George and Iris Owen 55 Charles St. West, #1201 Toronto, Ontario, Canada M5S 2W9
THE MYSTERIOUS CROP CIRCLES.
A New Horizons Note.
Copyright: New Horizons Research Foundation.
December 1991. INTRODUCTION.
About ten years ago a fascinating, strange, and unusual phenomenon made its appearance. It seemed to have little relationship to any other known phenomena, and it didn't seem to belong under any category. However, a number of claims have been made over the years which attempt to categorise the phenomena as belonging to some one particular belief or category. We refer to the phenomenon known as "crop circles". The discipline concerning the study of these circles is known as "cerealogy".
This paper is an attempt to draw together some of the information about crop circles, and arrive at an understanding of what is known, or believed, about them, today.
The phenomenon has become widely known during the last ten years, and in fact, in its present form, it has apparently only been occurring during the last ten years. It takes the form of large patterns of perfectly bent corn being formed in fields of growing corn. Initially, these were large beautifully formed circles, hence the name; but, more recently, the patterns formed have changed, increasing in complexity from year to year; the patterns that appeared last year, for instance, were not only extremely complex, but they were indeed beautiful. Most of these patterns in the cornfields have appeared in a certain part of England, namely the south western area, in the vicinity of Stonehenge, and near the other various ancient monuments of that region. However, they have also been reported in other parts of England, in the Midlands and eastern regions, and in other countries, notably Canada, Australia, Germany, New Zealand, France, The United States of America, and one has been reported in Sweden. Although the phenomenon has caused so much excitement in recent years, stories of simple circles appearing in fields of growing corn go back many centuries. There are legends dating from the Middle Ages that talk of circles being formed in the fields overnight, and these were attributed to fairies dancing through the corn, or it was said mowing devils came in the night and cut the corn in'rings.
One of us (IMO) grew up on an arable farm in eastern England and remembers such simple circles being found in the cornfields in the 1920's and 1930s when harvest time came around. Local farmers attributed them to weather disturbances, the general theory being that they were caused by miniature whirlwinds. This was before there were many aircraft in the sky, and as the circles were usually situated right in the middle of the cornfields, they were not usually discovered until the machines cutting the corn reached them. They were invisible from the sides of the fields, and so it was impossible to know exactly when they were formed. It goes without saying that nobody walks through a field of growing corn, it is so easy to damage it. In these early circles it was sometimes obvious that the circle had been formed earlier in the growing season, as the com was lying flat, but the ears had continued to grow, and were putting out green shoots.
The present rash of phenomena started with just such circles, a simple round, some twenty or more feet in diameter, with the corn laying down, all in one direction. It was laid perfectly smoothly, as if it had been swept downwards by some strange force. Typically the roots were not damaged, and the stalks were not broken, only bent. The corn continued to grow in this position. The circles usually appeared overnight, and were quickly noticed, not only by pilots of aircraft overflying the fields, but also because in many cases they appeared in fields overlooked by hilly country.
However, this simple pattern was not to remain for long. During succeeding years the patterns grew more and more elaborate, first taking the form of double and triple rings around the original circle, and then acquiring further extensions that in many cases looked like hieroglyphics, or some primitive attempt at communication. Recent circles were also considerably larger, some being as many as hundreds of feet in diameter, and occupying in their design some thousands of feet of cornfield. For many farmers these formations proved more than a minor problem. Not only were large tracts of growing corn badly damaged, and in some cases ruined, but the rest of the farmer's crops were usually trampled by the rush of the sightseers and investigators who came in the wake of such sightings. Many farmers had their whole crops destroyed. If, as has been suggested, some of these circles are caused by hoaxers and pranksters, it is a very irresponsible course of action indeed, and one would think that the farmers who suffer would have a claim for compensation.
THE PHENOMENA.
This current rash of phenomena started in 1980, when a large number of circles were found in the south western part of England. They appeared to be cut with almost surgical precision in fields of growing wheat and barley. The earlier circles were always reported as being in cornfields, but of recent years the patterns have been found in fields of rape, among growing root crops, such as sugar beet, in hayfmelds, in sugar cane in Australia, and there are even a few reports of the formations occurring on the sides of dusty roads, and one is reported as happening in a field of prickly thistles.
An interesting feature of many of these patterns is that even after the crop has been harvested, the place where the circle has been is still obvious, as if the ground surface were in some way changed. It seems, in these cases, as if something more than just the bending of the corn took place. However, all traces of the patterns disappears when winter comes and the soil is prepared for next year's crop. Also, there are no reports, as far as we are aware, of a circle, or pattern, reappearing on the same spot the following year, or of the crop being in any way affected the next year. If the soil had been affected by either chemicals, or burning, one would expect subsequent crops to show evidence of this.
As we have said in the Introduction, originally these patterns took the form of simple circles, some small, and others huge in diameter, but during the succeeding years the patterns took on the forms of oblongs, squares, rectangles, and triangles, and other geometrical patterns. Circles would be connected, or there would be a large circle with several small ones grouped around, or a series of rings would surround an inner circle. Later still the patterns became even more complicated, and some took on insect-like forms, others produced squiggly patterns, likened to brain patterns, and others resembled a child's scribbling. We have attached a number of examples to this paper. During the ten years from 1980 - 1990, almost every conceivable kind of pattern emerged.
In all the reported cases the corn is bent in one direction only, it is not trampled haphazardly, it is not crumpled, it lays straight. The stalks, although bent, are not broken, and the corn continues to grow in the ground. Corn continues to grow and ripen when the stalks are bent, if the roots are not pulled out and the stalks not broken. The corn will continue to ripen, and when ripe, because of the contact with the ground,the wheat in the ears will sprout green shoots. The loss to the farmer lies in the fact that he is unable to reap the harvest.
A further word of explanation. In all the published pictures there are straight lines running the length of the fields. These straight lines are furrows made by the wheels of the tractors that sowed the seed in the springtime. The seed does not fall in these furrows, and so these lines are left when the corn grows. Because modern machinery is often very large, and the vehicles have wide wheels these furrows can be wide enough for persons to walk in without damaging the corn. In the days when horses:pulled the seeding machinery these furrows did not exist. If one deviates from the tracks, however, the corn will be damaged. Growing corn is very heavy in the ears, and it is easy to damage it, and the damage is obvious. This is a factor to take into consideration if one is looking at the possibility of these patterns being the result of hoaxes or practical jokes. The patterns usually have occurred during the months of July and August, when the crops are ripening. However, in 1991, the weather in England in the springtime was cold and wet, and so the harvest was later than usual. However, in many cases, the circles started forming at the usual time, appearing in the young, still green, corn. This would cause one to wonder if the time of year was a factor, and not necessarily the ripeness of the corn.
This last year, 1991. saw some very varied and complicated pattern formations indeed. The earlier ones were labelled 'insectograms' by the investigators, as they resembled some weird kind of insect. About a dozen 'insectograms' appeared in the counties of Wiltshire and Hampshire during the early 1991 'season'. As the summer went on further strange shapes and patterns appeared. A report in the Fortean Times giving the types of formations during that summer lists 'dumb-bells', (consisting of two or sometimes three plain or ringed circles joined by a narrow corridor)} haloes were sometimes added to these shapes, or perhaps rectangular 'coffins' or other appendages. Also listed are 'curly* patterns resembling a child's scribblings, a 'wiggly pipe cleaner man', an "Irminsul" (described as the ancient long-suppressed symbol of German paganism), and an elaborate maze-like formation which was aptly named "The Brain". Another type of formation was in the shape of a whale, and some of these had' flippers. One of the most striking of this year's'crop' appeared at Barbary Castle, in a wheatfield just below the ruined castle. It was first spotted by the pilot of a light aircraft flying in the neighbourhood, A copy of the picture of this geometric pattern is found in the Appendix. Before it was damaged by storms and visitors it was surveyed accurately, and an analysis made of the meanings conveyed by its geometry, ahich are said to be startling and revelatory.
We quote from an article entitled "Geometry & Symbolism at Barbefry Castle" which appeared in The Cerealogist (No. 4, Summer 1991).
"The Barbary Castle formation is a regular but previously unknown form of geomatrical diagram We are presented with a novel and/interesting lesson in geometry, but the implications of this figure are not merely academic, for the numerical structure behind its dimensions are familiar to students of the ancient numerical science, and have deep cosmological significance.
There are many mysteries about this formation, including the method and sequence of its creation, and the strange lights and other phenomena which coincided with its appearance. Also enigmatic is the impressive ratchet spiral at the south-east corner. Yet the basic geometrical idea behind the diagram is clear enough. It demonstrates the principle of Three in One by means of a central circle which exactly contains the combined areas of the three circles around it. Moreover, the sum of all the four circular areas in the diagram is 31680 square feet. The significance of this number in arithmetic, cosmology, ancient theology, and temple architecture, was first explored in City of Revelation (1972), and is summed up in a section of The Dimensions of Paradise (Thames and Hudson, 1988). In traditional cosmology, 31680 miles was taken to be the measure around the sub-lunary world, and the early Christian scholars calculated the number 3168 as emblematic of Lord Jesus Christ. The same number was previously applied to the name of a leading principle in the pagan religion."
The writer of the article goes on to say "Neither physically nor intellectually, does this figure give signs of being a human creation. To identify its author seems therefore to be a problem for theology. One's rational mind shrinks away from the implication that this diagram constitutes a divine revelation".
The Barbery Castle figure was spotted on the 17th July. During the second week of August, however, an even more startling geometric figure appeared in a cornfield in Cambridgeshire, at a village called Ickleton, and some considerable distance away from Barbary Castle. According to a report in Fortean Times, "this figure shocked the minds of every mathematician who saw it. George Wingfield, an initiate of fractal geometry and chaos theory, flew over it, and recognised the perfect, unmistakeable form of the Mandelbrot Set, the bug-like figure from the world beyond the surface of apparent forms, which Dr. Mandelbrot discovered about 15 years ago. Its image in the wheatfield was beautifully executed, the wheatstalks "being laid in multiple swirled layers to create the characteristic plaited effect which no human imitators of crop circles have been able to reproduce". (A note on the Mandelbrot Set is found in the Appendix).
One accompaniment to the phenomena we have not as yet mentioned is that of lights and sound. While the majority of the patterns appear during the nighttime, and apparently without any sign of their happening, in some cases people have reported hearing noises, such as bangs, and seeing lights of various kinds. In many of these instances people nearby have often jumped to the conclusion that visiting UFO's have caused the patterns to occur. There have, of course, also been a few reports of close encounters with aliens, and . strange people, allegedly from outer space. These same people believe that the patterns are some form of attempted communication with people on earth. Nevertheless, in most of the cases the phenomena have appeared overnight, and completely silently. In some situations investigators have actually been camped out overnight in the fields, watching hopefully for the circles to appear, only to discover, to their surprise and chagrin when daylight arrives, that the circles and patterns have formed during the night in the next field, or just behind them, and they have seen or heard nothing.
Although we have said earlier in this article that the circle phenomenon has mainly developed during the last ten years or so, there were many reports of circle formation during the 1970's, and these came from many other countries. Crop circles were reported in Langenberg, Sask, Canada, for instance, in 1974, and they continue to occur in both Saskatchewan and Alberta. They were usually at that time attributed to UFO's and were frequently designated as 'landing rings'. Their physical appearance was exactly like the patterns of recent times - the corn was bent, not broken, and the stalks all lay perfectly swirled in one direction.
They also occurred in Australia, where they were commonly referred to as "UFO nests". They appeared there from the beginning of the 1970's, These "UFO nests" not only appeared in growing corn, but they appeared in other crops, such as hayfields, and fields of sugar cane. In one particular instance the 'nest' was found deep within a field of prickly saffron thistles. It is worth quoting the account given by the investigator from the local UFO society who visited the 'nest'.
"It was a day of dry heat, of bush fires, there was a sense of nothing living in the pale yellow dried grasses, and the endless asphalt road shifted and shimmered with mirages of the Mythical Inland Sea.
At first sight the UFO nest seemed from a distance to have been scorched into the field by the gigantic red sun hanging low in a heat-erazed sky. The circle could not have been placed in a more inhospitable environment, gouged as it was deep within a field of prickly saffron thistles. Fortunately for our legs, but rather inconveniently for the recording of the circle for posterity,Mr. Viv Huckel, the proud recipient of this nest had ploughed up to the circle at the request of the ABC (The Australian Broadcasting Commission) the day before. And for a very down-to-earth Aussie farmer he had plenty to say on the subject.
What hit me when I first saw it was that it was very similar to nests found in the sugar cane beds up in Queensland" said Viv; "I drove around, finished ploughing, and came back and had another squizz at it. What intrigued me was that the centre portion (about four feet across) was almost completely bare, and after that you started getting little stumps of saffron thistles - they'd been shredded. The further you got to the perimeter, pieces of stalk were just broken up; but the last couple of feet of thistles - and these are two feet high minimum - were completely knocked down in an anticlockwise direction, twisted up, and some had been completely torn out by the; roots'. It's pretty hard to pull a green saffron thistle by the roots,mate."
The farmer went on to say that the damage could not have been caused by cattle. He had had only 5 head of cows in that paddock during the previous six months, and he stated that neither cows nor sheep would 'bull' their way into saffron thistles. Neither was it a 'hares playground' he said. He also discounted the idea that it could have "been a whirlwind - "It doesn't stay in the one spot - it sweeps straight through a paddock and leaves a dread straight scar of knocked down thistles, say, all heading in one direction. Anyhow, it hasn't been hot enough for a whirlwind".
This is a very interesting account. The thistles evidently were more broken than corn is usually under such circumstances, but thistle stalks are stiffer, and one would expect them to break, wheras corn will bend. However, we have here, as in all the other circle phenomena the crop all swept down in the one direction, and not trampled around as one might expect were cattle responsible,
This site was two hundred miles west of Sidney, in the countryside. There can be no doubt that hoaxers were not at work in this remote corcner of Australia, in a field of thistles.
The above descriptions will, we hope, give the reader an idea of what has been happening with regard to the 'crop circle phenomenon' during recent years. At the last reckoning, researchers who have been studying these strange patterns estimate there have been over two thousand sitings reported; most of these have occurred in England, mainly in the south west corcner, but there have also been reports from East Anglia and the Midlands. The patterns have also been reported from America, Canada, Germany, Australia, New Zealand and France. There are many Japanese researchers interested in cerealogy, although we have not actually seen reports of the phenomena ... occuring in that country. One would presume that it may be that rice crops, which are the main staple in eastern countries,being shorter, and lighter, might not respond in the same way as the heavier crops of the west to whatever it is that is influencing these plants. However, given the weight of evidence we think it may quite properly be described as a world• wide phenomenon.
Dozens of magazine articles have been written, many radio and television programmes aired, a number of very beautiful books have been published, and many organisations have been set up to undertake research. These latter publish their own journals, and the subject of the research has acquired the title of "Cerealogy". Study of the phenomenon has become well established.
POSSIBLE CAUSES.
When the first circles appeared they were very simple, as we have said. They were generally just round areas of corn that had apparently been flattened overnight, and they usually occurred on clear calm nights. In other words unusual weather patterns, wind, rain, or storms did not appear to be a factor. However most people felt there was probably a simple explanation. This type of cirle had been known for many years, as we have mentioned there were references to these circles in the Middle Ages. Many farmers attributed the formations to small whirlwinds, similar to the 'dust dervishes' one can sometimes see forming on dusty roadsides. But, as we have just said, they often appeared in conditions, and in places, where such a wind was unlikely, There was speculation that at the sites in western England, where there are many hills, perhaps there were some unusual wind currents, but they appear also in the flat areas of East Anglia, where there are no hills. They are not 'fairy rings'. Such rings are caused by the outward growth of a fungus in the earth, and their mechanism is well understood. They also appear in the same place year after year, and the circle patterns do not usually recur in the same spot.
The simple explanation seemed less likely as the numbers of such patterns increased. During the 1970's there was a great deal of interest in UFOs, and the possibility that earth was being visited by aliens from other universes or stellar systems. Along with the descriptions of sightings of UFO's there were many reports of what came to be called 'landing rings'. UFO researchers believed these so called 'rings' denoted a place where an alien ship had landed, usually during the nightime, and then had taken off again at daylight. During the 1970's there were many reports of such phenomena. These 'landing rings' had many of the characteristics of the circles that appeared in the early 1980's, and so it was inevitable that sc-me researchers believe that all the current patterns are caused by visitors from another world. It is true that in some cases, lights and noises were reported as having occurred during the night when the patterns were formed, just as is reported in alleged UFO landings cases. However, in UFO landing reports it is often said that the earth itself is actually scorched, or burnt, and this has not been so in recent patterns. However, one common feature is the fact that the corn is bent, not broken, and is laid symetrically in one direction. Many UFO investigators believe that the intricate patterns of the more recent phenomena are some attempt at communication on the part of the alien visitors.
The whirlwind theory collapsed when the more elaborate formations started to happen. Whirlwinds certainly do not create the kind of patterns that were appearing.
Some scientists, however, are still looking for a 'natural' cause, and have put forward the theory that the hole that is presently in the earth's ozone layer is responsible for creating these patterns. They claim that the unprecedented atmospheric conditions caused by this hole have led to violent disturbances in the earth's magnetic field, and it has been claimed that these disturbances have been detected in the areas where the phenomena have appeared.
Other scientists have claimed that a mysterious, and as yet not understood, vortex system is responsible for the phenomena. Scientists from all over the world appear to have put forward their pet theories as to the cause of the figures, but none have succeeded in providing a convincing explanation.
It has been suggested, for instance, that the eleven year cycle of sunspot activity has some bearing on the formation of circles. Again, it was suggested that the frequency of power breakdowns in the vicinity of some of the phenomena has caused 'ionisation' of the crops, due to increased ionisation of the atmosphere under these conditions. In some mysterious way, it is alleged, this 'ionisation' causes the corn to bend over in these strange patterns.
As the patterns continued to become more complex, and especially after the formation of the geometric and hieroglyphic patterns, many people have begun to believe that they constitute messages. These messages could be coming from people on other planets or stars, or they could be 'religious' messages, as has been suggested in regard to the Barbary Castle figure of 1991. A comparison has been drawn between some of the figures and the famous lines drawn on the plains of Nazca in South America. These Nazca lines have baffled experts and investigators for many years. They cover large areas of countryside, and consist of long straight lines, drawn in special configurations. They can only be properly seen and appreciated from the air.
In a more lighthearted vein, because the figures are so pleasing, and set, usually in beautiful country, it has been suggested that they are meant as pictures, or art forms, created by these beings from outer space. They are using the background of earth's countryside as a canvas to paint pictures. Certainly many of the books full of pictures of these patterns make lovely coffee table books.
The facts of the matter are that nobody up until the present time has been able to come up with a reasonable explanation as to how these patterns are formed, or the reason for their existence.
A HOAX?
To many people the obvious answer as to their formation is that they are all a gigantic hoax. This may, indeed, turn out to be the case.
However, in fairness, one should put the points against the possibility of the whole affair being a hoax.
In the first place, the simple circles, which started up the cycle of patterns, have been in existence for many years. These early circles were almost certainly not hoaxes, and even then there was no satisfactory explanation for their formation. As we have said, it was assumed that they were caused by whirlwinds, but then, as now, there was no evidence for the presence of whirlwinds at the times and places they occurred. However, recently it has "been claimed by two landscape artists in England that they had been responsible for creating the various patterns and circles. They claimed that they went out at night, on many occasions, and personally created the patterns. Many people believed their description of what they had done, and assumed that the matter was closed. But one is left to wonder. In the first place, even in daylight, it would not be easy to create these patterns in the middle of fields of growing corn without leaving evidence of having been there. It is easy to see where people have trodden in fields of ripe corn, a stalk once bent cannot be stood up again, and even animals footprints are usually very easily seen if they have entered a field. To perform all these complicated manoevres and create all these complex patterns in the dead of night, and in utter silence, assumes a real dedication to a practical joke. But to carry out this hoax so many many times, in such very different areas (even if one were only talking about England), and over such a long period of time, assumes a tremendous dedication to a hoax - and for what purpose?
One might assume that many many people were involved in the hoax. That hundreds of people were creeping about the fields each night, drawing circles and patterns in the growing corn. But why would they do this? Until the two English artists 'confessed' nobody else claimed to have been responsible for the patterns. Over a period of more than ten years, one would have expected other people to have owned up, or one might expect their friends who were aware of what they were doing would have 'told' on them. One is certainly left to wonder why such pranksters were never discovered, especially considering that on many occasions circles and patterns appeared overnight in fields right next to where researchers and investigators were camped, in the hope of being present at the time these patterns were created. And what about all the phenomena that happened in other countries; were these also the result of practical jokes?
If the whole thing is really a vast practical joke, it has been very irresponsible behaviour on the part of the pranksters. Many farmers have suffered severe financial loss, due to having their crops destroyed, and one would imagine that these farmers might have a claim for compensation. Not only have the farmer's crops been damaged by the formations, but in most cases the rest of the crop has been trampled and ruined by the ho gardes of sightseers, and investigators.
Nevertheless, in spite of what we have just said about hoaxes, there can be no doubt that there have been some hoaxes. Investigators claim that they have usually been able to detect these hoaxes, and they are aware of them. That's as may be. In every type of investigation of this kind, and as we are especially aware when studying parapsychology, the hoaxer will appear. The effect, generally, is to 'muddy the water' and make it difficult to arrive at a proper scientific conclusion, and that is certainly true in this case.
It has been remarked that "if the whole phenomenon should turn out to be a hoax, it could tell us something about the state of society at a time when science and technology have achieved their present sophistication. Skeptics will claim that it proves public gullibility and fallibility, but the 'fall-out- from the destruction of trust could well blanket the scientific community as well. Trust, on which civilised society and behaviour are founded cannot survive such battering, whether in science, political and economic life, or in human relationships." CONCLUSIONS.
To sum up, the phenomenon of the formation of crop circles and patterns is a very interesting one indeed. It is entirely new in our experience.
We do not at this time, have an explanation as to how the patterns are formed, although it is accepted that a few, at least, are the result of hoaxes. However, on balance, it must be conceded that it is unlikely that the greater number are hoaxes} there has to be some other explanation.
It is a very widespread phenomenon, occurring in many areas of the world.
Described as 'the greatest of modern mysteries,' the study of the phenomena has been given the title of 'cerealogy' and this study has spawned a host of journals, many newspaper and magazine articles, as well as numerous books. Many investigators currently term themselves 'cerealogists'. International conferences have been held, as well as many regional conferences and symposia.
It may well rank with 'ufology' as one of the consuming areas of interest and speculation of our modern times.
Is it, perhaps, a modern urban legend? If it were not for the fact that the evidence is so visible and tangible, one might be tempted to regard cerealogy as such. But the evidence is there for all to see't We just do not know, at present, who or what, put it there. REFERENCES.
John Michell "This Year's Crop", Fortean Times, No. 59. Sept. 1991. p. 28-29
Bob Skinner "The Crop Circle Phenomenon", Fortean Times. No. 53 • Winter 1989-90 p. 32-37 Diane Kearns "An Australian UFO Nest". The Cerealogist, No.4. Summer 1991. p. 13.
ACKNOWLEDGEMENT.
The writers would like to express their gratitude to Mr. Barry Withers for the loan of extremely useful current material on the crop circle formations* and also for many stimulating and informative discussions. APPENDIX* Notes on some of the formations.
As has been said, recent years have seen an increase not only in the number of formations, but in their size and variety of patterns. The original geometrical simplicity when the patterns were restricted t© circles and straight lines has been lost. We may, like the poet in the Prelude to Goethe's Faust feel uneasy that "unrelated things that know no blending parade before us". In the beginning were simple circles, then annuli and concentric rings, then patterns composed of both circles and straight lines, then circles with appendages such as arrowheads and key shaped extensions. (Figures 7,2,3)• Then some formations appeared which seemed to be stylized reproductions of ancient pagan symbols — sungods, and also the ubiquitous "Great Goddess" of whom so much has been written in recent decades.
Worse yet was a somewhat repulsive "squiggle" which some observers nicknamed "The Brain", and an irritatingly "arch" formation dubbed "Curly". Next, biological elements intruded themselves — the "Whale", also a swarm of "insectograms" forms recognizable as depictions of insects. Some of these, like the Whale and Curly convey an impression of humour and a degree of "cockiness" and "cheekiness"j certainly they could be described as "jaunty". Very recently two antiquarian themes, in addition to the Sun God and the Goddess symbols have been represented. ©ne is an imitation of a design on the upland plain at Nazca in Peru. The other, relating to a totally different culture, has been recognized as the "Irminsul" copied from a monument to Arminus (Hermann) a German who defeated a Roman Army in 9 A.D. Lastly, or almost lastly, the huge and complex formation at Barbury Castle in Wiltshire is of considerable interest. This is not only because it represents a return, at perhaps a higher level, to the austere severity of pure geometry, but also because at least one contemporary scholar, John Michell, discerns an apocalyptic element in it. Finally, there came the "Mandelbrot Set".
No impartial commentator can fail to be somewhat disturbed at the apparent eclectic nature of the totality td the present day of the "crop circles". If ascribaBle to a single intelligence (terrestrial, extra-terrestrial, extra-dimensional, extra-temporal), the mind in question would appear to be at best diletante, like Shapkespeare's Autolycus — "a snapper up of unconsidered trifles", and at worst a pedant. The apparent diletanteism suggests, of course, the work of human hoaxers, some of whom may have a pedantic streak as well as being funsters. But as against taking the easy way out we should bear in mind the extraordinary increase in the incidence of formations, not only in Wessex but in eastern England also. The numbers are such that the affair could be said to have got beyond a joke! (Also, as mentioned previously, they have been found in many other parts of the world, including remote areas of Australia). We think it worthwhile to append some notes giving background on four of the formations — the "Irminsul", the Barbury Castle pattern, the vesica pisces, and the Mandelbrot Set at Ickleton.
The "Irminsul".
In 9 A.D. a Roman enterprise came to a sudden and catastrophic end. Augustus, the first Roman Emperor, had sanctioned the establishment of a new Province — Germania — across the Rhine. It was administered by the military governor — Varus — who was rather heavy- handed. H:ermann(= Arminius) was a German high-up in the Roman army in the Province. This is not at all surprising — the majority of the Roman legions consisted of barbarians - Goths, Germans, etc. Hermann's idiosyncrasy^was that he became a rebel — a German patriot, and raised a revolt against Varus. Rather cleverly he tempted Varus into an ambush in the Teutoburger Forest in which three legions were killed. The Teutonic nation principally involved in the victory were known to the Romans as the Cheruscans. Emperor Augustus, who was not used to failure, took the reverse very badly; all thought of Roman expansion beyond the Rhine was abandoned.
After some centuries (almost millenia) Hermann was instituted as a national German hero. Early in the,, nineteenth century some writers of the so-called "Sturm and Drang" movement (i.e Storm and Stress) stirred up both by the romantic movement of the eighteenth century, which idealized earlier periods in human history as times when men were nobler because closer to nature, and by the Napoleonic wars which evoked a previously hardly existent German national sentiment, rediscovered Hermann%
as a Teutonic hero. Prominent among these neo-Teutonic patriots were Johann H. Voss, who celebrated in poems the nobility of the "Volk Thuiskans" (i.e. the Cheruscans), the companions of Hermann, and also C.F.D. Schubart. He celebrated "freedom" as the force most revered by the ancestral Germans, and embodied in Hermann. He commenced his list of great German heros with Hermann, and concluded with King Frederick William, the founder of the Prussian state. Sometime later in the nineteenth century or early twentieth a memorial in stone was set up to Hermann somewhere near the location of his victory. We regret not to have the particulars, but it may be plausibly surmised that it was at the expense of private donors. The period was very much one for setting up statues and other memorials of patriotic characters, c.f. the Washington column in Washington, D.C., the statue of BoadecSa with chariot and horses at Westminster, and that of King Alfred at Wantage.
The actusl pattern of the Hermann standard (the"Irminsul") was probably the product of some nineteenth century artist's imagination. Amusingly enough, the nineteenth century German romantic patriots when seeking for an explanation why the early Teutons should have declined from their original moral status, blamed it on the corrupting influence of French civilisation*. However they were kind enough to exempt the Anglo-Saxons of England from any culpability! Whether the occurrence of the "Irminsul" in English Wessex has anything to do with this is a question we leave to otherst
The Barbury Castle formation.
The total length of the various sections of the Barbury Castle formation is of the order of a thousand yards, i.e. a kilometer, and if done by unassisted manpower would represent a very considerable physical effort, unless it were the work of a very large team.
The only interpretation of the pattern yet given is that in The Cerealogist (No. k, Summer 1991) by John Michell, the author of The View Over Atlantis (eds. 1969 and 1962), and other books which include City of Revelation, and The Dimensions of Paradise. Michell's interests are fairly eclectic; he ascribes reality to leylines, and a geomagnetic basis to fung-shui, as well as to the general tenets of the subject which in recent years has become known as geomancy. Michell's interpretation GEOMETRY AND SYMBOLISM AT BARBURY CASTLE Analyzing the geometric and numerical structure of this year's masterpiece in wheat, John Michell is stunned by its implications.
BARBURY CASTLE CROP FORMATION (17:7:91) SCALE 1:400 ORDNANCE SURVEY OUT) IH>. SU 152 761 atop TYPE WHEAT Surveyed (20:7SI) and drawn by J. F. Langrish. DIRECTION BEARING DISTANCE DIAGRAM TO SHOW XD 0* 105* 10* DIRECTION OF LAY XA 120" 106- 0* XB 234* 104* 0* AB 2M* 17T 0* BC 24" 91- 0" CD 30* 93' r DE 148* 65" 10* ue or t HA ISO* DQ 0* 38* 0- AY •oa* *S 6" BZ " 234*"'' '." 3ff r " Nouc A.B*D < - t ir i Tim peine of ds* otrai Hl>' aid— by afciag down tbcs) BImd BMIII pcH IBlrt'lf ™^ ttBt dc/Wootfa. .AHmiJuo potrt^ of. th o.a 'ttin3 (ha- nd*'yCAUM BO a £n DthA> and w«n dud ai th* Mm.
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THE BARBURY CASTLE formation is a central circle which exactly contains the histories revelation is said to have been the regular but previously unknown form of combined areas of the three circles around it source of all cultures and to have inspired geometrical diagram. John Langrish's survey Moreover, the sum of all the four circular successive renewals of the human spirit It shows that whoever made it lapsed from areas in the diagram is 31680 square feet The occurs, presumably, at times when it is most perfect symmetry, for the central feature of a significance of this number, in arithmetic, needed, and its content is always the same, circle and rings has slipped too far to the cosmology, ancient theology and temple being that cosmic Law, Canon or compilation north-west and, possibly in connection with architecture, was firstexplore d in City of of numerical, musical and geometric this, the triangle has become distorted. The Revelation (1972) and is summed up in a harmonies which provided the ruling dimensions of the various parts are nonethe• section of 77i<." Dimensions of Paradise (Thamestandars d of every ancient civilization. less consistent, and it is possible from the & Hudson, 19S8). In traditional cosmology, data to discover the basic construction of the 31680 miles was taken to be the measure figure. We are presented with a novel and around the sub-lunary world, and the early Construction interesting lesson in geometry, but the Christian scholars calcula ted the number A figure is said to be 'constructed' when it is implications of this figure are not merely 3168 as emblematic of Lord Jesus Christ. The drawn with nothing but a straight-edge and academic for the numerical structure behind same number was previously applied to the compass, the equipment with which, its dimensions are familiar to students of the name of a leading principle in the pagan according to the geometers' myth, the ancient numerical science and have deep religion. Creator designed the world-plan. The basic cosmological significance. There is a world of symbolism in this figure at Barbury Castle can be constructed There are many mysteries about this Barbury Castle wheat impression, some of it by the following stages, beginning with the formation, including the method and already apparent and some still awaiting most elementary geometric operation, the sequence of its creation and the strange lights recognition. Neither physically nor intellectu• division of a circle into six parts by its radius. and other phenomena which coincided with ally does this figure give signs of being a its appearance. Also enigmatic is the human creation. To identify its author seems Figure I impressive ratchet spiral at the south-east therefore to be a problem for theology. One's Draw a circle and, with the compass opening comer. Yet the basic geometrical idea behind rational mind shrinks away from the unchanged, mark off six arcs from the centre the diagram is dear enough. It demonstrates implication that this diagram constitutes a to the circumference. the principle of Three in One by means of a divine revelation. Yet in the traditional Draw a line from the centre of the circle
TWENTYFOUR • THE CEREALOGIST #4 . .c - > iv between two of the points on its to 3". and 3- Is stated ar->a or :hc ^: . and the combined areas of all three circumference and, with the same compass central circle is 31680 sq. ft. The fact that the number opening, divide the line into six lengths, each The average radius or the tnree outer 3 i oSO is of special significance within the length equal to the radius of the circle. circles is found by inspection of the survey to ancient scientific tradition is mentioned Draw a circle equal to the first with its be about 41ft. The radius of the central circle above. centre on the last of the six division points of is therefore 41 xV3 = 71ft., and this agrees the line. with the average of 71ft. shown on the survey Figure III Form an equilateral triangle on the line to be the radius of the larger circle. Note that Draw a second triangle with its angles at the between the centres of the two circles and 71/41 is a good approximation toV3. The points where lines from the centre meet the ciraw a third equal circle centred upon the calculated area of the larger circle is 15S40 circumferences of the three outer circles. This triangle's apex. corresponds to the triangle which appears in A comparison of the calculated with the the crop formation. Within that triangle inscribe a circle. Reduce the compass opening Figure II average surveyed lengths of the radii in the Find the centre of the triangle by drawing by half and draw the central circle. 3arbury Castle figure lines from the centre of each circle to the mid• Erase the original triangle, drawn in Fig.I, point of each opposite side of the triangle. together with other unneeded construction From the triangle's centre draw the circle Survev Calculated lines. Apart from the rims of circles, the contained within the triangle. Outlying circle (N) 41 41 widths of rings and lines and the outlying The area of the circle within the triangle is Outlying cirde (SW) 40.7 41 ratchet spiral, the Barbury Castle formation is exactly equal to the sum of the areas of the three Central outer ring 71 71 thus essentially constructed. >c.'es. Takins the radius of each of the Central inner nr.-.; 49.3 50.4 ; -raller circles to measure • unit makes 2-1-5 25.2 With thanks to John Langrish for his survey and to ibmed areas of the three circles equal ..'.•..-..i::.-v.- :::-,--^,: > d.~"/. :ng •:• diagrams.
i«bct4tp geometry at Barbury Castle. Wiltshire of the Barbury Castle wheat formation is however based on a theme that he developed and applied in The View Over Atlantis, and which the present writers find it convenient to entitle as numerological cosmography.
Numerological cosmography, of course, incorporates many of the methods of ordinary common or garden numerology. Many people for instance are familiar with the use of birthdates to obtain what is, in effect, the most inexpensive, though correspondingly limited, form of horoscope. If a person is born on 4 July 1919 = V7/1919 or 04/07/1919, then addition of all the digits in either the second or third of these representations yields the sum 4+7+1+9+1+9 = 31. The sum 31 is now itself subjected to the same process of reduction (or, as some would say cabalistic reduction), it is replaced by 3+1 = 4. The "buck" stops here; the number 4 is used to make a broad classification of the peron's character type. The number 4 is interpreted in terms of traditional number mysticism, based on a plenitude of associations -- literary, folkloric and religious (both current and defunct religions being of influence). In our present example 4 is usually associated with "four-square", that is to say the 4-number birthdate persoxi is depicted as somewhat of a "square", i.e. solid and dependable, but with the faults of their virtues — perhaps a little "stodgy".
The numerological cosmography of John Michell also involves an ancient system called gematria, which originated with Rabbinical commentators at approximately the dawn of the Christian era. The starting point was that the origin of all things in the created universe was the Word of Sod ( a principle echoed in the opening of St. John's Gospel — "In the Beginning was the Word"). In geomatria the rabbis attached to each letter of the Hebrew alphabet a number. As Greek was, in reality, the spoken and even literary language of most Jews in the Roman Empire, the system was transferred to the Greek alphabet (see Table) with multiple applications. Thus, if one was named George, i.e. georg in Greek, one's number was 3+5+70+100+3 + 181 = 1+8+1 = 10*1+0 = 1, which in traditional associations was considered a rather good number, being equated with God, the First Principle, Wholeness, creativeness, etc.
Although no human could aspire to divine virtue, none- the-less a man called George, bom July 4,1919, could feel pretty good about it! (c.f. The Secrets of Numbers, Lionel Stebbing). Gematria became a prime element in the cabala, a system of Jewish mysticism developed during the Christian era independently of Christian thought and theology. Cabala was not however purely gematria- it included many other ideas and images. During the Renaissance some scholars became aware of cabala and developed it in Christian terms.
Numerological cosmography had, it seems origins independent of gematria or cabala. We can only guess, but we can postulate with fair certainty that a powerful contributory stream of influence came from Freemasonry in its initial phase in the late seventeenth and early eighteenth century. Not being Freemasons, the present writers have not the advantage of such esoteric historical knowledge as may be possessed by enrolled Freemasons, and therefore have to take aim in the dark, as it were. The British founders of Freemasonry comprised such distinguished persons as the antiquary Elias Ashmele, after whom the museum at Oxford is namel, the architect, Sir Christopher Wren, and possibly Sir Isaac Newton, the great mathematician and physicist, together with persons actually connected with the trade of stone masons, particuarly the masons' lodge at Kilwinning in Scotland. It seems that from the outset the founders entertained ajtradition that Solomon's Temple in ancient Jerusalem was designed by great adepts or illuminati, possessed of a profound knowledge of the universe and of the divine plan on which it was constructed. Sir Isaac Newton, who was addicted to the study both of chronology and prophecy, worked out the measurememts for a model of Solomon's Temple which was exhibited in London some years before the Masonic movement was launched.
It was somewhat later that various antiquarians became interested in the numerical aspects of two other ancient but very prestigious monements. These werejStonehenge, Cwhich had been of general, though not numerological interest, since the time of John Aubrey in Restoration England): and the Great Pyramid of Egypt. In 18#7, Professor Piazzi Smyth, Astronomer Royal of Scotland, published a book, Our Inheritance in the Great Pyramid, in which he announced the discovery of several remarkable coincidences} e.g. the Pyramid's height was a thousand millionth part of the mean distance of the earth from the sun. Also the perimeter of the base was equal to the circumference of a circle whose radius equalled the V 6 Z £ XU I K L *
5 e f n # i K X u v 4 5 7 8 9 10 20 30 40 50 ? n S T V W-f^'V <> 80 100 200 300 400 500 600 700 800
Greek name Value
ek,uia, 'ev 215/51/55 Svo 474 rpeiR 615 906 nevre 440 65 "e« 'eirra 386 'OKTUi 1190 'evvea 111 Sexa 30
5 Ft X "K.
T::c -O shape sas discovered from the air by Ron Vtesi in 1990 and -is mvisii$sied by him and les Stacey. Its site was at Fordham Place, near CJicki$:er. Essex. The two circles, xuirled clockwise ^om slightly eccr~.:r.c wots, n-ere identical in size, measuring 10'6" across.
The oZo shape has had great appeal to waggish interpreters. According to Ralph, it has been taken as an advertisement for a new washing powder (OZO), while some have seen it as a reference to the defective OZOne layer. Far more interesting is the result which comes from analyzing the formation through its inherent geometry.
The plan of the oZo shape, constructed from Ron Wesf $ survey and photographs, is shown here in-black, and next to it is the completed figurewhic h naturally develops from it. It is plainly seen tha t the crop formation indicates a geometrically coherent religious symbol, only a part of which is visible on the ground. This raises the possibility that other crop circle patterns represented more elaborate, but uncompleted geometrical designs. In that case, they can be compared to masons' marks, traditionally carved by masons onto the stones they cut. A study of masons' marks shows that they too "epresen: she cores of more elaborate geometries. The figure that emerges from theoZo form is the nimbus of glory '•••'hich surrounds divinities and, in Christian art, is characteristic or :he \ -.rain Marv. In symbolic geometry it is known as the Vessel, symbolizing the female source of generation and also being associ• ated with the Grail. Its function in geometry parallels its symbolism, for it is the first figure by which are generated all the regular geomet• ric shapes. Its construction, by means of two equal circles passing through each other's centres, demonstrates the fruitful union of opposites. The construction lines (broken) in this figure form the square and the equilateral diamond or rhombus. They define the dimensions o: the two circles, and they also mark the limits of the swirled areas in the Z shape. No more elegant or economical way could be found of convevir.j unmistakably this most significant of aU symbols. This sureiv is rre mark of Ceres herself. Pyramid's height. If b is the length of one side of the base, and h is the height, then the last statement is equivalent to saying that *fb = 2 (Pi) h» i.e. that 2b/h = Pi, where Pi is the well known universal constant given numerically (to eight significant figures) by Pi = 3.1-+15927. The labours of Smyth and later writers have revealed further numerical coincidences, which have encouraged the development of numerological cosmography as a body of doctrine. In the present century the subject received a further stimulus from the study of church architecture, particularly the excavations of Glastonbury Abbey by Bligh Bond and his numerological interpretations of its original inferred dimensions. Besides publishing many reports and a book on the Glastonbury ruins,Bond, together with a Dr. T.S. Lea, authored two books, one on the cabala and one on gematria. His work has been an obvious source of inspiration to John Michell.
Whet/her or not they proceeded according!to the recognized methods of numerology, very many Christian exegetists have theorized over the centuries about the numbers in the Book of Revelation, whose authorship is attributed traditionally to the Apostle John, who is supposed also to have written the Gospel of St. John, and two Epistles. John was traditionally believed to have composed the Revelation while a prisoner working in the stone quarries on the island of Patmos during the persecution of Christians under the Emperor Nero (5^-68).
Revelation has some elements in common with the two other apocalypses in the Bible. With Ezekiel, it shares the "four living creatures" and also the "man with the reed", i.e. angel with the measuring rod. (Indeed, with admittedly a sense of stretching the parallel, the Barbury Castle formation might be regarded as including a reference to Ezekielfc the prophet saw the likeness^of a wheel within a wheel". Biit it might be unwise to pursue this particular hare.) With the vision of Daniel, Revelation shares the horned beasts. Of the numbers in Revelation, some of them, like 7, result from the actual historical fact of there being then the Seven Churches in &sia Minor — Ephesus, Sardis, etc. The seven angels, with their lamps, etc. were the guardian spirits appointed by God for those churches. The four living creatures are benign, and later were identified with the four evangelists and their gospels. But there are also four beasts which are nasty, was well as the Four Horsemen and two dragons. Twenty four virtuous elders sit on their thrones. A number of entitites occur in twelves, but ten is only represented by the beast with seven heads and ten horns. However the interesting numbers are the big ones. The best known is 666 — the 'number of the beast' with which much has been done by almost that number of commentators over the centuries. The 1260 days of prophecy are of course equivalent to the 42 months of 30 days each that "the holy city shall they tread under foot". The largest number is 144,000, being the "ransomed as the firstfruits of humanity for God and the Lamb". Without having read Sf. Thomas Acquinas on'the fewness of the saved1,* it is none-the-less hard to suppose that the reasoning by which he arrives at exactly that number was not influenced by its occurrence in Revelation.
The number however which particularly interests us in the present context is the measure of the New Jerusalem, the Golden City, which St. John in his vision sees "coming down out of Heaven from God". The angel with the gold measuring rod found the side of the City to be 12,000 furlongs; * its length and its breadth, and its height being equalV That is to say, it was a cube, with more than adequate accommodation even for a population of 144,000. Now, in his books, thfe City of Revelation and The Dimensions of Paradise, Michell arrives at the number 31680 as being both of cosmographic significance and religious import. Unfortunately neither book is to hand. We can reasonably accept however, that 3168 is, by gematria, applied to perhaps Jesus Christ Our Lord in Greek, or some similar title, emblematic of Jesus. &s will be seen by reference to his article "Geometry and Symbolism at Barbury Castle" Michell says that, "In traditional cosmology 31680 miles was taken to be the measure around the sublunary world". This statement is at first extremely puzzling. In medieval astronomy — that is the Ptolemaic system — the sublunary world was that part of the universe below the moon, i.e. the region enclosed by the sphere of the moon, Thus "the measure around the sublunary world" would be, in any ordinary linguistic usage, equal to 2(Pi)(distance of the moon from the centre of the earth). But if this were 31680 miles, then the distance of the moon from the earth's centre would be only 5042 miles (about) which would be an absurdly small estimate even for medieval man, who knew that the height of the moon was more than 1000 miles. (In the Ptolemaic system, as for Galileo and Newton, and for us, it was 60 earth radii).
Thus John Michell's statement has to be interpreted somewhat differently, indeed, like so much in numerological cosmography, in a special sense. Illumination comes as soon as we note a figure which he quotes for the earth's diameter in The View Over Atlantis,namely 7920 miles, which is tolerably accurate, even in these days of satellite and radar observations, which have produced new figures for the shapes of the earth. To this degree of numerical accuracy 31680 miles is just eight earth radii or four earth diameters. It is therefore the perimeter of a square circumscribed to a circle whose radius is equal to that of the earthl
Our troubles however are not yet over; we have still to reconcile 3l68?with or without an additional zero digit or so, with the dimensions of the New Jerusalem which was. is, or will be 12,000 furlongs on the side. Taking "furlong" at its face value equates it to 10 chains = 220 yards * 660 English feet. In yards therefore the cube which is the New Jerusalem has an edge length of 22 x 12,000 yards. We would like, somehow, to reconcile this figure with 31680 = 22 x 144 x 10. This distance 3168 yards is an extremely British, indeed Anglo-Saxon, measure. Squalling 144 x 22 yards it is "a gross of cricket pitches". However, except in the eyes of an older generation of Englishmen, it can hardly from that point of view be regarded as having sacred connotations, unless we regarded the Barbury Castle formation as having been perpetrated by the souls of great cricketers such as W.G. Grace or Jack Hobbs. However, getting back into Jerusalem the Golden, can we reconcile 3168(0) with 22 x 12,000?
As will be seen from various examples of the type of calculation permissable in numerological cosmography which Michell gives in The View Over Atlantis, it is legitimate to drop zeros, i.e. to alter the unit in terms of which a number is specified. In fact on page 144 (I synchronic!ty?) he says "It now appears that Glastonbury was originally conceived on the model of the New Jerusalem ... a square 12,000 furlongs on every side. If this is reduced to 12 furlongs, the furlong being equal to 660 feet, the area contained ... is 1440 acres exactly" [An acre = 10 square chains = 4840 square yards]. He goes on to develop some numerological reasoning especially appropriate to Glastonbury which need not concern us now. More general substitutions of one unit for another appear to be legitimate in numerology, which seems as a discipline ta.be immune to any accusations of inflexibility or dogmatic regidity. Thus, on page 139, also re Glastonbury, Michell equates a furlong (220 yards - 660 feet = 7920 inches) with 7920 miles, the earth's diameter. Even if the more conservative of us find this a trifle shock ing, we must be reconciled to the notion that in numerological cosmography perhaps it is the case riot that "anything goes" but that a surprising number of procedures are valid! Within this special context it might be legitimate to argue as follows. The floor area of the City in millions of square furlongs is 144 = 144 x 22 yard, furlongs = 3168 yard, furlongs = 3168 special units. This may seem a trick — indeed it is hard not to despise it, but it may be admissible in the present context.
There are of course (as must be the case in a discipline with such labile principles) other rationales which might seem to validate our apocalyptic number. In thousands of furlongs an edge of New Jerusalem measures 12 x 22 = 26.4| because a cube has 12 edges the total length 1 of the edges of the city is 1 * 12 x 264 *= 144 x 22 = 3168. This length 1 can be taken as being in some sense a measure of size. Another approach is to consider the ground plan only- its perimeter p. in kilofeet is p = 4x3x12x22 = 3168 (I) We should say that these last three arguments,which may savour less of ingenuity than of disingenuousness,have been devised*.by^ the: present writers and not, as far as we are aware, by Mr. Michell1 they seem however not to be entirely alien to the spirit of numerological cosmology, whose attitude to numbers resembles that to words of Humpty Dumpty in Alice through the Looking Glsss — there is no doubt as to who is to be master. As regards 1 as an indirect measure of volume we may note that the area of any plane polygon of perimeter P^is, because of the isoperimetric inequality, less than p2/MM). Applying this and the corresponding volume inequality we see that the volume of the City is less than l3/6(Pi) . These bounds, although the best possible,provide however only very poor estimates of size. 32
Even the ahove analysis, tedious though it may he, neglects one additional point of difficulty. The measure of the "Sity of Revelation is given in the King James Authorized Version of the Bible as 12,000 furlongs, as it is in the New English Bible. However, the translations are made-from the original Greek, where the word used is, of course, stadia, the plural of stadion — the original of stadium - the distance which gave its name to the Stadium at Olympia and to all stadiums ever since. To clinch the matter we need not look up the Greek New Testament; the Revised Standard Version says "twelve thousand stadia". Oddly enough it gives a footnote, "About fifteen hundred miles" which would be correct if a stadion were equal to a furlong. However this is not the case. The length of a stadion has been variously estimated; values given range from 19*+.17 to 200.25 yards.
Incidentally, on page 103 of The View Over Atlantis, it is claimed that the stadion is 0.1 of a nautical mile, i.e. about 202.9 yards, which suggests it has been stretched a bit in the interests of numerological cosmography. Be that as it may, it is helpful as indicating that John Michell is aware of the difference between the furlong and the stadion. Clearly therefore he has derived 31680 from the, in fact erroneous, concept of an edge of the W City as being 12,000 English furlongs. Within the general philosophy of numerological cosmography this derivation is, because of the apparently general principle of
— substitution of units, not necessarily invalid.
Even if it is only a coincidence, the fact noted by Michell that the total area of four of the circles in the ^ Barbury Castle formation approximates to 31680 square feet is a remarkable one, because he drew attention to this number before and not after the formation was made. Let
— R/ and R^ be the radii in feet of the central circle (central outer ring) and of the outlying circle (N).
Suppose that R; and Rz^in result of errors of formation and of measurement, are distributed in probability normally m with respective means and m and equal variances, then the least squares (op maximum likelihood)estimate of m is c'ft/(ir3+ T^J/h* • The observed values quoted by John
*" Michell are K, = 71, Ra = 41, and there is no reason not to accept them. In passing we may note that 71/41 = 1.7317073, which is a good approximation to »/T = 1.7320508.
Also,if we try to find values of Rt and Rz, that satisfy 1 the joint thesis that R, - RZV3 and 2 7TT R, = 31680, it turns out that they would he respectively 71.0072 and 40.9960, which are indistinguishable in practice from 71 and 41. A little more analysis is however in order. As 71 and 41 are given to the nearest foot only, each figure conceals an unknown rounding-off error, i.e. R/ lies between 70.5 and ft.5 and R^ between 40,5 and 41.5. We repeat the estimation of m, doing it for the two extreme cases (a) R = 70.5. R = 40.5. and (b) R/ = 71.5.
R2 = 41.5. We obtain (a) m = 40.652, Area = 31,151.10 square feet.(b) m = 41.336, Area = 32,20686 square feet. Each area differs from 31680 by 1.67% only. It would seem that the agreement with the apocalyptic number is good enough to allow that the assumption that it was being aimed at by an intelligence, human or otherwise, is an admissable one. Of course, the hypothesis is not proven thereby.
The Vesica Pisces.
In The Cerealogist (No 3. Spring 1991). John Michell commented briefly on the so-called 0Z0 formation discovered at Fordham Place, near Colchester, Essex, in 1990. As will be seen in Figure £T . a diagram of the formation can be completed to give a figure which is a good approximation to that known traditionally as the vesica pisees. This phrase sounds rather grand, but just means the "fish bladder", in pure reference to its shape. However it has been of •• symbolic importance to Christians from the beginning of the Christian era. The o*s in the 0Z0 formation are not traditional. The essential part of the symbol is the perimeter only? which consists of two circular arcs of equal radius cutting at an angle of 120°. (correspondingly the centre of each arc lies on the pther arc). Another characteristic feature is that the length is exactly equal to its width multiplied by \TS (i.e. by 1.7320508). &s shown in Figure ^ which is taken from The View Over Atlantis the vesica contains two equilateral triangles; also a one-tmr3 size replica of itself can easily be constructed within it by drawing two circles each with radius one-third of the bounding areas. Currently Michell says the vesica or "The Vessel" has been used as a symbol for the Virgin Mary. He also equates it with the oval nimbus around some early representations of the Virgin. As many readers will recognize, the vesica is identical with the outline of the symbol of the United Church of Canada. Here the derivation is probably from part of the Fish Symbol of the early Christians being 3 Vesica m W1WCATE EXISTS ^^u, FORMER EflST WILL the body of the Fish sans tail. In The View Over Atlantis Michell used the vesica as the starting point of a new theory of the Great Pyramid. He surmises that the vertical midsection of the Pyramid has the shape defined by the heavily printed triangle (Figure £> ). One of the present authors has examined this hypothesis in an earlier paper (New Horizons. 1, No. 2, Summer 1973. 102-108). The ratio of the bs^e to the height of the triangle was found to be ( ./3T-—3)/ S3,which is numerologically amusing, as it contains only the digit 3; the four 3's total 12, which by cabalistic reduction is 3 once more. The vesica seems to have achieved some prominence in cosmographical archaeological and architectural theory, as a result of Bligh Bond's excavations and numerological musings at Glastonbury in the early years of this century. Quoting William Kenawell (The Quest at Glastonbury)"Borid thought he had found two rival systems for determining the ground plans of medieval buildings .... They were (1) A system of commensurate squares, (2) A system of equilateral triangles,which, where combined in parallelograms gave a rectangular field or setting" To quote Bond himself (article entitled "The Geometric Cubit.. ) the principle involved was "... one of geometrical perfection, the object being the reproduction of the form of the Rhombus of two equilateral triangles in the greatest degree of accuracy consonant with practical methods of building, and having harmonious scales of measurement .... from very early times a peculiar respect even a sanctity — attaches to those proportions which most clearly accorded with the mathematical principles known to Master Masons". The Rhombus.is, of course, that formed by the two equilateral triangles inscribed to the vesica pisces. Bond saw in effect that a rectangle circumscribed to a vesica would be quite well proportioned. This is shown in Figure 71 , which incidentally shows another interesting feature of the vesica — that the circumscribing rectangle can be constituted from diagonals of a regular hexagon. (En passant, it may be noted that in the rectangle the ratio of length to breadth, /T as in the vesica, differs from that in a rectangle based on the "divine proportion" or "golden segment" — namely Phi = 1.618 ... as well as from the "sacred cuJrt" of Tons Brunes. However the eye may not find it easy to distinguish between these shape_s. (The present writers knew an architect who used -Jz - 1.4142 as an approximation Phi.) 36 In Bond's work there is much numerology including use of 666 = 18 x 37 = 9 x 74, so that lengths such as 74 feet are encountered; In addition7gematria establishes cross connections with the Great Pyramid and Stonehenge. Thus the Greek word for a house or a temple (i.e. the domicile of a deity) is oikos = 370 = the circumference of Stonehenge in the unit MY, i.e. megalithic yards, where 1MY = 2.72 feet, being a unit discovered by Professor Thorn to have been applied to the layout of many prehistoric stone circles in Britain. (See A. Thorn, Megalithic Sites in Britain). Also, as shown in Figure 8 . a vesica pisces can be fitted on to the ground plan of Stonehenge. If we speak of "The Temple" which is o OJKPS in Greek, gematria yields 440 = the side of the Pyramid in cubits, provided the correct choice is made from the variety of cubits in use in antiquity. The name cubit for the unit in question is derived from Latin, cubitus — the lower arm. The length is not only difficult to standardize but even to define, according as to whether the fingers, and, as well, the back of the hand, are included. Thus, while the Hebrew cubit is often quoted as about 19.05 inches = 1 foot 7.05 inches, at least two other cubits were in use in Biblical times. The latest to be adopted was called the "new cubit" and was about 20.6 inches, being the same as the Egyptian cubit, in which the Nilometer, a pole indicating the height of the Nile was marked. However y it seems that a sacred or Royal cubit of 20.88 inches was also used in Egyptian holy contexts. Taking the side (i.e. edge of base) as 755 feet, Michell converts it to 440 Egyptian cubits. That is he takes the cubit as 1.71591 feet = 20.5909 inches, which goes well with the figure quoted above for the "new" or ordinary Egyptian cubit. (See also Sir Charles Warren). m The Mandelbrot Set at Ickleton. The most surprising formation in 1991* and in fact during the whole history of crop circles, occurred at Ickleton, a village in the southeastern corner of Cambridgeshire. At first sight its ..outline (Figure ) resembles that of the cosmopolitan beetle Adalia bipunctata. This insect, which is judged harmless, and indeed beneficial to man because it eats aphids, is generally popular, being neat, clean, and shiny in appearance. Known as a ladybug or ladybeetle in North America, in England it is the "ladybird"} it is the subject of a nursery rhyme and its name has been given to a series of educational books for children. It could be presumed therefore that this formation should be classified as an ' insectogram*. However, unless the motivation for the Ickleton formation is overdetermined and involves a kind of visual pun on the resemblance to the ladybird, the figure is not primarily in the form of Adalia. Instead it represents a good first approximation to a famous mathematical object the so-called "Mandelbrot Set". The Mandelbrot Set is a region of the plane whose boundary is a curve of a special kind known as a "fractal" curve or set. It may be noted that the word set or curve may often (with regard to context) be used interchangeably. Thus a circle is a set of points equidistant from a given point (the centre of the circle). (While a curve can legitimately be called a set, the term "set" is more comprehensive than "curve". Thus the points interior to a circle constitute a set, as do the points within or on the circle, i.e. the interior points plus the boundary). Fractal sets are defined abstractly by mathematical rules, and, of course, cannot be drawn exactly; they can only be approximated to. But in that respect they are no different from better known curves such as circles, which, as pointed out by Plato more than two millenia ago, can only be drawn approximately and never exactly. Fractal sets (or curves) arise in many different ways. The essential feature is that, regarded as a curve, a fractal set differs from classical curves such as circles, ellipses, etc. by totally lacking the property of smoothness. 5 circle is not only continuous it is also smooth. At every point we can define the slope of the curve, i.e. we can draw a tangent. A tangent is a straight line which meets the curve only in the point of contact. It meets the curve in no second point. (There are exceptions to this last remark, but they are of ng> importance in the present context). A curve is said to be "smooth" at any point where a tangent can be drawn. At such a point the function (i.e. the mathematical formula f(x) defining the curved is said to be differentiable; the slope of the tangent there is called the derivative or differential coefficient, and is written f'(x). In most ordinary cases the derivative can be calculated from formulae discovered independently by Sir Isaac Newton and Gottfried William Leibniz in the 17th century. Thus if x and y are the Cartesian coordinates of a general point on a circle of radius 1 unit whose__£entre is at the point with coordinates (0,0), then y = v/?7~-X*) .and the slope of the tangent, i.e. the derivaxive of the function /T^X* J iS. (-xV^-x^ This is illustrated in Figure lO . Here s An important feature of the theory of functions and the curves they define is that smoothness (i.e. differentiability) is logically and factually independent of continuity. Wherever the curve has a sharpv bend in it is a point of continuity, but the curve is not smooth there. & simple example is constituted by f(x) = x if x is positive, f(x) = 0 if x is zero, f(x) = -x if x is negative. The graph is continuous everywhere but not smooth at x = 0, (see Figure It )• (From a mathematical point of view a straight line is also a "curve). It is also easy to construct functions where curves are "unsmooth" at an infinite number of points, such as those whose coordinates are integers, i.e. x = 1.2,3 , or 11,1, "rational numbers" e.g. fractions '(\J/%,xfr, fo>*-fa'/$7) Y^/s, T^'S*.» It is fairly easy to go further still and construct fInjections which are not differentiable at an infinite set of points, the set being "everywhere dense", i.e. there is no interval of x values, however small, which is free of such points. (See Titchmarsh, The Theory of Functions,for the method known as "the condensation of singularities", invented by George Cantor.) It was in the second half of the nineteenth century that pure mathematicians formed the ambition of defining curves that were continuous but not smooth at any of their points. This is the same as defining functions that are nowhere differentiable. The problem was solved "by the great mathematician Karl Weierstrasse (1815-1897) who announced his discovery in a paper read to the Berlin Academy in July 1872. (It was given greater fame however by Paul du Bois-Reymond in a paper in 1875). Weierstrass gave a formula for a function (defined as a special type of Fourier series) which was continuous but non-differentiable for all values of x. Weierstrass left the theory somewhat incomplete. Other eminent mathematicians such as Bromwich and Dini contributed improvements, but a major advance in understanding the Weierstrass function had to wait until Professor Geoffrey Hardy of Trinity College, Cambridge, applied new and more powerful mathematical methods in a paper published in 1916. In 1918 Knopp gave a general method of constructing non-differentiable functions; a further method was given by B.L. van der Waerden in 1930. (A function somewhat similar to that of Weierstrass was described by Darboux in 1875; both had been anticipated by the mathematicians Bolzano and Celleries in defining nowhere differentiable functions, but their results were not published, probably because they did not understand them.)(See Hawkins). To return to the Weierstrass curve, it needs to be said that it is extremely difficult to visualise it. It has been described as consisting of an infinite number of infinitesimal crinkles. What we can say is that between any two points on it there are an infinite number of "wiggles", thus there cannot be any unique line which is a tangent to the curve. (As we.shall see, this infinite crinkling or crenulation is intimately related to the connotations of the word "fractual) . Figure 12. shows an approximation to a section of a Weierstrass curve. Although some parts appear to be smooth this is an illusion. If the curve could be plotted exactly (which is, of course, impossible), and then magnified, the parts that now appear relatively straight would be revealed as just as crinkled as the "jagged" parts do on the present scale. It is characteristic of fractal curves, i.e. of non- differentiable functions, that each part on a smaller scale repeats the same kind of crinkling as is visible in the larger scale presentations. Such curves are said to be approximately self-similar. (Some, indeed, are exactly self-similar, e.g. von Koch's "snowflakes", but these constitute a minority.) Reverting to the basic feature of a Weierstrass curve, the function may be described as one unbounded variation (or fluctuation or oscillation). The "variation" of a function over an interval of x, i.e. for x ranging between the extremes x = x0 and x = x^ may be visualised in the following way. Let A and B be the points on the curve y = f(x) corresponding respectively to x = x0 and x = . LetX/.Xj X«„, be any points on the curve between A and B. Let L be the length of the polygonal arc AX,XZ ... X^. B. Then, if L is less than a constant independent of the mode of division, (i.e. of the positions of X, , Xz, ... X& and the choice of N) the function f(x) is said to be of bounded variation over the interval (AT©,***-) . The facts of interest are that a function of bounded variations is smooth at all but an infinitesimal fraction of points, i,e. it possesses a de fin ed *^lopeu or derivative" f\x), for "almost all" values of x. In addition the curve y = f(x) is "rectifiable"j i.e. any section of it has a defined and finite length. However no length can be defined for the arc of a function of unbounded variation; in other words the length is infinite. By making the dissection AX/ Xz .... X^!E> ' finer and finer the length L can be made to exceed any prescribed value. The curve is not rectifiable, and any piece of it has to be regarded as of infinite length. ("Rectifiable" means that a section of straight line can be found with the same length as any section of the curve). An elementary way of describing a continuous* curve which is not of bounded variation it to say that any part of it, however short, has an infinite number of maxima and minima. (zv^^ TJiis characterization.however, is inadequate because continuous curves can be definedthat oscillate infinitely often in any interval and yet possess a derivative, (See Kopcke, Pereno, Broden,in Hawkins; Broden showed that such functions can be defined via Cantor's method of condensation of singularities). Mathematical truth is elusive and belongs to a field in which intuition — or prejudice —is a bad guide and which over the centuries has been often discredited. Returning however to non-rectifiable curves of infinite length — why are they called "fractal?" This name results from an attempt to characterize them mathematically in terms of a quantity called "Hausdorff- Besicovitch dimension". The two mathematicians commemorated in this phrase are pelix Hausdorff, who at the beginning of this century made great contributions to the thjory of sets, and Professor A.S. Besicovitch of Trinity College, Cambridge, who in the period 1930- 1950 made profound contributions to the theory of fractals. The definition of Hausdorff-Besicovituh dimension, (v/hich for convenience we shall just call Hausdorff or H-dimension) is relativelysophisticated. But, of course, it has some relation to our ordinary or intuitive idea of dimension. In ordinary parlance we would all agree that a piece of straight line is one-dimensional and would be prepared to say the same of an "ordinary" curve such as the arc of a circle. Similarly we would regard the interior of a square or of a circle as two-dimensional. Likewise we would grant that any part of the surface of a sphere is of dimension 2. This intuitive approach to dimensionality of spaces or of sets is in fact perfectly sound. It has been vindicated by a variety of different approaches by mathematicians. However it needs to be stressed that the dimension ascribed by the intuitive or naive approach represents only one kind of dimensionality. The "intuitive" dimension is called the "topological dimension" of the set. It is in fact the case that both ordinary curves like circles and the majority of fractal curves such as "snowflakes" and Weierstrasse curves have all the same topological dimension, namely dimension 1. However this seems to leave the situation incompletely described. Non-rectifiable curves, being of infinite length, are^we feel,"thicker" than simple arcs, and, in a sense, occupy more space than the latter, even, though they have the same topological dimension and contain the same number of points (i.e. in the sense of the Cantorian infinite). Hausdorff's work led to a way of characterizing the "thickness* of these curves, as well as the "thinness" of other anomalous sets such as Cantor's "ternary set", which is of topological dimension 1, and contains the same number of points as any simple line or arc, but is "thinner" than such curves. A preliminary attempt had been made in 1914 by Caratheodory, but the first effective approach was by Hausdorff in 1919. Hausdorff drew on the theory of "measure" developed in the early years of this century by the French mathematician Henri Lebesgue^ who published important papers in 1901 and 1902. -or For Lebesgue the introduction of "measure" as a more sophisticated form of "length" or "area" was essentially a stepping stone to an improved definition of the mathematical process known as integration. "Lebesgue ifl^§»gration" proved to be a principle which increased the unity and power of modern mathematics to an astonishing degree (see Hawkins or Titchmarsh), Integration need not concern us» "but Lebesgue's definition of measure is highly relevant. Consider first a set S# of-points located entirely in 8 section of a straight line. Bach point can be labelled by a coordinate x, its distance from a fixed origin 0. Form a "covering set" J of non-overlapping intervals. (An "interval" is any piece of the line such as the points with x between say, 0.316 and 2.827). A set of J such intervals "covers" the set S; which we wish to "measure" if, and only if, every point of S/ is inside one of the intervals of J. We now imagine J to "shrink" as much as possible, subject only to the condition that Jalways covers S/. The minimum value M of the total length L(J) of J (i.e. the sum of the lengths of the separate intervals of J) is then the "Measure" of the set S. Actually this last statement requires to be qualified in two respects. Firstly* L(J) may not have a minimum value in which case the "infimum" of L(J) is taken to be the measure of Sf, An infimum is however for practical purposes hardly any different from a minimum. The infimum is, as it were, a "floor" beneath which L(J) cannot sink. (A minimum is an infimum which happens to be attained). The second qualification is that S( be 'measurable". Fortunately most sets are! Indeed it is not known for certain whether any non-measurable sets exist; the question constitutes a rather arcane branch of set theory involving the so-called Axioj» of Choice,introduced into mathematics by Zermelo. In the foregoing M is the Lebesgue measure of the set S/ which is of topological dimension 1. If however the S/ were spread over part of a plane and therefore of topological dimension 2, the calculation of its Lesbesgue measure would have to be modified. We imagine such a set — call it S2 covered by a set J consisting of squares instead of intervals. The Lebesgue (planar) measure of Sa is then the infimum of the total area of the squares which J comprises, as we imagine J shrinking while still covering 3 2. • If by this means we calculated the Lebesgue planar measure not of S j, but of S/ the result would be zero. This simple fact gives the clue to a method of defining dimension via calculation of measure. Following Hausdorff, for any Set S we can define (admittedly in a slighly complicated t'c •»» way) a Hausdorff d-dimensional measure which we denote by H(d,3). It can be shown that K(d,S) is^capable of only three possible values, namely, 0, 1 or 'infinity For a segment S of a straight line or of any "ordinary" smooth curve H(0,S )is infinite, H(l,£ ) equals one, and H(2;S ) = 0. Also H(d, S,).,is infinite^ for d less than 1 but zero for D greater than 1, "fhus d = 1, is correctly the Hausdorff dimension of any smooth curve, agreeing with its topological dimension. Similarly for «£. Sz , such as the interior of a circle H(2> S2) = 1 with H(d.Sj) =0 for d greater than 2 and infinity for d less than 2. Hausdorff applied this method to Cantor's ternary set which, although it has as many points as a line-segment^has Lebesgue measure zero because it is "too thin". However the Hausdorff dimension is d = (log 2)/{ (log 3) = O.6309, so that it has some thickness; indeed it is thicker than the "everywhere dense" set of "rational 1 points" — '(t/fij% whose Hausdorff dimension is zero. (For tne theory of H(d,S) see either Falconer or Barnsley, or Harrison). Von Koch's "snowflake", although of topological dimension 1, is, like all non rectifiable curves,"too thick". Its Hausdorff-Besicovitch dimension is twice that of Cantor's set and is (log -+}^og 3)= 1.2619. When speaking above of the Weierstrass curve we were being slightly imprecise. There is in fact a whole family of them. Any individual one is characterized by the values of two constants and g. The first constant X is greater than 1 while € lies between 0 and 1. The Weierstrass function in Figure IZ has A * 2 and S = 0.5. On the basis of a formula given by Hardy in his 1916 paper the Hausdorff- Besicovitch dimension of a Weierstrass curve is just (2 - £) which, properly enough, is greater than 1 (its topological dimension) but less than 2 — the topological dimension of any planar region. The name "fractal" for curves and sets whose dimensions are non-integral and therefore "fractions" was proposed by Professor Benoit B. Mandelbrot. Bom in France and educated in Paris he is a scientist with a very broad and diverse knowledge of many topics of current interest and has held many research and teaching positions in France, Switzerland, and the U.S.A. In the late 1960's he became interested in the shapes of various natural objects. For example, if one tears a sheet of paper the broken edges will be highly irregular. Furthermore, optical magnification increases the irregularity. Continuing to increase the magnification, one will get the impression^ that the profile is approximately self- similar. Indeed it is very like that of a non-rectifiable curve. Similar results are got by breaking a rock or looking at the edge of an actual real snowflake. On account of these resemblances Mandelbrot proposed that such curves and profiles be called "fractal", both those mathematically defined and the approximations found in nature. The term would seem to be appropriate in relation to the jagged or broken character of these curves. In French fraction means breakage or fracture, as well as arithmetical fraction. In English there is rthsoalmost Archaic word "anfractuous" = "tortuous, having many winding passages or grooves, sinuous", last used by T.S. Eliot in his poem The Waste Land — "Picture an anfractuous waste shore ..." However this is not Mandelbrot's intention; he wishes "fractal" to refer to the Hausdorff-Besicovitch dimension, particuarly to the fact that this dimension is not a whole number, being an "improper fraction", (i.e. greater than 1) for the curves of infinite length and a proper fraction (i.e. less than 1) for sets of zero Lebesgue measure such as Cantor's. Mandelbrot's own preference went further. He wished to restrict "fractal" to the former class (greater than 1) approximated by rocky coastlines, the edges of snowflakes, and torn materials. However this restriction was impossible to maintain, and all the curves and sets we have mentioned are lumped together as ?!fractals". Besides Mandelbrot's own books, of which the most recent one is The Fractal Geometry of Nature, numerous others have made fractals popular with a part of the literate public outside the ranks of pure mathematicians, and also with photographers, artists, and computer buffs, because of the beauty and interest, both of the natural objects, and of computer simulations of fractals, (see also Schroeder, Peitgen, Barnsley, Harrison, Gleik). The existence of what came to be known as the Mandelbrot Set was first announced in 1979 by Brooks and Matelski who published a computer drawing of it. Independently and at about the same time Mandelbrot also discovered the Set. It was however Douady and Hubbard'who in 1982 named it in his honour. The Mandelbrot Set emerged from a line of research quite unconnected with natural objects and having no relation to the Weierstrass, Koch, Cantor traditions of non-rectifiable curves, or lacunary sets. The field jof investigation can be called the theory of functional iteration. We can illustrate this in its simplest form by an application of "New*ton's rule" for extraction of a square root. Suppose that we require the square root y/HT of a number K. Then Newton's rule is summarized by the equation - A"/*>iJ , Let XQ be any reasonably good approximation to \TK7 then we can use the above formula to calculate xl - "ifc^ + ^A*) which, in fact, is a better approximation to /K". If kt is not good enough > we can proceed in the same way and obtain the successive further approximations <*z ; X>, Xq.; ... ad libitum. A nice example is provided by K = 10. Take Xo = 3»* we obtain Correct to five decimal places we infer \flO = 3.16228. Newton's rule for square roots is in fact a special case of what is called the Newton-Raphson process, Raphson being a younger contemporary of Sir Isaac Newton. Applied to finding a value of x satisfying an equation F(x) =0 the process consists in doing the functional iterations defined "by the equation , 1 * , where F' (X) is the derivative of the function F(x), which clearly has to be smooth for the process to be applicable. Applying this to the square root of K is to solve the equation X1—K = 0, which is F(x) = 0, with F(x) = X* - K. Using the well-known result that F ' (x) = 2xjthe iterative equation is which is Newton's rule for 4 K. us Unfortunately all problems are not so simple as but require very complicated iterative procedures, not only to get an answer, but often merely to find out whether an answer exists, or what kind of answer exists. The field of application comprises almost every area of human intellectual endeavour whether theoretical or m practical — mathematics, physics, chemistry, biology, meteorology, the environment, statistical inference, economics, as well as prediction and forecasting of all ^ kinds. Of course the contemporary dominance of numerical methods results from the advent of high speed computers, but the explosion of science in the last decades of the nineteenth century engendered at that time an interest in the m "convergence" problems that antedated the modern computer era. Thus, it had been realized that the Newton-Raphson procedure had' its quirks. If our choice of X0 is "bad" mt i.e. if we do not start the iteration "sufficiently close" to the solution x of the equation F(x) = 0, then the sequence X0 s ><,} ^} X3. , may not converge to x but to another solution (irrelevant to our problem), or it might not converge at all, or it might converge to a number which is not a solution at alll. (See Brftnner for an ^ example). It was in I897 that Arthur Cfiyley, Professor of • Mathematics in the University of Cambridge, and of Trinity College, some three centuries after Isaac Newton, studied the Newton-Raphson iterative,formula in the form Here Z^ =r Xn ~ir Ifa . Hi O In the eighteenth century mathematicians had generalized the concept of a number. The motivation for this was the discovery — that every algebraic equation can be solved if we are prepared to allow solutions in terms of so-called complex numbers as valid. The eq,action j mt Thus in the field of complex numbers the equation Z^ - 6Z + 25 = 0 has the two solutions (approximate to an equation of "degree" two), namely Z = 3+i+i and Z=3-4i. Prior to Cayley's time, mathematicians had developed an extensive theory of functions of the complex variable. This theory is very elegant; it is also very"powerful" in the sense that it embodies many applicable theorems. Of course every result obtained for iterations of Z can be displayed as relating to the simultaneous iteration of separate ordinary variables x and y. Thus if the corresponding iterative equations for and V are For example, Newton's rule for zzrf-~ is 2 with fu -Xv? + ^ibecause ''jz* - £ X* - £-J* )/f*li . In working with complex mumbers it is convenient to use a graphical representation as terms of "Cartesian coordinates" x and y. The complex number z = x + iy is represented by a point P in a plane (see Figure fO ); the distance of P "east" of a given origin 0 is x; the distance "north" of 0 is y. 'We write r for /^c-H-y* , the distance OP. Supposing an iteration to start al; the point Po represented by Zo, the results of the iteration can be visualized as the sequence of points Ti^T-x^Ty, . » , where corresponds to Zn in the iteration Zn+1 = F(Zn). The systematic study of such complex variable iterations was initiated by Gaston Julia in 1919- It was found that even the simplest possible functions F(Z) were characterized by extremely complicated behaviours of the point Often it turned out that the plane is divided into two distinct regions. If Po is in one of the two regions then the sequence To/P-f\. exhibits "chaotic" behaviour; the poinx Pn neither converges towards a limiting position, nor moves on a periodic or almost periodic orbit; instead it wanders in no discernable pattern — without rhyme or reason like a particle in a turbulent fluid, e.g. in a saucepan of boiling water. The region of chaotic behaviour is called the "Julia set", in honour of Gaston Julia. The shapes often adventitiously have a chance resemblance to natural objects. Thus Figure /3 shows (in black) the Julia set of the iteration which, in the trade, is nick-named "Douady's Rabbit' in honour of A. Douady (See Kleen). (Yes, Virginia, mathematicians do have a sense of humourl). While there are numerous and different Julia Sets there is only one Mandelbrot Set. This set is related to the Julia sets, although it is not, strictly speaking, a Julia set. The Mandelbrot Set is defined in terms of the following iteration: and ^ - fractal dimension is not known to us at time of writing but it is between 1 and 2. Figure /5(a) shows the whole boundary at low magnification; figure /S(b) shows a detail in a higher magnification. The importance of the City of Cambridge in the general scheme of things may be judged from the fact that, unlike Oxford, it is served by two railways from London. One of them, on the west, approaches through the towns of B6yfetph and Hitchin, and could be regarded as the more plebian of the two. The journey is usually rather dull unless one departs from London fairly early on a day when races are o 57 Filled Julia set of. 2w*/ - 2vi -0.12256117+ .74486177/("Douady's Rabbit',) 13. Blow up of the marked area quasi-self similarity. mi Ik at Newmarket. At these times announcers over the public address system at King's Cross railway station warn passengers to beware of confidence-tricksters and card-sharpers. Indeed in previous decades one could get, for quite a modest investment, an excellent demonstration of the three card trickl Ickleton is, however, far from this railway line. If from a location on the western railway line, e.g. Shepreth, or Foxton, one chooses in summer to drive eastward to Ickleton, ' one will approach the village over rolling downs which are among the most beautiful agrarian landscapes in Britain. Planted with grains, principally barley (whose local variety is the highest rated by British brewers)* on a sunny day the view cannot be bettered. On coming to Ickleton one finds that one has arrived almost at the other railway line, which has an important stop at Audley End. Of fame in the past (e.g. the Mastership of Magdalene College in the University of Cambridge was, and perhaps still is, in the gift of the Lord of the Manor there) Audley End has for some decades been a place for financial persons in the City of London (i.e. England's Wall Street or Bay Street) to use as the centre of a dormitory suburb. The area, which includes the historic borough of Saffron Walden, has long seen the conversion of medieval cottages to bijou residences, as well as the rehabilitation of Tudor-esque half timbered houses. Indeed if one chose to go to London moderately early in the day, and return likewise, one could enjoy the dominance of the train by these financial persons, dressed comme il faut in bowler hats, black jackets, pinastriped trousers, and with tightly rolled umbrellas. Perhaps nowadays their costumes are different — designer jeans (?), but their attitudes are as genial as ever. These gentlemen seemed always extrovert and happy to the point of "jolliness". We cannot judge whether financially they had much to be jolly about, but even if this was not the case, they have to be given credit for putting a good face on it. Possibly two decades ago these gentlemen were not especially numerically literate but times have changed, and doubtless a high proportion of these bankers, investors, brokers, etc. etc. not only employ computer operators, but are themselves financial analysts, and also wellread in contemporary mathematics and computer science. Taking these considerations into account we have to retain in our mind that some genial IS 15 la) TILE MECHANICS Since the mid-1970s, Roger Penrose, one of the world's top mathematicians, has been fascinated by patterns of tiles. It is easy, for example, to cover a floor with square tiles but impossible to cover a surface with pentagons all of the same size. Gaps would occur because the spaces between pentagons are not pentagon-shaped. Penrose found, however, that he could cover a floor with pentagons and three extra shapes—a star, a diamond, and a hat The pattern at first appears to be regular. Butacloser look shows that the motif never repeats itself, though it is always close to doing so. The phenomenon is of interest to those studying crystals, which grow—atom by atom—with the sort of regularity found in tile patterns. If a person were tiling a floor with pentagons, stars, diamonds, and hats, he would have to step back and look at the entire floora s if it were a jigsaw puzzle, before deciding where the next oteca should oo. — * _ "' of Audley End and its environs may have perpetrated a jovial jest* Other possible culprits in respect of .Ickleton are, of, course, the students and graduate students of Cambridge University. Besides mathematicians and physicists the University has many brilliant engineers and computer scientists, as well as biologists/meteorologists, geophysicists, and economists, all of whom are highly numerate and play on computer keyboards as easily as a Mpzart on a piano. Even in earlier years the University has been the setting for many brilliant student feats of an amusing nature, but which also demonstrated considerable technical ability. Twentyfive years ago an unknown group of students (presumed to be engineers) in the course of one Spring night set an automobile (admittedly small, but beyond most people's abilities to hoist) on top of the roof of the Senate House. A firm of contractors took some days to get it down. Apocalypse Soon? Turning back to the general picture, we first give attention to the direst interpretations of the crop patterns. It is not hard to stitch together the images into an apparently coordinated prediction, if not of doom, at least of stormy waters ahead for humankind! To paraphrase a well-known hymn> "On Avon's bank the prophet's cry Persuades us that the End is nigh"* The argument goes as followsi- First, simple geometry — a clearcut message — "We are rational like youI" Then, variations to show that the communicators have a programme annuli, keys, side excrescences. Next, historical reflections on the progress of humankind — "These were thy gods, 0 Israeli". — sun god, fertility goddess" CAlso perhaps the Brain = snake cdeity; c.f. Serpent Mounds in Ohio or at Peterborough, Ontario), Perhaps also the imitation of the spiral on thejplain of Nasca, Peru (a pattern in loose stones and visible as a whole only from above) has a message, "Did you believe in gods in the sky?". Alternatively, do they mean to say, "We can see youl" Or is this symbol overdetermined?with yet other layers of implication? Spirals will inevitably bring to mind the symbolism of William Butler Yeats' poem of 1921 (which for many of us only became alive about 1937 in the Hitler years,when it seemed to exemplify the superiority of the poet's intuition over that of the politicians^. Yeats' poem The Second Coming commencesi Turning and turning in the widening gyre The falcon cannot hear the falconer .... Things fall apart; the centre cannot hold Mere anarchy is losed upon the world, Surely some revelation is at hand " The poem goes on to say what many of our contempories would agree with, in view of the violence of so many current fanatacisms. "The best lack all conviction, while the Worst burn with a passionate intesnity." On this line of thought it might be possible to gather in even the Irminsul as having an admonitory purpose. The descendants of Hermann's people resisted the Emperor of Rome and centuries later the Emperor of the French, but then aided the imperialisms of the Kaiser William and Adolf Hitler, the ripples from whose failure brought down all colonial imperialisms. We have now to contejta with a legacy of nascent nationalisms. If we are looking for rebukes to humankind among the crop patterns, then what of the Whale? Is it an injunction to "save the whales" or a monition to the effect that "whales are people too"? Science fiction fans or their parents may recall the charming movie Star TreklV in which extra-terrestrial entities check on the well-being of Earth's cetecea. Following this line of thought we must admit that the "insectograms" are somewhat ambivalent in their import. Should we beware that insects do not become'-the meek that shall inherit the earth?'' Or is the megsage a warning not to give ourselves airs? Or is it that the communicators, whoever they may be, wish to indicate that their view of us is like that expressed to the Deity by Mephistopheles in Goethe's Fausti- "The little god of earth (i.e. humankind) ... To me he seems to be, with deference to your Grace, One of those crickets, jumping over the place". Later, Mephistopheles speculates as to whether we are "fit for overthrow" so that our "toy-worid" can be sterilized! What of the Mandelbrot Set at Ickleton? It is not altogether easy to read an apocalyptic meaning into it, Possibly it says "In spite of the intellectual success of your species as exemplified by your mathematics, physics, and technology, you need to be humble! In all fields of endeavour you have already encountered limits to the efficacy of your reasoning and have to resort to reliance on the computer" Perhaps it is also saying "Beware that slave does not become your master, in the way that slaves can bel" Again, possibly, Ickleton says "We are more like computers than you are" (This is a science fiction theme though not one of those most commonly encountered). Although it is John Michell's analysis of the Barbury Castle formation that has inspired the hypothesis that the crop circles, taken in sequence, constitute g progressively developing series of warnings, it is fair to eay that the reasoning is circuitous and the conclusions very narrowly based, depending in fact on a single numerical coincidence. If the formation is correctly interpreted as related to Revelation, this only implies that its authors, whoever they may be, are making a monitory use of the prophecies. After all we have to recollect that Revelation was based only on a vision (albeit of a good and saintly person). He doubtless took it literally in all respects, but it was clearly conditioned by the expectation of the imminent end of the world as it was, and the Second Coming of Jesus (c.f. the Gospels, 8 Mark, 16 Matthew, 9 Luke). As this did not happen (a difficulty as awkward for the Christian Fathers prior to Augustine, as the delay in the coming of *tyesaiah was for the Hebrew prophets after Jeremiah) we are at liberty to accept Revelation as purely metaphorical. As ever, we should beware of the first of the four horsemen —— "the man on the white horse", the conqueror "who rode forth to conquer and he conquered", (though it seems difficult to picture President Bush, the last remaining candidate for this role in world history, as riding through Washington or Badhdad in that persona as representative of the world order as a Pax Americana). We may also be advised of the possibilities of war, pestilence, and death while forms of economic imperialism, regional conflicts, and arms sales perturb the Third World. Before resigning our flirtation with the apocalyptic view of the crop circles we should visit once more with our good Sir Isaac Newton of Trinity College, Cambridge. Newton was in many respects a successor to the Puritans of the mid-^ seventeenth century, "those Cromwellians of the eastern counties of England who constituted the backbone of the Afew Model Army. In religion they were Independents and the forerunners of modem denominations such as the Congregationalistg and Unitarians. In the field they defeated both Anglican and Presbyterian armiesi they gave to the world what some would regard as the moat previous of all ideas — that of toleration. Long after the Civil War some scholars and divines in the University of Cambridge, even though ordained in the Church of England, kept alive the spirit of toleration. Ralph Cudsworth and Henry More attempted to bring the ehurch to "latitudinarianisra" i.e. a breadth of toleration in religious belief. This attitude left its mark on Cambridge which in both "town and gown" is remarkably free of the "odium theologicaura". In religion Newton was a "Subordinationist", a Unitarian, in fact an Arian, and therefore technically a heretic. Gn this account he refused ordination in the Anglican Church and thereby sacrificed appointment to the academically prestigious post (then as now) of the Mastership of Trinity College. Inspired by Cudworth, More, and also Joseph Mede (see, A.R.G. Owen, 1963), Newton put together a book which was published only posthumously, Observations upon the Prophecies of Daniel, and the Apocalypse of St. John. For us the most interesting of Sir Isaac's findings may be the prediction that the Beast of Revelation, which was equated sometimes with Milton's "She of the Seven Hills" i.e. the Church of Rome would come to an end in 1867 A.D. Newton based this prediction on the 1260 years in Revelation and the date 607 A.D. when, in his opinion, the worship of Antichrist reached its peak. As this date has come and gone, we may perhaps breathe again (or not, in case of error). Final Remarks. We are in a position like David Hume re the universe — is it a machine, a work of art, a theatre for moral improvement? If we do not know what it is, we cannot deduce its purposed The reader may object that this is a truism if we do not know its purpose then we do not know its purpose; yesi quite so — that is the problem — which is just what Hume saidt Unless we can divine the purpose, if any, of the crop circles we have to account them "a riddle, a mystery, an enigma". If we can catch large numbers of pranksters in flagrante delicto, well and good! Until thenr serious students should reserve judgement. In the history of seifence too many counter-examples are on record of cases where a genuine phenomenon has been rejected as false. Thus we should keep in mind that the crop formations may be an occult phenomenon. Here we use the word "occult" in its primary and classical sense, which was"".also that of Newton's critics re the "force" of gravitation. "Occult" meant without known cause. In this sense "occult" forces may, for all we know, to the contrary, be wielded by humans who wish to be secret, whether or not they are the "illuminati" — Rosicrucians, freemasonic, etc. of the kind recently burlesqued in Umberto Eco's amusing novel Foucault's Pendulum. (Among those listed at various times as being concerned in that benign though hypothetical conspiracy have been Francis Bacon, and, more rarely, Newton himself; Bacon was, of course, an alumnus of Trinity College, but we do not wish to pursue that avenue of enquiryi). Are the communicators via the crop formations non-human. — alien, though not ipso facto or necessarily inimical to man? Perhaps science-fiction buffs are. as well is ant*, qualified to answer; a century of so of science-fiction writing has performed as a "thought laboratory" for "thought experiments" concerning the nature and psychology of non-human life. Trivial pursuit as this may seem, it has the merit of encouraging open-mindedness, which is possibly humankind's chief virtue. Science fiction parades many types of aliens before us. Kumanoidg, very like ourselves in temperament •— all too humanl Androids.very moral, untemperamental, —• inclined to subordinate emotion to reason. Cyborgs, beings sharing a group mind and therefore without individual judgement. Energetic lifeforms«with a mode of existence profoundly differring from biochemically based life and with varying degrees of empathy with humanoids. Their attitudes vary according to their species. Some are totally benign to humans and will help them in an emergency. Others are benignly inquisitive as to the nature of human living, and may try on occasion to understand it. Yet other lifeforms without being per se evil, are yet juvenile or immature in their outlook and may do us minor mischief on account of insensitivity or ignorance. A few non-humans, of course, are represented tfn Star Trek and other science fiction as totally callous and' destructive to other species;, hut these are^_a minority and usually humourless. The humour of the authors of the crop patterns is a reassuring feature. On the five hundredth anniversary of the discovery of America, it is too late for the previous inhabitant of the New World to pray that they will not be discovered. If the crop circle commtoiicators are from elsewhere in our space, or in some other dimension, then it is too late for us to utter the same prayer^and we must hope for the best{ Supposing that the crop circle communicators represent a higher civilization than our own, and choose to indicate the fact to us, how could they do it, while still restricting themselves to such a crude, indeed clumsy, mode of communication as they are now using? It would be very significant if they would indicate some mathematical or scientific fact which we would recognize as such, but would be of a revolutionary or startling nature. It would need to be of the stature of Becquerel*s discovery of radioactivity in I896, or of Go'del's theorem of 1930 on fommally undecidable propositions. However it is hard to see how abstract matters can be applied by simple diagrams. The only kind of thing which comes to mind is the recent discovery of irregular tiling of an infinite plane by Professor Roger Penrose of the University of Oxford (See Figure f& ). REFERENCES. John Michell Geometry & Symbolism at Barbury Castle. The Cerealogist, No. 4, Summer 1991- p. 24-25 The View Over Atlantis, Garnstone Press, London, ed. 1969. City of Revelation. The Dimensions of Paradise, Thames & Hudson, London 1988. Lionel Stebbing The Secrets of Numbers. New Knowledge Books, East Grinstead. U.K. C. Piazzi Smyth Our Inheritance in the Great Pyramid. 1577. P. Bligh Bond & Gematria, Blackwell, Oxford, 1917. T.S. Lea Cabala, Blackwell, Oxford. John Michell "A Sacred Symbol in an Essex crop mark". The Cerealogist, No. 3. Spring 1991. p. 16-17. William W, Kenawell The Quest at Glastonbury. Helix Press, Garrett Publications. New York, 1965. F. Bligh Bond The Geometric Cubit as a Basis of Proportion in the Plan of Medieval Buildings." Jl. 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