Squareness Reconsidered

Posted by on March 6, 2017


The East Field formation of 12th July 1990 (1) was a momentous event in many ways. Having arrived to enormous excitement (2) in East Field, Alton Barnes, it established that location as the cherished historic centre of the crop circle phenomenon. It also granted the Carson family, farmers and custodians of East Field the role of crop circle nobility, a role they fulfilled with grace and dignity for some years. But above all, Alton Barnes 1990 marked a step-change in the nature of the phenomenon.

1. East Field, Alton Barnes, 12th July 1990

1. East Field, Alton Barnes, 12th July 1990


2Traffic jam at East Field

2 Traffic jam at East Field

Prior to this event the crop circles were exclusively round. There were circles and sometimes rings but they never presented even a hint of a straight line. In 1990 the deluge of geometrical form that we were about to witness through the coming quarter century was unimaginable.

The pre-1990 circles and rings occasionally grouped themselves into the hint of a significant pattern but it was really the quintuplets that carried most meaning. And then only with the benefit of hindsight! Diagram (3) is a single page from Alan Brown’s voluminous and still astonishingly unpublished, masterwork, Patterns of Coincidence. It shows sixteen silhouettes of the late ‘80s. His study, about which more later, examines in scholarly detail almost seventy formations.

3A page of Allan Brown’s Quintuplet diagrams

3 A page of Allan Brown’s Quintuplet diagrams

Years before Allan’s work I was intrigued with the increasing number and variety of these simple formations. Their sizes and proportions continually changed but they consistently retained an inherent grace and order.


I discovered that the quintuplet formations were obeying, not slavishly but generally, four rules (and two extra) and these were the rules that generated the range and elegance of the “quintuplet type”.

The First Rule

The Central Circle (often known as the Mother Circle) is larger than the four satellite or “Daughter” Circles.

The Second Rule

The four Daughter Circles are substantially the same diameter.

The Third Rule

The Daughter Circles are the same distance from the Mother.

The Fourth Rule

The Daughters are set out on orthogonal axes, that is the axes are at

rightangles to each other.

There are two further rules relating to the “Celtic Cross” formation that is a quintuplet where the satellites or Daughter Circles are linked by an orbit ring.

The First Celtic Cross Rule

The orbit ring runs through the centres of the satellites.

The Second Celtic Cross Rule

The orbit ring is normally thin.

The First Rule has never, to my knowledge, been broken and the Second Rule only rarely. The Third Rule is fairly consistently obeyed while, as can be seen in diagram (3) liberties are occasionally taken with the Fourth!


I returned to California after the 1997 season perplexed by the Headbourne Worthy quintuplet (4) of 7th July. After devising the quintuplet “rules” I was confronted here by an entirely new model. First, the daughter or satellite circles crashed into the body of the mother circle. This was without precedent. Second, almost as though wishing to draw our attention to the first irregularity, the four daughters were conspicuously emphasized with round frames.

4Headbourne Worthy, 7th July 1997

4 Headbourne Worthy, 7th July 1997

As so often with important ideas, after they have been noticed, they are obvious. Headbourne Worthy was telling us, screaming at us, “Look! Pay attention. Please notice. The circle is squared!” The diagram (5) shows how, if lines are drawn to connect the centres of the four Daughter satellites, the perimeter of that square will be equal to the circumference of the Mother circle.

5 Headbourne Worthy diagram

5 Headbourne Worthy diagram

It is my suspicion that the Crop Circle Phenomenon has chosen to establish an entirely symbolic conversation. There is substantial evidence that it wants us to concentrate on the symbolism of Squaring the Circle (or more precisely Circling the Square). What an optimistic thought in these bleak times!  The Square might merge with the Circle, the Material could approach the Divine, the Earth, perhaps, is to become a little more Heavenly.



John Michell, (6) author, philosopher, geometer, antiquarian and mystic  died in 2009. He was one of the most influential of early crop circle observers setting up and editing the Cereologist, an early and influential magazine devoted to the phenomenon. His absence is keenly felt, especially among those of us who believe that geometry and number are central to the work.

6John Michell

6 John Michell

John’s legacy is enormous. He published over forty books but I want to concentrate here on his seminal Earth/Moon diagram (7). I have spent years contemplating and digesting many diagrams but John Michell’s Earth/Moon is always the most compelling.

7John Michell’s Earth/Moon diagram

7 John Michell’s Earth/Moon diagram

He said that, reading a newspaper on the upper deck of a London bus he saw, in a moment of epiphany, the complete and immaculate diagram. He quickly drew it in the margins of the paper.

The diagram is remarkably graceful but requires close attention. It is based fundamentally on the ratio of the Earth to the Moon, 7920 miles diameter to

2160 miles diameter. This ratio, when reduced to its lowest common denominator is 11/3. The LCD unit is 720 miles. The red numerals on the diagram represent units and the letter “r” means radius.

John’s insight on the bus was this. The Moon is pulled down to kiss the surface of the Earth and a square is drawn to enclose each of the two spheres. Another circle is drawn, centred on the Earth and passing through the centre of the Moon. The square enclosing the Earth precisely squares the new circle.

This elegant construction has two further astonishments. First, the triangular “shoulders” which act like book-ends supporting the tiny moon turn out in fact to be Pythagorean 3.4.5 right-angle triangles. Second, the diameter of the Earth, 7920 miles, is the product of 8 x 9 x 10 x 11 while the diameter of the second circle (half Earth + half Moon = 5040) is the product of 1 x 2 x 3 x 4 x 5 x 6 x 7. Where did this come from? Who organised this precision? Is this what scientists would dismiss as “coincidence”?

The bus-trip insight that brought this humbling diagram could not have occurred to just anyone, but John Michell was an almost mystical Master of Number. Before the revelation of the diagram itself he would have been fully aware of the two magical numbers, 7920 and 5040, and the eleven to three ratio of our planet and its satellite.

I have often reflected on John Michell’s Earth/Moon squared circle diagram. It remains an inspiration. It has for me the elegance and accuracy of a perfectly assembled old Swiss watch.


John Martineau (8) was studying philosophy at Bristol University when he was snagged by the mystery of the crop circles. He became aware of Dr Gerald Hawkins’ work on the geometry of the early formations. Hawkins had shown that the circles expressed the exact diatonic ratios of musical intervals.

8John Martineau

8 John Martineau

In the early ‘90s John had his epiphany! He realised, and demonstrated the extent to which crop formations are constrained, contained, sized and positioned by invisible geometries. There is no obvious indication within the crop of the underlying guidelines; which are subsequently discovered by drawing board work.

9Upton Scudamore, August 1987

9 Upton Scudamore, August 1987

The Upton Scudamore “fat” quintuplet (9) became John’s revelation. He discovered that this initially unremarkable formation was designed and constructed (10) to comply with triangular (threefold), square (fourfold), pentagonal (fivefold) and octagonal (eightfold) geometries. It is easy to understand how a quintuplet, based on four, might accommodate the square and the octagram but how are the proportions of four different number systems organised to synchronise so immaculately on a single formation?

10 Upton Scudamore geometries

10 Upton Scudamore geometries

Diagrams (11) & (12) explore the pentagram geometry in closer detail. 

11 Upton Scudamore pentagram

11 Upton Scudamore pentagram

12 Upton Scudamore pentagram detail

12 Upton Scudamore pentagram detail

(11) shows how the pentagram positions and sizes the Mother circle by location on the “arm-crossing” of the star. Diagram (12) shows how the crucial distance between the Mother and the four Daughter circles is established by another pentagonal protocol. In this case the red circle, instead of seeking the arm-crossing position, is simply contained by the five sides of the pentagon at the centre of the small star. The red circle then becomes the standard distance between Mother and Daughters.

I have included here only a tiny sample of John’s crucial work of the early ‘90s. He demonstrated that a hidden system of geometric arrangements constrains and orders the crop circles and controls their size, position and proportion.    

Historically, I hold this to be one of the few true advances in our studies of this mystery.  John remains intractably (and in my view, inappropriately) modest and, over more than twenty years, has implacably refused to acknowledge the importance of this crop circle geometry achievement!

Moving deftly sideways (or upwards) image (13) shows an eclipse, another of John Martineau’s long-term preoccupations. The photo shows the way the Sun and the Moon, during an eclipse and when viewed from planet Earth, are seen as precisely the same diameter. Who worked this out? Who designed it? Is it yet another one of those coincidences?


13 Eclipse


At the end of the millennium I returned to England after several years in California. By that time, season by season, I had drawn many dozens of surveys or silhouettes of the circles and held, in the back of my mind, the idea of publishing a selection of them as posters. The problem for me was that, though I had this archive recording years of obsessive work, my computer skills then were limited. I met Allan Brown (14) who is a designer and polymath with a well-honed range of computer skills. We agreed to work together on the crop circle poster series.

14 Allan Brown

14 Allan Brown

The project took several weeks and, looking back, we both remember it with real pleasure. During this period Allan knew my interest in the allegory of the squaring of the circle. Its manifestation at Headbourne Worthy encouraged him to start an exploration of the full range of quintuplet and Celtic Cross events to see whether the squared circle was touched on elsewhere.

The results of his research were remarkable. He assembled a collection of almost seventy quintuplets, the substantial majority of which, by various methods and protocols, squared the circle. A handful revealed absolutely novel geometric methods which had been unknown prior to this study.

This material, Patterns of Coincidence, has never been published though a few copies exist. Allan went on, working with John Michell, to publish Crooked Soley: a Crop Circle Revelation and How the World is Made. These are, I believe, two essential texts for anyone with any interest in the crop circle phenomenon.


This piece was going to deal with the square and I promise it will, it will. But, as often, I have been pulled sideways!

The circles have brought me a great deal of joy but among the most precious of gifts has been my knowing, working with and friendship with Allan and the two Johns. As I have tried to illustrate, each of them has made an enormous contribution to our understanding. Tellingly, in every case they leave us with more questions than answers.

John Michell’s Earth/Moon diagram, John Martineau’s geometric studies and Allan Brown’s quintuplet analyses are the gifts of giants. 3 Geniuses


I want to cite first the West Kennett square fractal of 4th August 1999 (15). This is without doubt one of the more elaborate and enigmatic crop formations we have ever been given but, before looking at it in detail we must examine its forebear, the Silbury Hill Koch formation (16) of 23rd July 1997.

15West Kennett, 4th August 1999

15 West Kennett, 4th August 1999


16 Silbury Koch fractal overlay

16 Silbury Koch fractal overlay

The Swedish mathematician Helge von Koch proposed an early protocol, known as the Koch Snowflake fractal (17) which was surprisingly, and precisely, reproduced in a crop circle at Silbury Hill.  The overlaid diagram (16) will explain the principal more clearly. The “Mother/Daughter” idea which has thus far been useful in the discussion of the quintuplets becomes even more so here.

17 Koch Snowflake diagram

17 Koch Snowflake diagram

The first step is the placing of the main, the “mother” equilateral triangle, shown here in black. Each of the three sides of the black triangle is divided into thirds and, using that 1/3 dimension, three blue “daughter” triangles are placed on the centre of each black side.

This method is repeated. Each side of the blue daughter triangles is divided into three (the second iteration) and, again using the 1/3 division, twelve red “grand-daughter” triangles (the third iteration) are placed at their appropriate location.

Were this protocol to be followed meticulously, we might expect 48 yet smaller triangles (the great-granddaughters and the fourth iteration) to surround the formation and to complete the sequence. Why are they not there? This question, like many others is impossible to answer definitively, but I suspect that the medium (wheat) and the scale (increasingly small) made the tiny equilateral triangles difficult to create. Instead, at every point where the fourth triangles were expected, there is a small circle. The circles’ diameter is exactly the size the triangles should have been.


We return to the marvellous West Kennett formation of 4th August 1999 (15) and (17) and the overlay diagram (18) follows much the same procedures as the triangle-based Silbury Koch just discussed. Again, the Mother, this time square, is black, the daughters are blue and the grand-daughters red. Here again, the fourth iteration great grand-daughter squares are avoided to be replaced by 288 small circles. While Silbury maintained a constant 1/3 ratio between its iterations or generations, I have failed to discover that the West Kennett squares showed any reliable proportional relationship.

18West Kennett overlay

18 West Kennett overlay


The West Kennett formation was as rich and momentous as any we have ever had. It displayed many meaningful characteristics and, of course, posed more questions than answers. Firstly, it was a rare negative/positive twin. A little while earlier on 16th July 1999 at Windmill Hill (19) an exact predictor of the West Kennet formation appeared; exact that is except for the fact that the standing crop in Windmill Hill was laid in West Kennett while the laid was standing. And vice versa. The only other pair of laid/standing reverse twins that I know of were at Clatford 21st June 1998 (20) in barley and Firs Hill 22nd June 1998 (21) in wheat. It is not common, even when formations are so clearly linked, for them to arrive on following days.  

19Windmill Hill 16/7/99

19 Windmill Hill 16/7/99

20Clatford 21/6/98

20 Clatford 21/6/98


21Firs Farm 22/6/98

21 Firs Farm 22/6/98


Andreas Muller entered the West Kennett crop circle on the morning it appeared and he called it “The Thistle King’s Castle”. The name has endured ever since. The Thistle King itself (22), mature and around three feet tall, was perfectly placed in the centre of the swirl of one of the 288 small surrounding circles. The only reasonable explanation of this poetic positioning is that the whole formation must have been gently and delicately lowered around the thistle. Alternative suggestions welcomed.

22 “Thistle King”

22 “Thistle King”


Allan Brown produced a diagram (23) showing that each one of the diminishing squares relates to its relevant circle using the small surrounding circles as guides. The squares are circled. The circumference of the circles is equal to the perimeter of the squares.

23 West Kennett squares circles

23 West Kennett squares circles


The authors of this phenomenon are clearly capable of taking over the Earth’s conventional media to deliver the “message” of their choice. They know us too well. They know us to be an aggressive, volatile and impulsive bunch and so they give us allusions, glimpses and hints rather than suggestions, statements or injunctions.

The story of fractals in the crop circle narrative leaves much more to be explored but the central point here is the significance of the awesome West Kennett Square of Squares which touches on so many aspects of the circle phenomenon. (The almost uncanny similarity between West Kennett and the quintuplets will not have escaped you!)

My whole premise is based on the view, widespread both culturally and historically, that the circle symbolises God, the divine and the spiritual while the square represents the material realm and, more particularly, Earth.

We have had decades of formations reiterating the Squaring of the Circle. I think that we are being advised that reconciliation between Earth and Spirit is, if not imminent, then at least possible. The circle cannot be squared mathematically without the application of Pi. This is perhaps the symbolic key to the door between squareness and circularity and, again, for decades the formations have restated Pi in several formats.

West Kennett and the massive variety of crop circle squares (and cubes!) both preceding and following it suggest range, diversity and playfulness.

The final image (24) shows a random selection, from the last few years, of what our cousins feel might be developed here. They symbolise coming attractions and show what might be achieved as Earth and Spirit find a new way to reconcile.

24A few recent squares

24 A few recent squares


  1. complete highlight receiving these emails , thanks you so much

  2. marvellous! Thank you for enlightening ke to the geometry, whereas I gazed before somehow feeling these designs were speaking to me , but I didnt know the language.
    Many many thanks for sharing your wisdom.

  3. There were two more crop circles that displayed the reversal of the image, both based on flower of life geometry. June 17 and 24, 2007 in Germany.
    I have wanted to be able to review Allan Brown’s work on the quintuplets of years. Any chance of convincing him to publish it?

  4. Michael, many thanks again to the mathematical wizardry of your mind and heart. There is an inordinate amount of intuition behind what you write here. On the topic of positive/negative, a few nights ago I watched a documentary, circa 2009 by Alan Foster at the UFO Congress. In it, he discussed the formation of July 2000, known as the silicone chip. With over 1600 clumps of grain left standing, the fact that if lines were drawn either up and down or across the centre, each side perfectly fits into the other as a plug inserts into a receptacle. Alan referred to this as ludicrously complex. As they say in Jamaica: Yeah man!

  5. Classic to the Core… along with some Angelic Intelligence (AI) threaded throughout…

  6. Thank you Michael. As usual, well researched, thoughtful, questioning, thorough, reasoned, eloquent. Definitely not square 🙂

  7. Great, as always.
    A big hug to my old friend.

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