Free Alien Dawn: A Classic Investigation into the Contact Experience by Colin Wilson Page B

and totally unnoticeable in a rock group.
    But after 1986 things changed. From this time on circles appeared with rings around them. Hawkins found the simple fraction was now given by the area of the ring divided by the area of the circle. From schooldays we recall that ancient area formula, pi-r-squared. Modern computer experts would say, ‘Ha! Data compression. We get a larger ratio from the same sized pattern’.
    The first step in his reasoning was that, if the first circles had been made by Meaden’s vortices, then all patterns involving several symmetrical circles must be ruled out, since a whirlwind was not likely to make neat patterns. Which meant that the great majority of circles since the mid-1980s must be made by hoaxers. But would hoaxers take the trouble to give their patterns the precision of geometrical diagrams? In fact, would they even be capable of such precision, working in the dark, and on a large scale? They would be facing the same kind of problem as the makers of the Nazca lines in the desert of Peru—the great birds, spiders and animals drawn on the sand—but with the difference that the Nazca people worked by daylight and had an indefinite amount of time at their disposal, whereas the circle makers had to complete their work in the dark in a few hours.
    Hawkins began by looking closely at a ‘triplet’ of circles which had appeared at Corhampton, near Cheesefoot Head on 8 June 1988. To visualise these, imagine two oranges on a table, about an inch apart. Now imagine taking a small, flat piece of wood—like the kind by which you hold an ice lolly—and laying it across the top of them. Then balance another orange in the centre of the lolly stick, and you have a formation like the Cheesefoot Head circles of 1988 .
    Now even a nonmathematical reader can see that the lolly stick forms what our teachers taught us to call a tangent to each of the oranges. And, since all the oranges are spaced out equally, you could insert two more lolly sticks between them to make two more tangents. The three sticks would form a triangle in the space between the three oranges.
    That is a nicely symmetrical pattern. But, of course, it does not prove that the circle makers were interested in geometry. Perhaps they just thought there was something pleasing about the arrangement.
    When he had been at school, Hawkins had been made to study Euclid, the Greek mathematician, born around 300 BC, who had written the first textbook of geometry. Euclid is an acquired taste; either you like him or you don’t. Bertrand Russell had found him so enjoyable that he read right through the Elements as if it were Alice in Wonderland.
    As an astronomer, Hawkins had always appreciated the importance of Euclid, in spite of having been brow beaten with him at school. So now he began looking at his three circles, and wondering if they made a theorem. He tried sticking his compass point in the centre of one circle, and drawing a large circle whose circumference went through the centres of the other two circles. He realised, to his satisfaction, that the diameter of the large circle, compared with that of the smaller ones, was exactly 16 to 3.
    So now he had a new theorem. If you take three crop circles, and stick them at the corners of a small equilateral triangle, then draw a large circle that passes through two centres, the small circles are just three sixteens the size of the larger one.
    He looked up his Elements to see if Euclid had stumbled on that one. He hadn’t.
    Of course, it is not enough to work out a theorem with a ruler: it has to be proved. That took many weeks, thinking in the shower and while driving. Eventually, he obtained his proof—elegantly simple.
    After this success, he was unstoppable. Another crop-circle pattern, in a wheatfield near Guildford, Surrey, showed an equilateral triangle inside a circle—so its vertices touched the circumference—then another circle inside the triangle. Hawkins soon worked out that