Posted by Michael Glickman on April 17, 2012

A LITTLE GEOMETRY AND NUMBER

Fig.1 - Radial division of circles

Seven is considered to represent spirit, revelation, the eternal feminine and that which lies behind the veil. Eleven is the first number of the Master Number series that is said to indicate degrees of contact and interaction with other dimensions. Given these credentials we cannot be surprised that these two noble numbers choose to hold themselves slightly aloof from such as the ordinary fours and nines.

Fig.2 - Fundamental contained shapes

Let us take this one step further by considering the contained and containing circles. In these diagrams the containing circle is blue and has a value of 1. The containing circle always holds the contained shape or polygon by its points, corners or vertices. The contained circle, by contrast, is always held by the sides of the polygons. Thus the yellow circle is held by the three sides of the triangle, the red by the four sides of the square and the green by the five sides of the pentagon. Notice that the point of contact is always the precise centre of the side.

Fig.3 - Pentagrams and heptagrams

A in Diagram [Fig.3] shows the circle contained within the central pentagon of the pentagram. Read this again. Slowly! This contained circle has a diameter of 0.309017 in relation to the main blue circle’s diameter 1.

B in Diagram [Fig.3] introduces another variant – the arm-crossing circle, shown here in paler blue. Instead of being contained by the five sides of the central pentagon, the arm-crossing circle is sized by the places where the arms cross. The relationship to the main blue circle is 0.381966.

C in Diagram [Fig.3] shows a heptagram, a seven-pointed star. Its central heptagon contains a pink circle (diameter 0.222521). The arm-crossing circle here, in paler pink, has a diameter of 0.356896.

This needs to be understood because it is an essential feature of the crop circles’ design strategy. Each of these shapes, and indeed every regular polygon, is wedded to a particular proportion and harmonic. It is my view that a major part of the crop circles’ visual appeal is their inherent order and harmony. The authors of this phenomenon understand the importance of these matters.

A FEW EXAMPLES
The contained yellow circle in the triangle in Diagram [Fig.2] is exactly half the diameter of the blue containing circle. This is Euclid’s Second Theorem and is a crop circle regular.

Photograph [Fig.4] shows Crooked Soley, 27th August 2002, which certainly was a canonical formation. Diagram [Fig.5] shows the precision with which that Euclid’s ideas are incorporated here.

Fig.4 - Crooked Soley 27th August 2002

Fig.5 - Crooked Soley diagram

Fig.6 - Etchilhampton double 30th July 1997

Fig.7 - Etchilhampton diagram

Photograph [Fig.8] shows the swirling crescents formation which appeared at Barbury Castle on 20th April 1997 and diagram [Fig.9] shows how the pentagram locates and sizes the central ring. It is curious that, though Gypsy Patch and Barbury Castle were both six-fold spinners, they are based on different geometrical systems.

Fig.8 - Barbury Castle 20th April 1997

Fig.9 - Barbury Castle diagram

OLIVER’S CASTLE
On 11th August 1997 in the fields below the steep escarpment of Oliver’s Castle outside Devizes a famous six armed star formation appeared. For many, this was the most articulate and revealing crop circle ever. However, it was to become perhaps the single most contentious formation in crop circle history.
It started out with two disadvantages. First it arrived during a spectacularly wet weekend and second it positioned itself on the land of John Butler, a famously angry farmer. This set up an immediate paradox. Butler adamantly refused entry to anyone under any circumstances but, Oliver’s Castle escarpment, being possibly the best crop circle viewing position in Wiltshire, was crowded for days with an audience of people staring down into his forbidden field.

Fig.10 - Oliver’s Castle 11th August 1997

Then one day, not too long after the formation’s appearance, we arrived to see that a wide avenue had been cut through its centre. John Butler, who was harvesting several neighbouring fields, had run the combine through the heart of the circle. I sympathise with the frustrations of the bewildered farmer who can only believe that he is the victim of malicious vandals. However I have never heard of a harvester being used to deliver so gratuitously spiteful a message. The machine had cut through the heart of the crop circle and destroyed two of its arms. The other four arms were intact. I knew that the stubble always holds the exact dimensions of destroyed elements so, hoping that farmer Butler’s act of destruction had lessened his ire, I simply offered him five pounds to let me go in and measure. He happily accepted. Yet another farmer convinced we were all mad!

INSIDE THE OLIVER’S CASTLE FORMATION

Fig.11 - Oliver’s Castle diagram

The second conundrum was width of the pathways. The connections from A to the six Bs was exactly and unerringly four feet. We measured in several locations and it was always precise. From the Bs to the Cs and Ds the widths were similarly exact. The paths reduced to three feet with the same uncanny precision, though the six outer paths were not as straight as the inner. I was shocked by this accuracy. I have neither before nor since observed such consistency in crop circle dimensions. I felt there was some pressing significance in this four and three. It would be some months before I found out.

THE OLIVER’S CASTLE VIDEO
There were rumours about an extraordinary video taken at Oliver’s Castle. A young man called John Weyleigh had spent the night at Oliver’s Castle on a crop watch and was woken in the early hours by a strange noise. He reached for his video camera, went to the edge to look down and saw balls of light moving above the field. He was aware only of the luminosities and did not notice that a circle was forming beneath them until he later reviewed his video at home.

The video shows a graceful loop over the field carried out by a pair of balls of light. As they pass, the Oliver’s Castle crop circle forms beneath them, working from the centre outwards. Moments after the first, a second pair of lights performs the same circuit, as though to inspect the quality of the work. The four lights disappear into the line of bushes at the far edge of the field.

The brief video, which lasts only seconds, is astonishing for the revelation of the way the wheat falls below the luminosities and captivating for the sweeping elegance of its movement and choreography.

This footage was everything we had been waiting for. For my part, it was – and remains to this day – one of the most important gifts we have ever received from the circlemakers.

The film was shown twice, to general excitement, at the Barge but it was not long before the strident noise machine of the hoax boosting fraternity raised its head. As a believer in the veracity of the footage I was a target for prolonged attack and thus began one of the most disgraceful episodes in the history of the circles. The legacy of that period is still with us. Fifteen years later too many remain obstinately convinced that crop circle research is a branch of forensic investigation.

DISCOVERY OF THE SEVEN
I returned to California after the season, distracted by the viciousness of the attacks on the film’s provenance but more curious than ever about the Oliver’s Castle formation and its fours and threes. I worked for weeks searching for a hidden constraining geometry. Pentagonal and hexagonal failed and, remembering the insistence of four and three, I tried square and triangular without success.

Amid the tumult of debunking the video was studied by Jim Dilettoso, a video analyst of great repute who has worked for NASA. He found no evidence of falsification or tampering. He produced an immaculate slowed-down version that showed the swirling energy of manufacture laying down the formation sequentially from the central circle.

This verified our observation that the quality of lay dropped significantly as it moved outwards. Jim Dilettoso’s work confirmed this. I suddenly realised that perhaps I should base my geometry on a ring around the Bs rather than a total enclosure around the Ds. This then would include the perfect A and the ring of Bs and bypass the sloppy Cs and Ds. I had spent fruitless months trying to crack Oliver’s geometry. The nagging four and three instantly made sense. It meant four PLUS three. It meant seven!

Fig.12 - Oliver’s Castle diagram with heptagram

I had, for some time, been preoccupied with the geometry of seven. The circles had never produced a manifested sevenfold pattern and when this constraining seven appeared so accurately at Oliver’s Castle I read it as a hint that we might expect a materialised sevenfold soon. Bear in mind that the number seven is held to symbolise – among other things – revelation and what lies behind the veil. Absolutely appropriate for the Oliver’s Castle events.

I was thrilled and in my column in SC Magazine, I rashly predicted the arrival of a real sevenfold crop circle in 1998.

EAST FIELD

Fig.13 - East Field 9th July 1998

I returned that evening to great excitement. People had gone in and confirmed that it was, in fact, sevenfold. It had a superficial similarity to the 1997 Koch fractal but the reiterative nature of the fractal system, never having been seen before, was hard to grasp.

Fig.14 - East Field diagram

The formation itself was gigantic. At approximately 1.6 acres it was, and is still, the largest single flattened area within a crop circle.

The weather had not been good and the large number of visitors coming to visit this giant caused further damage. But even after more than a week, Andreas Müller, one of our most experienced field researchers said, “It was massive and the impressive lay was still readable. It was so big that I had the sense that, if we could only start to understand the mechanics of the way it was laid down, it would bring some positive information for humanity. There had clearly been an energetic system at work here.”
A historic comparison with the Koch sixfold fractals of 1997 is illuminating.

The first, at Silbury Hill, arrived on 23rd July. It had a large, flattened central area. The second Koch sixfold appeared on 8th August at Milk Hill. It was virtually identical but it had a large and ornate central feature, an inverted fractal.

The East Field sevenfold fractal appeared on the 9th July 1998 with a huge laid central area. Exactly one month later, on 9th August at Tawsmead Copse, a formation appeared which, again, had the same overall silhouette as its predecessor and again had a large and decorative central rosette. This Tawsmead Copse formation is widely regarded as one of the classics of the phenomenon.

THE DANEBURY SPINNER

Fig.15 - Danebury 5th July 1998

It was a very simple and cleanly made design. There was a laid circle, of about 102’ in diameter, with a central standing circle of half the size or about 51’ diameter – Euclid’s Second again [Fig.16].

The outer circle was surrounded by seven laid and swirling elements. The seven elements around the

Fig.16 - Danebury diagram

This was a spinner and the first seven-fold spinner.

Acknowledgements and thanks
Photography: Steve Alexander, Werner Anderhub, Patricia Murray and Karen Ryan
Photography: Dolf Braat and Ofmil Haynes Jr