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 'Julia set' at Stonehenge in 1996 and the 'Triple Julia set' 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?

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.


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.

By taking a second tetrahedron and putting it upside 
down 'in' the first tetrahedron, we get a so-called 
star-tetrahedron. See image.

When we take this star-tetrahedron and 'pull out' 
the red tetrahedron, we are left with a tetrahedron 
with its 'heart' taken out. The empty space in the 
tetrahedron is where the two tetrahedrons were 
overlapping. See image.

This empty space we can fill up again. 
See image.

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.

The next image 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 next image 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!

(c) text and diagrams: Bert Janssen.

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Bert Janssen & Janet Ossebaard

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