The Fractal Field
The Fractal Field
Every crop circle season’s end is both remarkable and bewildering. It takes some months to find what might be the most effective approach to understanding. I have come to rely on two guidance systems.
The first and most obvious is the crop circles themselves and I use drawings as the main gateway. I have drawn the formations every year since 1990 but this year, for the first time, I felt impelled, almost instructed, to draw every crop circle as soon as it appeared. This has never happened before and the result is that my living room wall still displays a complete set of 2017 formations. Living with a comprehensive visual record of the season, I am very impressed with the quality and variety of this year’s gifts. I believe that 2017 gave us a notable and possibly historic display.
The Crop Circle Teatime meetings offer the second system, almost a litmus test. I must admit that I have found these visiting international groups bring an accurate yearly sense of how the human spirit shifts. Fifteen years ago the Q&As were overwhelmed by doubts generated by the man-made nonsense; this is no longer even mentioned!
Despite my enthusiasm for this year’s offerings, I felt unable to discuss the year usefully for a while. Karen Alexander suggested, with her usual prescience and wisdom, that having posted The Circle and Its Centre, 23rd “January 2017, Squareness Reconsidered, 6th March 2017 and Phive Pointed Star, 10th April 2017 the fourth and final component of the set had to be crop circle fractals. As always I am grateful; this is an under-explored aspect of the crop circle story.
THE ICKLETON MANDELBROT
We were giddy at the end of the 1991 crop circle summer. After the surprises of 1990’s East Field event and its attendant siblings, the 1991 revelations of the now classic Barbury Castle equilateral triangle were almost indigestible. And then, on 17th August 1991 at Ickleton near Cambridge, a majestic and beautifully made cardioid-shaped formation (1) appeared.
It was a further mystery. It had been assumed that the season was ended and this new event had occurred far beyond the phenomenon’s usual territory. Somebody recognised it as fractal and a part of the Mandelbrot Set, but the word “fractal” and the name “Mandelbrot” were in those days new to most of us. Of course it was quickly suggested that, because of the proximity to Cambridge, the formation must have been made by one of the University’s mathematics departments. The nervous, bless them, will forever respond thus!
Look carefully at (1). This was one of the most important and most carefully crafted formations we have ever had. Each half of the heart-shaped main area is contained by a subtle multi-curve line and there was the first example of a semi-transparent “curtain” (2) the thickness of two or three standing stems of wheat. This is no longer unusual but it was first demonstrated here.
Diagram (3) is the silhouette of the Mandelbrot crop formation, named after its creator mathematician, Benoit Mandelbrot. Diagrams (4) and (5) are randomly selected moments from the many images to be found showing the growth and development of fractal systems. The silhouette chosen here as a crop circle is very useful for, if you watch the animated development of the fractal, this symbol insistently recurs at every scale.
This was the first fractal formation we were given and it was surely a Herald and a Harbinger of the many fractals to appear since. I have pulled three definitions, or parts of definitions, from Google and Wikipedia. They might be helpful.
“A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process.”
“A fractal is a curve or geometrical figure, each part of which has the same statistical character as the whole. They are useful in modelling structures (such as snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth and galaxy formation.”
“A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known as expanding symmetry or evolving symmetry. If the replication is exactly the same at every scale, it is called a self-similar pattern. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.”
My own contribution would be that a fractal both contains and is contained by a component of itself. Having stated that, I must freely admit that much of the scientific and mathematical theory is beyond me though I find it astonishing. But even more astonishing is the way that so much of science and mathematics leaks out around the edges to show real relevance to matters of consciousness, spirit and occasionally poetry.
The next fractal crop circle to appear (6) was opposite Stonehenge on the other side of the A303, the main artery to the west of England. In the summer months the 303 is crowded with traffic heading to holiday destinations in Devon and Cornwall. Hundreds of travellers stopped to experience what must have been their first circle.
The Stonehenge formation became, if not the most visited crop circle ever, then certainly a contender for that title. It was rumoured that the farmer was happy with the income that was generated. I was there on the first day and on several subsequent occasions including two or three night visits. Even in the dark, it was never empty. In daylight the formation was overwhelming in scale. There were one hundred and fifty-one beautifully swirled circles of various sizes and someone, seeing it for the first time, likened it to a “confetti storm”.
The Stonehenge crop circle was based on a fractal formula discovered by mathematician Raoul Julia and, accordingly, it is often referred to as the Julia Set. Few of us fully understood the mathematical principals that it embodied and, for some years, it was regularly (and wrongly) assumed that any formation made of many circles was a fractal. This formation, more than twenty years later, is one of my most memorable crop circle experiences. Beyond its fractal features it exhibited in the field a significant number of numerical, geometric and angular characteristics.
SILBURY HILL 23/7/97
This formation, a Koch fractal (7), was dealt with in more detail in my posting of 23rd January 2017 The Circle and its Centre, but I include a further diagram here (8) to illustrate its fractal nature. The first image shows the equilateral triangle on which the system is based. The image shows how each of the triangle’s three sides is divided into thirds and a smaller triangle is placed on the centre third of each of the three sides. Diagram (8) shows that the repetition of this sequence makes the form more complex as the length of the perimeter increases. In principle, this operation could be infinitely repeated.
This illustrates one of the many paradoxes that fractals offer: while the perimeter is able to increase beyond limit, the area of the form remains contained.
MILK HILL 8/8/97
About two weeks after the arrival of Silbury Hill, its sibling arrived at Milk Hill. As drawing (9) illustrates, the two formations were virtually identical apart from Milk Hill’s central feature, a decorative rosette. Whereas the Koch fractal develops, as we have seen, by adding smaller and smaller triangles to its periphery, the central “rosette” at Milk Hill demonstrates a novel procedure. Silbury starts with an equilateral triangle but, at Milk Hill, the central device starts with a hexagon. While the fractal programme (for the main silhouette at least) requires the addition of small triangles to the form, the central feature here reverses the method. It starts with a hexagon and then removes small triangles from the shape as though it had been pecked by birds!
EAST FIELD 9/7/98
We had long awaited the arrival of sevenfold geometry and the appearance of the enormous heptagram in East Field (10) produced great excitement. It also brought two misapprehensions: first, that this was the first seven-fold. Wrong! In fact the exquisite little Danebury Ring (11) formation of 5th July beat it by a few days. Second, it was believed for some time that the East Field heptagram was the largest single area of laid crop. Again wrong! After some time it was realised that the record would be taken by the Roundway “Splash” (12) of 31st July 1999. Coincidentally and curiously these contenders for the largest area were both seven-fold.
A Dutch architect produced a brilliant and complex diagram (13) exploring the fractal possibilities of this gargantuan East Field crop formation.
TAWSMEAD COPSE 9/8/98
Precisely one month later the lovely Tawsmead Copse formation (14) and (15) appeared. Like its predecessor cousins of 1997, Silbury Hill and Milk Hill, the later arrival, Tawsmead, had developed a central feature. Tawsmead’s overall profile was not an exact twin of East Field’s and nor did it seem to carry new geometric information. It was however one of the most and meticulously made crop circles I can recall. The numerous circles surrounding the body of the formation were arrayed like display cases, showing an astonishing range of swirls, weaves and nests.
WEST KENNET 4/8/99
The beautiful West Kennet formation (16) was an articulate and immediately comprehensible fractal square of squares. Figure (17) shows the construction system. Starting from the black “mother” square, the fractal develops four smaller pale blue “daughter” squares on the corners and then sixteen red “grand-daughter” squares on the daughters’ corners.
The Silbury triangular Koch fractal went through three iterations and then presented circles where yet smaller triangles might have been expected. The West Kennet fractal similarly went through three iterations and then resolved into circles.
What could this mean?
In both cases the diameter of the circles seems to represent the size the fourth iteration triangles or squares would have been. But it is clear that the medium here – standing crop – had reached its limit of scale. Smaller triangles or squares in wheat would clearly have challenged even the prodigious technical skills of our cousins and so they ended both sequences with groups of circles, a simpler shape which might be understood as a full stop or termination.
INFINITY, INWARDS AND OUTWARDS
This raises a fundamental area of fractal interest.
Observing the growth and development of a fractal set or system on a computer it is clear that it is able either to grow or to shrink without constraint. The boundaries just noted in the West Kennet formation were entirely a function of the material medium, the wheat. These systems demonstrate a boundless ability to increase or reduce, to become greater or smaller beyond limit.
CLIFFORDS HILL 28/7/01
The Cliffords Hill formation (18) raises many questions.
It is based on a single simple idea. A circle is drawn and two circles, each exactly half the size of the first, are placed within it. Applying the equation, 2πR (where R equals radius), to calculate the circumference of a circle it is clear that each of the half-size circles will have a circumference of exactly half that of the first, containing circle. Thus, referring to Diagram (19), the sum of the circumferences of the two green circles is precisely that of the main blue circle. And pursuing that logic, the circumferences of the four red circles equal the two green, the eight pink and the sixteen yellow. And, to remind ourselves, the one big blue. Thus, at every stage, at every doubling, the total perimeter is constant.
(This sequence, starting with the double circle, reminds us of the Dao Yin Yang symbol (20). The Chinese Dao philosophy holds as a basic principle that all things exist as inseparable and contradictory opposites. Examples might be dark-light, old-young and female-male. These phenomena could not exist without their contrasting twin. To emphasize this the Yin Yang symbol shows a seed of darkness in the light and a white seed in the black.)
The Cliffords Hill formation demonstrates, more clearly than any other, how free of dimensional constraints fractals are. If we simply double and keep doubling the number of circles we approach (theoretically) a line so thin (and so long) that it becomes invisible. Conversely, if we assume that our initial blue circle, rather than being the starting point here is, in fact the twentieth iteration, we might have to imagine a fractal the size of Europe!
THE STATIC AND THE DYNAMIC
Most crop circles seem to arrive as fixed diagrams. There have been occasions in which they were transformed or added to but, essentially, they have been static patterns. Perhaps illuminating the significance and importance of the fractals is that, alone among crop circles, they imply limitless expansion or contraction. Boundless and infinite change.
The “Infinite” is mentioned again and this reminds me of the word “continuity” which, in turn, brings up this season’s Battlesbury formation (21). For me, Battlesbury was perhaps the most arresting and articulate crop events of the year.
Over the years many formations featuring a single continuous ribbon-like path have appeared and Battlesbury is exemplary. The broad unicursal ribbon has been overlaid and underlaid to replicate a central mother circle and four satellites or daughters – a quintuplet!
In Diagram (22) I have taken one branch of that quintuplet and “fractalised” it through a couple of iterations while (23) shows a condensed view of Battlesbury. Diagram (24) shows the Lemniscate (the symbol of Infinity) while (25) shows a Mobius strip. If you take a strip of paper and simply join its ends to form a loop, you have created two surfaces, an internal and an external. If before joining you add a single twist to one end of the strip you have created a single continuous surface. The enigmatic formation of 16/6/2010 at Chirton Bottom (26) is possibly the closest the circles have ever come to a direct representation of a Mobius strip.
Each of the four diagrams, (20), (23), (24), (25) and (26), though unrelated to fractals, have an illuminating relationship to each other and to both infinity and continuity. Perhaps these two words are particularly relevant.