A radio astronomer reports on the mathematical relationships within the elements of crop circles in England
An interesting interview (1992) with Gerald S. Hawkins, who earned a Ph D in radio astronomy with Sir Bernard Lovell at Jodrell Bank, England, and a D Sc for astronomical research at the HarvardSmithsonian Observatories. His undergraduate degrees were in physics and mathematics from London University. Hawkins’ discovery that Stonehenge was built by neolithic people to mark the rising and setting of the sun and moon over an 18.6year cycle stimulated the new field of archaeoastronomy. From 1957 to 1969 he was Professor of Astronomy and Chairman of the Department at Boston University, and Dean of the College at Dickinson College from 1969 to 1971. He is currently a commission member of the International Astronomical Union, and is engaged in research projects in archaeoastronomy and the crop circle phenomenon.
Monte Leach (interviewer): How did you get interested in the crop circle phenomenon?
Gerald Hawkins: Many years ago, I had worked on the problem of Stonehenge, showing it was an astronomical observatory. My friends and colleagues mentioned that crop circles were occurring around Stonehenge, and suggested that I have a look at them.
I began reading Colin Andrews’ and Pat Delgado’s book, Circular Evidence. I found that the only connection I could find between Stonehenge and the circles was geographic. But I got interested in crop circles for their own sake.
ML: What interested you about them?
GH: I was very impressed with Andrews’ and Delgado’s book. It provided all the information that a scientist would need to start an analysis. In fact, Colin Andrews has told me that that’s exactly what they intended to happen. I began to analyse their measurements statistically.
The major scale
ML: What did you find?
GH: The measurements of these patterns enabled me to find simple ratios. In one type of pattern, circles were separated from each other, like a big circle surrounded by a group of socalled satellites. In this case, the ratios were the ratios of diameters. A second type of pattern had concentric rings like a target. In this case, I took the ratios of areas. The ratios I found, such as 3/2, 5/4, 9/8, ‘rang a bell’ in my head because they are the numbers which musicologists call the ‘perfect’ intervals of the major scale.
ML: How do the ratios correspond with, for instance, the notes on a piano that people might be familiar with?
GH: If you take the note C on the piano, for instance, then go up to the note G, you’ve increased the frequency of the note (the number of vibrations per second), or its pitch, by 1 1/2 times. One and onehalf is 3/2. Each of the notes in the perfect system has an exact ratio — that is, one single number divided by another, like 5/3.
ML: If we were going to go up the major scale from middle C, what ratios would we have?
GH: The notes are C, D, E, F, G, A and B. The ratios are 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, finishing with 2, which would be C octave.
ML: How many formations did you analyse and how many turned out to have diatonic ratios relating to the major scale?
GH: I took every pattern in their book, Circular Evidence. I found that some of them were listed as accurately measured and some were listed as roughly or approximately measured. I finished up with 18 patterns that were accurately measured. Of these, 11 of them turned out to follow the diatonic ratios. Colin Andrews has since given me accurate measurements for one of the circles in the book that had been discarded because it was inaccurate. That one turned out to be diatonic as well. We finished up with 19 accurately measured formations, of which 12 were major diatonic.
The difficulty of hitting a diatonic ratio just by chance is enormous. The probability of hitting 12 out of 19 is only 1 part in 25,000. We’re sure, 25,000 to 1, that this is a real result.
ML: Could this in some way be a ‘music of the spheres’, so to speak?
GH: I am just a conventional scientist analyzing this mathematically. One has to report that the ratios are the same as the ratios of our own Western invention — the diatonic ratios of the (major) scale. We have only developed this diatonic major scale in Western music slowly through history. These are not the ratios that would be used in Japanese music, for instance. But I am not calling the crop circles ‘musical’. They just follow the same mathematical relationships.
Who done it?
ML: You’ve established that there’s a 25,000 to 1 chance that these ratios are random occurrences. What about natural science processes?
GH: Natural science processes, left to their own devices, like whirlwinds, rutting hedgehogs, and bacteria have no relationship to the diatonic ratios. They (the diatonic ratios) are humaninvented. They are the human response to sound. The only place I can find diatonic ratios in nature are bird calls and the song of the whale. I don’t think the birds made the circles, nor did the whales.
ML: So we’ve eliminated natural phenomena. What about Douglas Bower and David Chorley (Doug and Dave), the two Englishmen who claimed last year that they created the circles. Could they have formed these diatonic ratios?
GH: They could have, if they knew about the diatonic scale, and wished to put it in the circles. But I think we have to quote their reason for making the circles. They said they “did it for a laugh.” That’s fine. If they did it for a laugh, then it doesn’t fit with putting in such an esoteric piece of information. I did write to them. They never replied.
ML: You wrote to them saying what?
GH: “Why did you put diatonic ratios in?”
ML: And they didn’t reply.
GH: No. I think we can eliminate them. It’s so difficult to make a diatonic ratio. It has to be laid out accurately to within a few inches with a 50 foot circle, for example.
ML: And many if not all of these circles were created at night.
GH: Yes. Mostly they seem to be created at night.
Intellectual profile
ML: That eliminates natural processes and Doug and Dave. What’s left?
GH: Lord Zuckerman [former science adviser to the British Government] wrote a review of Colin Andrews’ and Pat Delgado’s book. He said that before we start building theories we should first investigate what would be perhaps the most pleasant solution for scientists, which is that the formations were made by human hoaxers. In a way, he’s not stating that that is his notion. He thinks it would be the simplest explanation. In fact, I am not supporting the theory that they are made by hoaxers. I am only investigating it.
ML: You’re investigating the theory that it’s done by hoaxers to see if that makes sense?
GH: Yes, but now I’ve upgraded the investigation, because I’ve found an intellectual profile. This means I’ve eliminated all natural science processes, so I don’t have to consider any of those any more. The intellectual profile narrows it down.
ML: What have you found in terms of this intellectual profile?
GH: My mathematical friends have commented on my findings. The suspected hoaxers are very erudite and knowledgeable in mathematics. We have equated the intellectual profile, at least at the mathematics level, as senior high school, first year college math major. That’s pushing it to a narrow slot. But there’s more to this than just the diatonic ratios.
Undiscovered theorems
ML: How so?
GH: The year 1988 was a watershed because that was when the first geometry appeared. It is in Circular Evidence. These geometrical patterns were quite a surprise to me. There are only a few of them.
ML: These are in addition to the circles you investigated in terms of the diatonic ratios?
GH: The geometry is really ‘the dog’, and the diatonic ratios of the circles are ‘the tail.’ That is, there is much more involved in the geometry than in those simple diatonic ratios in the circles, although, interestingly, the diatonic ratios are also found in the geometry, without the need for measurement. The ratio is given by logic — mind over matter.
ML: What did you find from these more complex patterns?
GH: Very interesting examples of pure geometry, or Euclidean geometry.
ML: You found Euclidean theorems demonstrated in these other patterns?
GH: These are plane geometry, Euclidean theorems, but they are not in Euclid’s 13 books. Everybody agrees that they are, by definition, theorems. But there’s a big debate now between people who say that Euclid missed them, and those that say he didn’t care about them — in other words, that the theorems are not important. I believe that Euclid missed them, the reason being that I can show you a point in his long treatise where they should be. They should be in Book 13, after proposition 12. There he had a very complicated theorem. These would just naturally follow. Another reason why he missed them was that we are pretty sure that he didn’t know the full set of perfect diatonic ratios in 300 BC.
ML: These are theorems based on Euclid’s work, but ones that Euclid did not write down himself. But they are widely accepted as fulfilling his theorems on geometry?
GH: Only widely accepted after I published them. They were unknown.
ML: Based on your analysis of these crop circles, you discovered the theorems yourself?
GH: Yes. A theorem, if you look it up in the dictionary, is a fact that can be proved. The trouble is, first of all, seeing the fact, and then being able to prove it. But there’s no way out once you’ve done that. The intellectual profile of the hoaxer has moved up one notch. It has the capability of creating theorems not in the books of Euclid.
It does seem that senior high school students can prove these theorems, but the question is, could they have conceived of them to put them in a wheat field? In this regard, we’ve got a very touchy situation in that there is a general theorem from which all of the others can be derived. I stumbled upon it by luck and accident and colleagues advised me to not publish it. None of the readers of Science News [which published an article on this subject] could conceive of that theorem. In a way, it does indicate the difficulty of conceiving these theorems. They may be easy to prove when you’re told them, but difficult to conceive.
ML: And I would assume that the readers of Science News,would be pretty well versed in these areas.
GH: It’s a pretty good crosssection. The circulation is 267,000. We found from the letters that came in that Euclidean geometry is not part of the intellectual profile of our presentday culture. But it is part of the culture of the crop circle makers.
ML: What about the more recent formations?
GH: Now we enter the other types of patterns — the pictograms, the insectograms. Exit Gerald S. Hawkins. I don’t know what to do about those.
ML: Your investigations leave off at the geometric patterns.
GH: The investigations are continuing, but I haven’t gotten anywhere. I see no recognizable mathematical features. I’m approaching it entirely mathematically, because there is the strength of numbers. There’s the unchallengeability of a geometric proof of a theorem, for example. The other patterns involve other types of investigation, such as artistry and images. But everything I’ve told you here shows that we’ve got a developing phenomenon, starting from the very simple arrangement of diatonic ratios, to a very intricate way of showing diatonic ratios in the geometries, and now to something which I think hardly anybody would claim to understand — the pictograms, insectograms, and so forth.
ML: So the major focus of your work right now is looking into these?
GH: Yes. It’s totally absorbing. It’s not a joke. It’s not a laugh. It’s not something that can be just brushed aside.
ML: Is there anybody else who is investigating it seriously in terms of your scientist colleagues?
GH: No. It boils down to two factors. You wouldn’t get a grant to study this sort of thing. And, two, it might endanger your tenure. It is as serious as that. There are whole areas in the scientific community that are not informed about the crop circle phenomenon, and have come to the conclusion that it is ridiculous, a hoax, a joke, and a waste of time.
It’s a difficult topic because it tends to raise a kneejerk solution in people’s minds. Then they are stuck. Their minds are closed. One can’t do much about it. But if they can keep an open mind, I think they’ll find they’ve got a very interesting phenomenon.

McWillis  Some are made by humans but I think some are made by ET's


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