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 below on the left.

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

 
 
 
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 above on the right. The empty space in the tetrahedron is where the two tetrahedrons were overlapping.

 
 
 
This empty space we can fill up again. If we just look at the shape of the filling-up material, we see an octahedron! See images above.

 
 
With other words: the heart of a star-tetrahedron is an octahedron. The same you get when you fold a drawing of the 1999 West Overton formation together.
The image below 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 above 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.

 
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