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Oral session, Wednesday, January 07
[35.06] From Euclid to Ptolemy in English Crop Circles G. S. Hawkins (Boston University Research) The late Lord Soli Zuckerman, science advisor to several British governments, encouraged the author, an astronomer, to test the theory that all crop circles were made by hoaxers. Within the hundreds of formations in Southern England he saw a thread of surprising historical content at the intellectual level of College Dons. One diagram in celestial mechanics involved triple conjunctions of Mercury, Venus and Mars every 67 2/3 years. Ptolemy's fourth musical scale, tense diatonic, occurred in the circles during the period 1978-88. Starting on E, Ptolemaic ratios make our perfect diatonic scale of white notes on the keyboard of the piano or church organ. For separated circles the ratio was given by diameters, and for concentric circles it was diameters squared. A series of rotationally symmetric figures began in 1988 which combined Ptolemy's ratios with Euclid's theorems. In his last plane theorem, Euclid (Elements 13,12) proved that the square on the side of an equilateral triangle is 3 times the square on the circumcircle radius diatonic note G. From the 1988 figure one can prove the square on the side is 16/3 times the square on the semi-altitude, giving note F(3). Later rotational figures over the next 5 years led to diatonic ratios for the hexagon, square and triangle. They gave with the exactness of Euclidean theorems the notes F, C(2) and E(2), and they are the only regular polygons to do so. Although these 4 crop theorems derive from Euclid, they were previously unknown as a set in the literature, nor had the Ptolemaic connection been published.