Early Reflections on Crop Circles
These pages contain my early work on crop circle geometry and crop circle reconstructions.
My later work can be found on my website Crop Circles and More


  Geometry | Reconstructions | Photos | Crop Circles and More
 

Crop Circle Geometry - 3 D fractals

     
 
In the mid 90s some of the found crop circles were based on fractals. For instance the so-called 'Juliaset' at Stonehenge in 1996 and the 'Triple Juliaset' at Windmill Hill the same year. Or the Koch-fractals in 1997. In later years, this resembling disappeared and was replaced by other features. In 1999 many of the crop circles looked three-dimensional. But did the fractals really disappear?
 
 


West Overton - England 1999

 
 
If you make a drawing of the 1999 West Overton formation, cut it out and fold it together, you get a three-dimensional shape: an octahedron. See image on the right. This octahedron forms the core of a three-dimensional fractal. To understand this we shall first have a look at an other Platonic solid; the tetrahedron. See image below.

 
 

 
 

By taking a second tetrahedron and putting it upside down 'in' the first tetrahedron, we get a so-called star-tetrahedron. See image above. The image on the left shows this star-tetrahedron viewed from above.
 
 
When we take this star-tetrahedron and 'pull out' the red tetrahedron, we are left with a tetrahedron with its 'heart' taken out. See image on the right. The empty space in the tetrahedron is where the two tetrahedrons were overlapping.

 
 

 
 
This empty space we can fill up again. See image above. If we just look at the shape of the filling-up material, we see an octahedron! With other words: the heart of a star-tetrahedron is an octahedron. See images on the right.

 
 

The image on the left shows the tetrahedron with the filling. You are looking at one side of the tetrahedron. All the other sides look like this. What we see is an upside down red equilateral triangle (=octahedron) in a bigger equilateral triangle (=tetrahedron).
 
 
We can repeat the same procedure with the three green equilateral triangles, which are in fact three tetrahedrons. The image on the right shows the result. We are now looking at the second step of the iteration that forms a Sierpinsky Gasket, which is a well-known fractal!

 
 
The procedure described above can be repeated again and again forming a three-dimensional Sierpinsky Gasket. A true three-dimensional fractal! Remember: the red fillings are in fact octahedrons, the shape formed by folding together the West Overton 1999 formation.
 
 

 
 
So the fractals did not disappear, at least not up until 1999. They were just hidden. And more: they were hidden in a three-dimensional form!