Crop circles started as a prank, but now has become a medium for geometric designs. Here are some example of beautiful ones.

For high school students, you can learn geometry by trying to duplicate these designs on paper. You can use the computer software GeoGebra to draw them interactively. Most crop circle designs are geometrically simple, so that you should be able to look at it, and figure out exactly the center and radius of each circle or triangle or other key points. You'll learn a lot trigonometry and planar geometry.

Once you became familiar with the software, you can make your design such that there are several parameters, so when you drag a point, the design changes. In effect, creating a template that creates many variations of the same style of design.

For example, many design features circles touching circles. Suppose you have a circle of radius 1 centered at (0,0), and a circle or radius 0.66 centered the (0.33,0). So, this circle touches the larger circle. You have now a moon shape (called crescent or lune). Now, create more circles between these circles, all touching the two circles and each other. Your job is to find the center and radius of these circles.

Now, once you have done this, you can create in your GeoGebra software so that you can drag the smaller circle around, and instantaneously see all the touching circles change accordingly. For some math of this, see: Problem of Apollonius.

For info about GeoGebra, see: Great Software for Plane Geometry.

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