The Circle and Its Centre
THE CIRCLE AND ITS CENTRE
Rereading my last piece (Antsy [but not nasty] about Ansty) of November 23rd 2016 I returned to the eternally recurring question of the circle’s centre. Ansty was a formation with eight or ten concentric circles. Concentricity, by definition, requires that the many circles share a single centre. The centre at Ansty (1) was an immaculate three-leaf device in a totally undisturbed area. How was this done? Somebody must answer this question.
I have been wrestling with the paradox of the centre-less circle for some time and it seems appropriate now to discuss it.
GEOMETRY AND BASICS
Two useful definitions of the circle:
“A perfectly round plane figure whose circumference is everywhere equidistant from its centre”
“the curve traced out by a point that moves so that its distance from a given point is constant”
The disc, the doughnut and the ring, (2), (3) and (4) have long been familiar elements of crop circle formations. But all forms based on circularity (including 3D spheres, cylinders and cones) require two essential precursors (5); a centre (location, the red dot) and a radius (dimension, the green line). Please reread the two definitions. They both emphasise these precursors. Statements of certainty make me nervous but I am sure that a circle cannot be made without these two fundamental components. In our dimension, anyway!
(6) shows a typical draughting compass. The screw adjustment maintains the exact distance between the point (the centre) and the circumference which establishes the radius, the size, the magnitude.
Diagrams (7) and (8) show some of the elements of circle geometry while (9) and (10) show the two basic methods of division.
THE UBIQUITOUS CIRCLE
We inhabit a world based on roundness. Wherever you are now, stop reading and, remaining in your seat, look around you. Without moving, you will be able to see at least ten circles or objects based on circularity.
And we were doing just fine on our obviously flat world until some smartypants showed up and suggested it was round, an orb! (In fact, for the pedantic, it is an Oblate Spheroid.) Curiously many historic cultures, though knowing the earth to be flat, assumed it was a disc. We are viscerally welded to the idea of the circle.
GOD AS A CIRCLE
We all know the poetic statement, widely attributed, but popularly credited to Voltaire in the Seventeenth or Eighteenth Centuries. “God is an infinite circle whose centre is everywhere and whose circumference is nowhere.”
This is a beautiful and arresting notion but a little superficial research shows that as early as the fourth-century it was credited to Marius Victorinus. He was followed by Alain de Lille in the twelfth century and eventually by Blaise Pascal in the seventeenth. These three, and probably others, said “Nature is an infinite sphere whose centre is everywhere and whose circumference is nowhere.” Around Voltaire’s time the sphere reduced itself to a circle.
BLACK ELK SPEAKS
We move suddenly to the other hemisphere, North America, where Black Elk, (11) medicine man and visionary of the Oglala Lakota, speaks lyrically of the circle.
“Even the seasons form a great circle in their changing, and always come back to where they were.”
“Everything the Power of the World does is done in a circle. The sky is round, and I have heard that the earth is round like a ball, and so are all the stars. The wind in its greatest power, whirls. Birds make their nest in circles, for theirs is the same religion as ours.”
“The Wasichus have put us in these square boxes. Our power is gone and we are dying for the power is not in us anymore. You can look at our boys and see how it is with us. When we were living by the power of the circle in the way we should, boys were men at twelve and thirteen. But now it takes them very much longer to mature. Well, it is as it is. We are prisoners of war while we are waiting here. But there is another world.”
AN EXTRAORDINARY COINCIDENCE
The Silbury Hill Koch fractal formation of 23rd July1997 (12) hosted two simultaneous events. First, people entered the formation and sat down together. Second, Patricia Murray, overflew the formation and photographed the group where they were. Now, neither of these two occurrences is unusual in itself but an aerial photograph of a formally arranged group of visitors happens only occasionally.
I was curious about the positioning of the group. It was very close to the centre of the formation but not on the centre. I drew in the horizontal and then the vertical centre lines of the formation (13) and found, to my surprise that the seated circle was placed precisely within one of the four corners formed by the two axes. Look carefully at the photograph. These white lines did not exist on the field to guide the visitors to their significant place. Look at the accuracy with which the group touches the lines.
Who were they and where were they from? What is their recollection of the forming and positioning of the group? What moved them to take that exact position at the precise moment the photograph was to be taken? I have tried unsuccessfully to contact them for nearly twenty years.
This event, for me is charmed and loaded well beyond the possibilities of that old copout coincidence and beyond happenstance or accident.
Let me take it further. If you see a picture of children holding hands in a circle, it will rarely be circular. It will tend to be elliptical or egg-shaped. We are poor at forming accurate life-size circles.
I enlarged and perspective-corrected the picture and found that the group was geometrically circular. There were 22 individuals. 22 is the second in the Master Number sequence which symbolises varying degrees of contact with other dimensions. The Silbury formation was a replica of the Koch fractal, a mathematical diagram originated by Helge von Koch and based on sequences of diminishing equilateral triangles. Image (14) shows that the formation is surrounded by 18 third iteration triangles. A further bewilderment is the discovery that the diameter of the circular group is the exact dimension of the eighteen triangles.
So here I am, arguing against myself! Having stated that a circle cannot be made without the two essential components, centre and size, I present a centre-less circle sized only by comparison with separate triangular features. In mitigation, I did say “In our dimension, anyway!”
But who or what caused or persuaded twenty-two individuals to form a remarkably perfect circle, of a specific dimension, and then to place that circle at an exact but unmarked position?
Images (15) & (16) are astonishing. (15) was taken by a lady holidaying in northern Scotland. She kindly sent me the image because she knew of my interests. I saw (16) in a local newspaper and pursued the journalist for a copy. I believe this picture was taken in the West of England or Wales.
Most people, faced with the strange, the implausible or the incomprehensible will simply shrug it off and ignore it. For me, the questions posed by the Silbury formation and the sheep circles remain irritatingly unresolved. We will always find an enthusiastic “explanologist” who insists that the farmers put the food out in perfect circles and then asked the sheep not to cluster but to stay positioned in single file as they eat!
CROP CIRCLES ARE NOT CIRCULAR
It is some years since I actually measured a formation in the field but I was always concerned to confirm their circularity. A simple method is to check several diameters, through the approximate centre. I used to find that most of them were oval. It was generally a small ovality, five percent or less between the long and short axes. They would be, as a patent attorney might say, substantially circular, but the consistency of the ovality was noteworthy.
The formation at Waden Hill, Silbury of 3rd July 1998 (17) & (18) was a simple enough formation, a second cousin to the Flower of Life geometry. As can be seen from the drawing and the photograph one of the six petals was about half the size of its five sisters. The smaller petal faced North-East and a line drawn through it it from the formation centre seemed to tangent the ditch surrounding the Avebury stones.
I include this rather un-noticed formation here because the flattened circle at its centre was the most accurately circular I ever found. Several cross-measurements tallied to an inch.
CROP CIRCLES AND PI
The crop circles’ fascination with Pi has persisted over a number of years. One possible explanation for this is their parallel interest in the ancient conundrum, the Squaring of the Circle. This holds that, if the Square represents the world, the material realm, and the circle is symbolic of God, the Infinite, then the Squaring of the Circle might be a signal of an imminent reconciliation of Spirit and Matter. And of course Pi, the arithmetical constant involved in all circular calculations, might be seen as the master key to that transformation.
Figure (19) shows, first, the “real” Pi number which has an infinite number of digits. The second number, 22/7, is a simple formula which, as can be seen, produces a very similar result.
On 1st June 2008 an exquisite formation arrived below (20) Barbury Castle. It was beautifully and precisely made and it quickly revealed that, by its brilliant exploitation of both the radial and concentric systems of the circle, it was encoding Pi, 3.14159264…
Prior to this seminal Barbury Castle formation there had been no reference to the “real” Pi number, but several formations had referred to 22/7. Perhaps the most significant of these was Picked Hill (sometimes referred to as Woodborough Hill) of 13th August 2000 (21) which again used both 44 radial and 14 concentric geometries. The brilliance of this exemplary crop circle is confirmed by the implicit heptagram (22) which restates the seven.
The complexity of Picked Hill (23) required at least 45 centres of curvature. On site there was no evidence of a single one. As the image shows, 44/14 is equal to 22/7.
THE CROP CIRCLE AND ITS CENTRE?
The crop circles have never been comprehensively surveyed although www.cropcirclecenter valiantly tackles the issue and maintains an admirably scrupulous catalogue of events. My statements therefore can be no more than an opinion.
So here is my opinion! The majority of formations that I have seen have had a token centre-ikon, a tuft, a whirl or a bird’s nest. But I have never known the centre symbol to be on the actual or geometrical centre of the formation.
Karen Alexander disagrees with this statement having seen several formations in which the centre marker appears to coincide with the actual centre. Both positions must hold!
Photograph (24) is a groundshot of the remarkable and enormous Milk Hill formation of 12th August 2001. Though there are no standing features it is clear that the swirls here are not centrally placed.
ALMOST CENTRAL TRAMLINES
A double formation appeared at Gypsy Patch Etchilhampton on 31st July 1997 and (25) shows its northern component, a swirled six-pointed star. The diagram (26) illustrates the elegant geometry underlying it.
The points of the star are perfectly contained by the green ring and that in turn is held by a red square. The corners of the red square size and position the laid outer ring surrounding the formation. The sharp-eyed will have noticed the Squaring of the Circle.
The following year, on the 5th July 1998 another swirling formation (27) arrived at Danebury Ring. Like Gypsy Patch it was precisely and cleanly formed and, again like Gypsy patch it was based on a powerful guiding geometry (28), in this case Euclid’s 2nd Theorem. Mr Euclid tells us that the circle contained by an equilateral triangle will be half the size of the circle containing that triangle. The crop circle narrative often returns to this theorem and the ratios it generates.
I refer to these two formations to draw attention to another remarkable feature they share. Look back at photographs (25) & (27) and note the position of the tramlines which cut across both their standing centres. They nearly bisect the discs and they almost taunt us with how close to the centre they are!
TWO HILLS, TWO DISCS, TWO BRACELETS
The Silbury Hill formation (29) of 4th May 1998 was lovely, elaborate and very difficult to draw. The crop was oil seed rape and the image of this elegant golden artefact on the hillside was almost shockingly beautiful. A large disc of standing crop was set within a broad, almost moat-like laid ring from which emerged thirty-three standing features.
These components were delicately shaped curved forms like flames (30). The inner and outer banks of the “moat” were slightly scalloped to echo the curving geometry of the flames. To place thirty-three elements equally around a circle of 360° is not a task to be lightly undertaken for it requires that each pair is separated by a precise angle of 10.9090°…
There has been no crop circle comparable to Silbury until Cley Hill 30th July 2016 (31). For me these two share so many characteristics that I consider them to be siblings. Cley Hill was made in wheat and was dominated, again, by a large standing central disc. This disc was surrounded once more by a broad laid ring, in this case smooth sided. From the flattened ring emerged twenty standing forms. Their shape reminded me of old-fashioned school pen nibs and that is what I call them though there is no more significance to the word “nib” here than there was to “flame” at Silbury. There were 20 of these nibs and so their angle of separation was a much simpler 18 °.
It is my strongly held view that neither the Silbury flames nor the Cley Hill nibs (32) could have been placed so meticulously around their respective “bracelets” without the use of a centrally placed theodolite. Prove me wrong.
We have all made our personal relationship with the phenomenon and for me, from the start, drawing the circles brought rewards. I was always anxious to respect their inherent quality and regularly found that I was spending an inordinate amount of time on seemingly unimportant details.
The flames and nibs are a case in point. The thirty-three Silbury Hill flames were identical as were the twenty Cley Hill nibs but, in both cases, I found their simple shape extremely difficult to draw. I learned a great deal.
AN INCONCLUSIVE CONCLUSION
I am, at the end of this rambling and discursive piece, more bewildered than I was when I began! We have looked at the widespread and ancient reverence for the circle and we have seen the crop circles’ paradoxical avoidance of true centres.
But how are large and small standing discs formed without reference to a middle? Is the force that organises sheep circles the same as that which controlled the human circle at Silbury? I have opened a box which, for me at least, is unclosable.