S. Hawkins who found
a 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 fifth
theorem. The fifth theorem is a geometrical theorem which he found by his mathematical and
geometrical analyses on crop circles. Prof. Hawkins' geometrical theorem is Euclidean geometry
in 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 the
formations, he discovered that those regular polygon patterns or circumscribed circles conveying
diatonic ratios are the triangle, square and hexagram. Circumscribed circle is a circle that draws by
the vertex of polygon. By using the same center as circumscribed circle, decreasing the radius so
that the circle's circumference touches each of the inner-aspects of the triangle's sides, one draws

6. its inscribed circle. This is a new theorem which neither Euclid had discovered nor appeared in the
world 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 triangle's circumscribed circle to the diameter of the circles at each
corner is 4:3. Let say there are three equal circles which share a common tangent and those
tangents create an equilateral triangle. The circumscribed circle is on the vertex of the triangle.
2nd Theorem
For 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.

7. 3rd Theorem
For a square, the ratio of the areas of circumscribed and inscribed circles is 2:1
4th Theorem
For 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)

8. The 2nd, 3rd, 4th theorems are the special of regular polygons. Only triangle, square, and hexagon
will give a diatonic ratio from the circumscribed and inscribed circles. The 5th theory involves a
triangle and various concentric circles touching the triangle's sides and corners. As the
relationships between circles (rings) and triangles, different circles or rings are explored using
various kinds of triangles (i.e., equilateral, isosceles, right). Each triangle discloses different
geometrical relationships.
Some of the crop circles also follow other geometry rule in Mathematics, for example, crop
Mandela pattern which have rotational symmetry and give diatonic ratios, which satisfy the rules
of Satellite Circles (1981) and Concentric Circles (1986).
The geometry crop circles have a good influence in teaching and learning area. Crop
circles were used by the teacher in class. It led to students developing more of a sense of the
Mathematical area of thinking, for example you might ask: Can you describe the patterns of the
crop 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

9. 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 find
information on YouTube about geometric crop circles.
Although the mystery of crop circles has never been explained fully, we had managed to
learn from them. Scientists get the opportunity to get together and work with other experts. In
Mathematics, 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 are
published 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 the
crop circle and was inspired by them and we can expected more founding while many scientist are
having their attention and interest to the crop circles.