CROP CIRCLE GEOMETRY


by Bert Janssen 
In 1998 two crop circles with sevenfold geometry appeared quite close
to each other. The first one appeared in the East Field of Alton Barnes
and the second one near Tawsmead Copse. Their appearance showed many
similarities.
The following geometrical analysis will illustrate that not only did they show many similarities, they were also very strongly related to each other. The two formations are like brother and sister, they have the same geometrical genes so to speak. Furthermore I will show you that the contruction lines found in both formations will turn out to be of unexpected value! But let us first take a closer look at the Tawsmead Copse formation.
Note: All diagrams came into being by means of construction, that is, without measuring anything.
(I will show you in a future article how sevenfold geometry can be constructed. For now, please consider it true that indeed it can be done...)
Diagram 1 
Diagram 2 
In Diagram 1 a sevenfold star was constructed.
Within that star, a circle and a heptahedron can be constructed. This was done in Diagram 2.
Diagram 3 
Diagram 4 
Furthermore, as you can see in Diagram 3, another sevenfold star can be constructed within the star of Diagram 1.
In Diagram 4 a heptahedron was constructed around the sevenfold star of Diagram 1.
Diagram 5 
Diagram 6 
With the aid of the heptahedron of Diagram 4, another sevenfold star can be constructed.
The intersections of the heptahedron and the radians have the function of construction points. Diagram 5 shows the result.
Around the circle  in which the heptahedron lies  another sevenfold star can be constructed. This was done in Diagram 6.
Diagram 7 
Diagram 8 
Together, the sevenfold stars of Diagrams 5 and 6 form the outside of the Tawsmead Copse formation. The centre of this formation is formed by the heptahedron and the sevenfold star of Diagrams 2 and 3.
The most fascinating part of this construction is that it doesn't matter where you start. You can start with the smallest heptahedron in the centre of the formation and work your way towards the outside, or you can start with the largest sevenfold star and work your way towards the centre. In fact, you can start with any sevenfold star/heptahedron and subsequently work your way towards the inside and/or outside. It really doesn't matter at all where you start, you will always end up with the correct end result. The only thing you must do however, is take the right steps...
For instance, in order to  from the centre of the formation  construct the two sevenfold stars that together form the outside of the formation, you absolutely need the sevenfold star of Diagram 1 and the heptahedron of Diagram 4.
In Diagram 9 these are drawn as white lines in the formation.
Diagram 9 
As you can see immediately, the necessary heptahedron (of Diagram 4) and the necessary sevenfold star (of Diagram 1) actually formed part of the formation in the field. That is, in the form of underlying 'construction lines'. They could clearly be seen in the formation. On most areal shots they can also well be recognised.
East Field formation
It is far more complicated to construct the East Field formation.
However, with the help of the Tawsmead Copse formation things become much easier. There turns out to be a strong connection between the two formations.
Some elements of the Tawsmead Copse formation are necessary in order to be able to reconstruct the East Field formation. The following Diagrams show you what I mean.
Diagram 10 
Diagram 11 
Firstly, one constructs a circle. The centre of this circle is the intersection of the outside of the Tawsmead Copse formation and the radians. The outside of the circle touches the circle that was drawn around the heptahedron in the heart of the Tawsmead Copse formation. Diagram 10 shows how this is done.
Subsequently an identical circle is constructed in the centre of the formation.
See Diagram 11.
Diagram 12 
Diagram 13 
Then, another circle is constructed. This one touches the central circle and has as its centre the intersection of the radians and the sevenfold star of Diagram 1.
See Diagram 12.
Note: all circles constructed so far could only be made with the aid of the Tawsmead Copse formation!
In Diagram 13 a circle is constructed that goes through the intersection of the radians and the circle of Diagram 12.
Diagram 14 
Diagram 15 
These intersections also function as a starting point for the sevenfold star as constructed in Diagram 14.
This sevenfold star crosses the circle of Diagram 12 at two different points.
With the aid of the 'second' intersection, the sevenfold star of Diagram 15 is constructed.
Diagram 16 
Diagram 17 
The intersection of the circle of Diagram 10 and the circle of Diagram 12 functions as the centre of a small circle, which is shown in Diagram 16.
The outside of this little circle crosses the intersection of the outside of the Tawsmead Copse formation and the sevenfold star of Diagram 15.
Diagram 17 shows an enlargement of this construction step.
Diagram 18 
Diagram 19 
The same little circle crosses the sevenfold star of Diagram 14.
Together with the intersections of Diagrams 17 and 18, the centre of the circle forms two lines which have  by definition  the same length.
This is shown in Diagram 18.
The same intersections form the centres of two new little circles, which are identical to the circle of Diagram 16.
Diagram 19 shows the construction of these little circles.
Diagram 20 
Diagram 21 
One of the circles crosses  among other things  the outside of the Tawsmead Copse formation.
The other circle crosses  among other things  the sevenfold star of Diagram 14.
Because of this, two more lines are created, as shown in Diagram 20.
These have  by definition  the same length as the first two lines.
We now have four small lines with the same length.
On the right side of the formation, we can repeat constructing these lines.
By connecting the ends with the intersections of the sevenfold star of Diagram 14, we get a total line as shown in Diagram 21.
Diagram 22 
Diagram 23 
By repeating this entire process of line construction, we can construct a long, continuous line that forms exactly the outside of the East Field formation.
The result can be seen in Diagram 22.
In Diagram 23 the redundant lines were removed, after which the continuous line was filled up.
What you now see is the East Field formation.
For this construction we needed (among other things):
 the circle of Diagram 2,
 the circle of Diagram 12,
 and the subsequently created larger circle of Diagram 13.
Diagram 24 shows these elements in the formation.
Diagram 24 
Once again we see that the elements needed for the construction were actually there in the field as the onderlying construction lines. Some areal shots clearly show these lines.
Diagram 25 
Diagram 26 
Diagram 25 shows  apart from the construction lines of the East Field formation  also the construction lines of the Tawsmead Copse formation.
Diagram 26 shows both formations in correct proportion.
Diagram 27 
Diagram 27 shows Diagrams 25 and 26 fused together.
The East Field and Tawsmead Copse formations are strongly related. Not only do they have the same basic geometry, the one also evolves from the other.
It is relatively difficult to construct the East Field formation. However, when we take the Tawsmead Copse formation as a starting point, the construction of the East Field formation becomes quite easy all of a sudden. In other words, starting with the Tawsmead Copse formation, the East Field formation is relatively easy to construct!
This makes the first appearing of the East Field formation very strange indeed! After all, it appeared a month before the creation of the Tawsmead Copse formation. The logical order, as we have seen, is exactly the other way round!
Furthermore I used a construction technique that made some elements strictly necessary. These elements also appeared in the fields as underlying construction lines. However, let me repeat what I wrote in my article "Back to Basics".
The construction techniques that I used cannot be repeated in the fields without leaving their traces. Look for instance at the large number of sevenfold stars that I needed and that can no longer be found in the end result. It is absolutely impossible to erase those figues in the fields. And yet, the underlying construction lines indicate that exactly the same construction techniques were used when the formations were actually created!
Of course, hoaxers might claim that they actually can mark out those large figures in the fields  like for instance the large sevenfold stars  without leaving a trace.
Allthough this is quite unlikely to be true, let's give them the benifit of the doubt.
Just for now... Suppose it is indeed possible to make those constructions without leaving a trace. This means one can make absolutely fascinating formations: formations without underlying construction lines!
From a hoaxer's point of view, those are the ultimate formations!
In other words, if a hoaxer has the possibility to leave out construction lines, he will definately leave them out!
If he can make large sevenfold stars without leaving any construction lines, he will never make smaller sevenfold stars which do show construction lines!
This puts any hoax claim in a catch22 situation.
1. Either hoaxers can make large sevenfold stars without leaving construction lines, but that means we would not have found any construction lines in the formation at all. However, we did find them!
2. Or the hoaxers cannot make sevenfold stars without leaving construction lines, but that means we should have found the construction lines in the large sevenfold stars as well. However, we didn't. They weren't there!
Whichever way you look at it, any hoax claim drops out for good.
Which leaves us with the question: Why did we find those construction lines? Because of what I've written above, I can only draw one conclusion:
This however is not only an astonishing, but also a farreaching conclusion! The presence of the construction lines inside the formations and the absence of those lines outside the formations, are precisely what proofs the formations are NOT manmade!
In other words:
The construction lines that make most people suspicious,
are indeed the ultimate proof that a formation is NOT a hoax.
You could even consider the construction lines to be a message of the 'circle makers'. A message with which they show us that a formation is genuine and not the work of a couple of hoaxers. If the construction lines would have been absent, any hoax claim would be much more difficult to refute.
My final conclusion therefore is:
In a future article I will explain how sevenfold geometry can be constructed. Furthermore I will evaluate the sevenfold geometry crop circle formation at Danebury Ring.
Bert Janssen, 1998, Email: ossebaard@wxs.nl
All diagrams © 1998 Bert Janssen. (However, do copy, as long as you mention my name...)