email Martin Keitel

Geometry of the
"Joker" crop circle in Sompting, West Sussex
found June 3, 2002

A diagram revealing the circles defining the shape
(actually the large circle doesn't belong)

I got interested in this one for three reasons:

1. It is located close to the house of Lyn Collin, one of my dear friends in England, where I was visiting just few weeks before.

2. The shape reminds me of some thoughts I was recently processing, concerning gravity, acceleration and centrifugal force.

3. Based on the aerial photo, the centers of the circles defining the crescent arms seemed to be positioned on standing crop.

To get away with the perspective angle in the aerial photo, this time I used a reliable new system I recently develop. Using this method, instead of turning the image to match overhead view, I turn the camera in virtual space (using a 3D modelling software) to match the angle of the real camera that was used. This way I'm able to work on the geometry as seen from directly above, while all the time seeing the perspective image in another window!

original aerial photo by David Russel

Above you can see the same diagram exposed over the aerial photo. As you can see, it's a perfect match! Some obvious observations:

1. All elements are circles (instead of ellipses, as estimated by someone else).

2. All circle centers are located on flattened areas - the crescent circle centers on the outer edge of the central ring (red dots). Both of these circles (and the corresponding ones on the opposite side) are equal in diameter.

3. The centers of the "satellite" circles are located on the large circles defining the outer edges of the crescents.

As for the relationships between the elements, at least a couple of interesting ratios are found:

1. The ratio between the central circle and the inner edge of the ring can be exactly defined with a regular heptagram (7-pointed star), as seen in the image above.

2. The size of the four circles used to outline the crescent arms (here positioned in the center), can be derived from the outer edge of the ring, using a square.

Note that all the geometry has been done without perspective - the perspective view is created by the virtual camera angle!

So, once again, the ratios of the circles follow the rules of Euclidean geometry, where the relationship between different circle elements are based on regular polygons. And once again, though technically speaking an easy shape to be executed by people, why bother for the mathematics - or could it be just co-incidential? With this accuracy, hardly so!