It so happened that his wife, Julia, had always had an ambition to play the harp. So Hawkins bought her one. And, although he was not a musician, he decided to tune the harp himself. Which in turn led him to teach himself the elements of music.
The musical scale, as everyone knows, has eight notes—doh, re, mi, fa, so, la, ti, doh. The top doh completes it, but also begins a new octave. These are the white notes on a piano keyboard—we also call them C, D, E, F, G, A, B and top C—and we feel that they are somehow natural and complete. But of course the keyboard also has black notes, the semitones. Hawkins found that the harp sounded better if he tuned it according to the white notes—which are also known as the diatonic scale—and used mathematical fractions.
Now the reason that each sounds different is that they all differ in pitch, which is the number of times the piano string vibrates per second. The doh in the middle of a piano keyboard is 264 vibrations per second. And the top doh is exactly twice that—528 vibrations per second. And since, to our Western ears, the remaining six notes sound ‘perfectly spaced’, you might expect each note to increase its vibrations by a jump of one seventh, so that re is one and a seventh, mi is one and two-sevenths, and so on. But our Western ears deceive us—the notes are not perfectly spaced. In fact, re is one and an eighth, mi is one and two-eighths, fa is one and a third, soh one and a half, la one and two-thirds, and ti one and seven-eighths.
However, the points is that these are simple fractions. The black notes are a different matter. While re is one and an eighth above doh, the black note next to it (called a minor third) is 32 divided by 27, which no one could call a simple fraction.
Now, as Hawkins went on measuring and comparing crop circles, he discovered that most of the ratios came out in simple fractions—and, moreover, the fractions of the diatonic scale, listed above. The 16:3 of the Cheesefoot Head formation was the note F. Three well-known circle formations yielded 5:4, the note E, 3:2, the note G, and 5:3, the note A. A few, where circles and diameters were compared, yielded two notes.
If Doug and Dave, or other hoaxers, had been responsible for the crop circles, then the odds were thousands to one against this happening (Hawkins calculated them as 25,000 to 1). In fact, Hawkins studied a number of circles that Doug and Dave admitted to, and found no diatonic ratios in any of them.
Now it is true that not every circle revealed this musical code. Out of the first eighteen Hawkins studied, only eleven were ‘diatonic’. The answer may be that Doug and Dave had made the other seven. Or that some of the ‘circle makers’ were not musical. Yet almost two-thirds was an impressive number, far beyond statistical probability.
So Hawkins felt he had stumbled on a discovery that was as interesting as any he had ever made. If the ‘circle makers’ were human hoaxers, then they had devised a highly sophisticated code.
But why bother with a code? Why not simply spell out the message, like the WEARENOTALONE inscription, which is claimed by Doug and Dave? An obvious answer suggests itself. WEARENOTALONE sounds like Doug and Dave. But a code of diatonic ratios is almost certainly beyond Doug and Dave.
Following the advice of Lord Zuckerman, Gerald Hawkins sent off an account of his findings to Nature, the prestigious magazine for publishing new and yet-to-be-explained discoveries. In 1963, Nature had published his work on Stonehenge, but now in July 1991 the editor balked, saying: your findings do not ‘provide a sufficient advance towards understanding the origin of crop circles to excite immediate interest of a wide scientific audience’.
It was like asking an astronomer to explain the origin of the stars, sun or moon. Hawkins, like any careful research scientist, does not rush in with a theory. He looks at the facts that are there. Edmund Hilary did not ask for the theory of origin of Mount Everest, he climbed it because it was there.
Even now as I prod and poke, I cannot get Hawkins to come out with an origin. For him, the game’s afoot. He continues to investigate the theory of hoaxers as Zuckerman suggested, but what he has found in the intellectual profile has made him even more cautious. Now he talks about ‘circle makers’, a tautology that neatly covers hoaxers and all possible origins.
But let us go back to the basic question. If there is a complex geometry hidden in the circles, does this not suggest that nonhuman intelligences are trying to communicate with us? If Hawkins’s geometry leads us to conclude that the answer is yes, then we have decided a question that has been preoccupying scientists for well over two centuries: the question of whether we are alone in the universe, or whether there are other intelligent beings.
But then, others have believed they had solved that problem, and have been proved mistaken. In 1877, the Italian astronomer Giovanni Schiaparelli studied Mars on one of its closest approaches to Earth, and believed he saw lines that looked like rivers on its surface. He called them canali and included them on his map of Mars. Canali means ‘channels’, but it was translated into English as ‘canals’, and soon half the civilised world believed that Mars was the home of a superintelligent but dying race who had built canals to bring water from the polar icecaps to the tropics; the idea inspired H. G. Wells to write his novel The War of the Worlds. Then the American astronomer Percival Lowell looked at Mars through a more powerful telescope, and saw the canals even more clearly. His books arguing that there is life on Mars caused great excitement. But all these dreams of Martian civilisation came to an end when the Mariner 4 space probe sped past Mars in 1965, and its photographs showed a surface as bleak and dead as the moon—and without canals. Our eyes join up dots that are close together to form lines, and Lowell and Schiaparelli had been victims of this illusion.
In July 1976, when the Viking 1 orbiter began sending back photographs of the surface of Mars, a NASA researcher named Toby Owens observed what looked like an enormous face in the Plain of Cydonia. Mentioned at a NASA press conference as a kind of joke, the ‘face on Mars’ was soon exciting widespread attention. Richard Hoagland, a former NASA adviser, argued that the face, and what look like other ‘man-made’ (or Martian-made) structures in the same area, prove that Mars was once the home of a technically accomplished civilisation. Sceptics replied that the face, and other such structures, are natural features, and that they are no more significant than the face of the man in the moon.
There is, however, a basic difference between the approaches of Lowell and Hoagland, and that of Hawkins. Lowell’s evidence depended on what he saw—or thought he saw—through a telescope, Hoagland’s upon blurry photographs taken from miles above Mars. Hawkins has argued that many of the crop circles show too much geometrical sophistication to be by hoaxers, and that, moreover, the underlying pattern of diatonic ratios is too precise to be an accident. For the crop circles, the odds against an accident are about 400,000 to 1, something like getting 7 sixes in ten throws of the dice’.
Moreover, Hawkins’s work on the four theorems made him aware that they were all special cases of a fifth theorem. One of his advantages as a scientist was a naturally visual imagination, the ability to see shapes and patterns in his head. (It is sometimes called eidetic imagery.) He drew the second theorem in his head, then contracted the circles and changed the shapes until he had the third and fourth theorems—then realised they all sprang out of a more general theorem about triangles in concentric circles.
The existence of this fifth theorem was revealed in Science News for 1 February 1992, in an article entitled ‘Euclid’s Crop Circles’, by Ivars Peterson. He begins by describing crop circles, and how farmers are plagued with ‘some enigmatic nocturnal pest’.
Peterson takes, of course, the view that the circle makers are quite definitely human hoaxers. After all, if a respectable science magazine is to agree that crop-circle theorems are not a matter of chance, then it is also forced to conclude that someone put them there. And, since extraterrestrial intelligences are unthinkable, then it had to be hoaxers.
The article goes on to say that Hawkins wrote a letter to Doug Bower and Dave Chorley, asking
them how they had managed to incorporate diatonic ratios into the ‘artwork’, and concludes, ‘The media did not give you credit for the unusual cleverness behind the design of the patterns’.
Doug and Dave, apparently, decided not to reply.
Peterson goes on to describe how Hawkins had discovered the 4:3 ratio in the Cheesefoot Head design, and how he had then discovered three more theorems in designs involving triangles, squares, and hexagons. But then he realised that these could all be derived from a fifth theorem.
The hoaxers apparently had the requisite knowledge not only to prove a Euclidean theorem but also to conceive of an original theorem in the first place—a far more challenging task. To show how difficult such a task can be, Hawkins often playfully refuses to divulge his fifth theorem, inviting anyone interested to come up with the theorem itself before trying to prove it.
What Hawkins now has [Peterson concludes] is an intellectual fingerprint of the hoaxers involved. ‘One has to admire this sort of mind, let alone how it’s done or why it’s done’, he says.
Did Chorley and Bower have the mathematical sophistication to depict novel Euclidean theorems in the wheat?
Perhaps Euclid’s ghost is stalking the English countryside by night, leaving his distinctive mark wherever he happens to alight.
(Peterson, ‘Euclid’s Crop Circles’)
Hawkins learnt later that the editor had cut out the reference to Euclid’s ghost, which was originally in the first paragraph. Somehow, the writer got it reinstated at the end of the article. ‘Euclid’s ghost’ clearly implies more than two hoaxers called Doug and Dave.
In September 1995, another magazine, the Mathematics Teacher, ran an article on Hawkins called ‘Geometry in English Wheat Fields’. This also accepted that the four theorems were not simply being ‘read into’ the crop circles. And, like Science News, it took it for granted that the circle makers were hoaxers. It challenged its seventy thousand readers to work out the fifth theorem from which the other four were derived (none succeeded), adding that, so far, the circle makers themselves had not shown any knowledge of it.
They were behind the times.
At one point, wondering what patterns would appear next, Hawkins had hoped that it would be four concentric circles, with a triangle whose sides formed chords to three of them—the basic diagram of his fifth theorem. In fact the circle makers went one better.
At Litchfield, Hampshire, on 6 July 1995, a 250-foot circle appeared with eight concentric circles, as well as a surround of half-circles made to look like intestines. Hawkins realised that the circle makers were ahead of him. They were pointing out that there was a whole geometry of concentric circles. He felt that it was as if they were sticking out their tongues and saying: ‘Get a move on!’
Hawkins contacted the Mathematics Teacher, and told them that the circle makers had now demonstrated knowledge of his fifth theorem. But the editor said that the article was already in the press, and it was too late to change.
Hawkins’s increasing conviction that the circle makers were intelligent was increased by the curious affair of the runic cipher.
In late July 1991, an American named John Beckjord cut out—with the farmer’s permission—a message, TALK TO US, in a field near Alton Barnes, Wiltshire. He wrote it on a double row of straight tractor lines. A week later, a circle spotter in an aeroplane saw a curious set of markings at Milk Hill, not far from Beckjord’s message, which was also on a double row of straight tractor lines. But the ‘letters’ made no sense; Beckjord thought the message was in Korean, while someone else suggested Atlantean.
Clearly, the obvious assumption was that this was the work of a hoaxer.
Someone sent Gerald Hawkins a copy. The first thing that struck him was that the whole thing showed a care and precision that argued that someone meant something by it. This was also the opinion of a number of trained cryptologists to whom he showed it. And three sets of parallel lines at the beginning, middle and end suggested that they were intended as word-dividers. In which case, the message seemed to consist of two words—the first of six letters, the second of five.
Meanwhile, one of the cryptologists had recognised that the symbols were runes—letters of the ancient Germanic alphabet. But that brought the solution no closer, for in runes the message spelled ‘DPPDSD XIVDI’, where the X is a non-runic letter looking like a capital I.
Sherlock Holmes had been confronted with a similar problem in a story called ‘The Dancing Men’, in which the newly married wife of a Norfolk squire is driven to despair by a series of strange notes containing matchstick men in various positions. They are sent by a former lover who is determined to get her back.
Holmes recognises that the dancing men stand for letters of the alphabet, and, since the most frequently used letter is E, is quickly able to identify the E. He also reasons that, when the figures hold a small flag, this symbolises the end of a word. A chain of similar reasoning gradually breaks the code—although not before the husband has been murdered and the lady attempted suicide.
In the Milk Hill message, the first word contains the same letter—a kind of square-looking I—at the beginning, middle, and end. And the opening letter is followed by two identical letters looking like square C’s.
We use double consonants—BB, CC, DD, FF—far more than double vowels. So the chances are that the first letter is a vowel, followed by a double consonant, followed by the same vowel, followed by an unknown consonant, followed by the same vowel again. It could, for example, be effete. And the same vowel occurs as the penultimate letter of the second word.
Julia Hawkins, an anthropologist, had Sylvester Mawson’s Dictionary of Foreign Terms, a book containing eighteen thousand common phrases in forty-two languages, from French and Latin to Hindustani and Russian in her library. Hawkins plodded steadily through it until he finally found a word of six letters that fitted. It was the Latin oppono, meaning ‘I oppose’.
If that was correct, the penultimate letter of the last word was O. And since that was followed by an unknown letter, the word ending was probably OS. That also gave the second letter, so the message now read ‘OPPONO _S_OS’. What would the writer be opposing? Another long search suggested astos, meaning ‘acts of craft and cunning’—or frauds. So the script could be translated as ‘I am against frauds’ or ‘I am against hoaxes’—a sentiment that might well be felt by a genuine circle maker.
But, if the circle makers were using a runic alphabet, why did they change its letters? The answer may be that they didn’t. For example, the runic P is a square C, as in ‘OPPONO’. And, in a slightly different version of the runic alphabet used by the Knights Templars, O is identical with the O used here, except that it is upside down. Runes were not standardised, like our modern alphabet; the circle makers may have been using another version of the runic alphabet.
Hawkins’s solution was published in the magazine The Cereologist (no. 3). The magazine also offered a prize of £100 for a more plausible rendering; many tried, but no one succeeded.
Even those who feel that Hawkins was wasting his time will acknowledge that his approach was impeccably scientific. Given his initial assumption that the script was not a meaningless jumble of symbols, the rest follows—although it is just possible that some other solution might be found in Korean or Armenian or Icelandic.
What may bother other readers is the assumption that the script was meaningful. If it was a hoaxer, perhaps responding to Beckjord’s TALK TO US, then he would be much more likely to write random nonsense. But if it was not a hoaxer, then we are asked to believe that some extraterrestrial intelligence was writing in Latin to express its annoyance with hoaxers like Doug and Dave. Surely that is too far-fetched to make sense?
But then, the same argument applies to crop circles in general. Is it not more likely that they were simply patterns made for fun?
Against this objection, two facts stand out clearly. One is that, for whatever reason, Lord Zuckerman decided to make a close study
of the circles, and ended by being more than half convinced that they were not the work of hoaxers or whirlwinds. The second is that Gerald Hawkins concluded that the circles showed signs of more care and intelligence than would be expected of hoaxers. If two scientists, both pre-eminent in their own field, decide that the circles are worth serious attention, then are most of us justified in dismissing them as unimportant?
For me, an equally interesting question is: why did Zuckerman and Hawkins take the risk of having their names associated with the ‘lunatic fringe’ by displaying an interest in crop circles? There can be only one answer: that both were carried away by a powerful first impression that this was no hoax—that something or someone was trying to communicate.
And, although Hawkins decided to hold back on publishing his investigation in 1993, after Linda Howe had published the first four theorems in her Glimpses of Other Realities, his communications with me up to the end of 1997 make it clear that he remains as actively interested as ever.
In due course, his fifth theorem became available on request from the National Council of Teachers of Mathematics. But the Mathematics Teacher decided against publishing it for the time being, perhaps feeling that mathematics in a cornfield was getting beyond the bounds of scientific propriety.
If so, they were not far wrong. Looking back over the history of crop circles since 1980, it looked as if the circle makers were engaged in a kind of argument with those who did not believe they were real. First the circles were blamed on midsummer whirlwinds, but, as the circle makers began making several at the same time, this explanation became more and more unsatisfactory. As multiple whirlwinds—and other such absurdities—were postulated, they began adding lines, which could hardly be formed by whirlwinds. When Hawkins recognised their geometry, and their diatonic ratios, they began making them increasingly complex, with triangles, squares and hexagons, and finally targets.