



introduction
What is it that makes crop circles such a fascinating phenomenon? No doubt the mysterious and
inexplicable aspect plays a mayor role here, but is that all? How can it be explained that people get so fascinated merely
by looking at them? Even when they don't know a single thing about the crop formations, the symbols seem to stir up interest nevertheless.
There's something about the pictogram's that has some kind of hypnotising effect on people. In these sections I will
explain some of the geometrical aspects of the patterns. These aspects could (partly) explain why the symbols have such influence on people.






external pentagonal geometry
The first formations to be studied on their geometrical characteristics were those of the early 90s.
People like John Martineau and Wolfgang Schindler studied these formations intensively. And not without result.
Both Martineau and Schindler looked mainly at the outer shapes and found many peculiarities. This section briefly shows the outcome of especially Schindler's work.




external pentagonal geometry
The first formations to be studied on their geometrical characteristics were those of the early 90s.
People like John Martineau and Wolfgang Schindler studied these formations intensively. And not without result.


Both Martineau and Schindler looked mainly at the outer shapes and found many peculiarities. This section briefly shows the outcome of especially Schindler's work.








internal geometry
Whereas most people only look at the outer shapes of the crop circles, I took a look inside. By trying to reconstruct different formations,
I found that these formations were all based on an intriguing geometry. This geometry was not visible at the outside but it was definitely there.
This internal geometry has some amazing implications. In the next sections those implications will be explained.
This section shows how this internal geometry functions. See the mainsection
Crop Circle Reconstructions
to get a complete overview of all the steps needed to reconstruct the different crop formations.




internal geometry
Whereas most people only look at the outer shapes of the crop circles, I took a look inside. By trying to reconstruct different formations,
I found that these formations were all based on an intriguing geometry. This geometry was not visible at the outside but it was definitely there.
This internal geometry has some amazing implications. In the next sections those implications will be explained.


This section shows how this internal geometry functions. See the mainsection
Crop Circle Reconstructions
to get a complete overview of all the steps needed to reconstruct the different crop formations.








size, placing and ratios
There is no coincidence in geometrical constructions. Every element is in size and placing determent by the previous steps.
That is the nature of geometrical construction. This means also that if it's possible to reconstruct crop circles
using geometrical construction techniques, the different elements within the construction (crop circle) will by nature have special ratios
to each other and their placing do follow strict rules. Neither the size of the elements nor their placing is coincidental.
This section shows two examples of how this goes for crop circles. It shows that different elements within a formation
are indeed not random in size and placing and do have by nature special ratios to eachother.




size, placing and ratios
There is no coincidence in geometrical constructions. Every element is in size and placing determent by the previous steps.
That is the nature of geometrical construction. This means also that if it's possible to reconstruct crop circles
using geometrical construction techniques, the different elements within the construction (crop circle) will by nature have special ratios
to each other and their placing do follow strict rules. Neither the size of the elements nor their placing is coincidental.


This section shows two examples of how this goes for crop circles. It shows that different elements within a formation
are indeed not random in size and placing and do have by nature special ratios to eachother.








construction points
Geometrical constructions can be made with the aid of a straight edge and a pair of compasses.
The compass is obviously being used for making circles. In order to do this you must put the needle of the compass in the paper.
While reconstructing crop circles, you have to do this many, many times. The places where you put the needle in the paper
you could call ‘construction points’. These construction points were in the formations I studied always on special places.
This section shows what it is that makes those places special and which conclusions can be drawn from it.




construction points
Geometrical constructions can be made with the aid of a straight edge and a pair of compasses.
The compass is obviously being used for making circles. In order to do this you must put the needle of the compass in the paper.
While reconstructing crop circles, you have to do this many, many times. The places where you put the needle in the paper
you could call ‘construction points’.


These construction points were in the formations I studied always on special places.
This section shows what it is that makes those places special and which conclusions can be drawn from it.








construction lines
If you take a closer look at the lay of the crop in a crop circle you will sometimes find pathways of crop under the general lay.
These pathways are on average a foot wide and it often looks as if these pathways represent the general outlay of the formation.
They look like 'construction lines' and these construction lines have for obvious reasons a very bad reputation. A lot of people
think that these lines are the ultimate proof of human activity. But it is not that simple.
I will show in this section that construction lines found in formations can turn out to be of unexpected value.




construction lines
If you take a closer look at the lay of the crop in a crop circle you will sometimes find pathways of crop under the general lay.
These pathways are on average a foot wide and it often looks as if these pathways represent the general outlay of the formation.
They look like 'construction lines' and these construction lines have for obvious reasons a very bad reputation.


A lot of people think that these lines are the ultimate proof of human activity. But it is not that simple.
I will show in this section that construction lines found in formations can turn out to be of unexpected value.








3 D  fractals
In the mid 90s some of the found crop circles were based on fractals. For instance the socalled 'Juliaset' at Stonehenge
in 1996 and the 'Triple Juliaset' at Windmill Hill the same year. In later years this resembling disappeared and was replaced by other features.
In 1999 a lot of the crop circles looked threedimensionally. But did the fractals really disappear? In this section I will show how the 1999,
threedimensionally looking, crop circle at WestOverton (England) also contained an element of a fractal. A threedimensional fractal.




3 D  fractals
In the mid 90s some of the found crop circles were based on fractals. For instance the socalled 'Juliaset' at Stonehenge
in 1996 and the 'Triple Juliaset' at Windmill Hill the same year. In later years this resembling disappeared and was replaced by other features.
In 1999 a lot of the crop circles looked threedimensionally.


But did the fractals really disappear? In this section I will show how the 1999,
threedimensionally looking, crop circle at WestOverton (England) also contained an element of a fractal. A threedimensional fractal.








3 D  crop circle models
Especially after the 1999 season, were a lot of crop circles looked threedimensional, many people got suddenly interested in the
threedimensional aspects a crop circle can have. Already before this 1999 season I looked at the possibilities to reconstruct
(twodimensional) crop circles into threedimensional shapes. This can be done in various ways, depending on the conventions
you make before you start. In this section I will show two different ways of making the twodimensional patterns threedimensional.




3 D  crop circle models
Especially after the 1999 season, were a lot of crop circles looked threedimensional, many people got suddenly interested in the
threedimensional aspects a crop circle can have. Already before this 1999 season I looked at the possibilities to reconstruct
(twodimensional) crop circles into threedimensional shapes.


This can be done in various ways, depending on the conventions
you make before you start. In this section I will show two different ways of making the twodimensional patterns threedimensional.


















