Mathematics Awareness Month 2014: Mathematics, Magic, and Mystery

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The Beautiful Geometry of Crop Circles

The idea that crop circles are mysterious formations of extraterrestrial origin may be evidence of the lack of geometry in the modern mathematics curriculum. A little practice with a straightedge and compass is all that is needed to understand that these works of art are made by very human mathematical artists. Like all good mathematicians, these artists are very happy to share their theories with the public. Here is a slide show of a bountiful geometric harvest.

In the video below you can watch a master crop circle artist creating a design with straightedge and compass. The finished product is shown at the end.

And this is what happens when the designers head out into the “field.”

Artist Simon Beck makes his geometric designs in snow using a different type of compass.

Taking it Further

Can you draw the diagram for the crop circles below?

easy crop
Image by Mark Bland

On this Coolmath post you will find some hints and challenges for some simple crop circles that look surprisingly complex.

The Underlying Mathematics

Compass and straightedge constructions of geometric figures have been studied for thousands of years. The three impossible geometric problems of antiquity: squaring the circle, doubling the cube, and trisecting an angle were straightedge and compass challenges that turned out to be impossible. It took thousands of years of mathematical development before this was shown to be the case. For example, the problem of squaring the circle was only shown to be impossible in 1882 with the discovery of the transcendence of pi.

An excellent reference for modern geometric constructions is Geometry Revisited by H.M.S. Coxeter and S.L. Greitzer.

In his article: “Coming Soon to a Field Near You,” physicist Richard Taylor theorizes that some crop circle makers are using modern technology to execute more complex designs.

Many crop circle designs are based on serious mathematics. Fractal designs are popular. A Julia Set design is shown here.

The Field Guide: The Art, History, and Philosophy of Crop Circle Making, by Rob Irving and John Lundberg (Strange Attractor Press, 2006), is the insider’s guide to the secret world of crop circles. Not that we are encouraging any mischief making by Mathematics Awareness Month participants.