Reprint Granted: April 14, 05
Dr. Colette M. Dowell ND
Moving Forward Publications
Note: This is a historical article.
Prof. GERALD S. HAWKINS CROP CIRCLE FIFTH THEOREM CHALLENGE
By Dr. Colette M. Dowell
Prof. Gerald S. Hawkins' diatonic geometrical theorems which he derived from his mathematical and geometrical analyses on crop circles, has been recently published in Volume 88, Number 9: December 1995, of the highly-acclaimed academic journal 'Mathematics Teacher,' : The National Council of Teachers of Mathematics.
This is a great achievement for all concerned Croppies and for the educational world at large. Crop circles, thanks to Prof. Hawkins, are now getting recognition in the so-called more sophisticated scientific academia realm.
Prof. Hawkins' geometrical theorems are Euclidean in nature but express diatonic ratios. ( Diatonic ratios are the ratios used to create the diatonic scale of music.) Through tabulating the diameters and areas of the earlier crop circle formations (eg., plain circles, circles with satellite circles and circles with
concentric rings), he discovered that the only regular polygons inscribed or circumscribed by a circle conveying diatonic ratios are the triangle, square and the hexagram. They are new theorems which Euclid had not discovered. However, Gerald being a Euclidean buff, can precisely indicate where the theorems should
have been listed and placed within the contents of Euclid's Elements.
Gerald has found five theorems. Theorems I-IV can be derived from the V theorem. He has a challenge for you. His challenge is to figure out what theorem V is and to prove it by Euclidean geometry, not trigonometry. His challenge was first
published in Science News (1992) but there were no correct answers to his puzzle and as of yet there has been no one who has figured it out. Remember theorems I-IV must be derived from theorem V.
Below is a chart I created of the diatonic ratios. The scale is based on seven intervals. The eighth interval is the octave (or twice the value of the first interval).For means of simplicity I will diagram the diatonic ratios in intervals of seven so that the ratio directly below any given ratio will be double the value, or known as the octave.
Note C D E F G A B
Interval 1 2 3 4 5 6 7
Oct. 1 (C) 1:1 9:8 5:4 4:3 3:2 5:3 15:8
Oct. 2 (C') 2:1 18:8 10:4 8:3 6:2 10:3 30:8
Oct. 3 (C'') 4:1 36:8 20:4 16:3 12:2 20:3 60:8
AND SO ON AND SO ON ............
Let three equal circles that share a common tangent make an equilateral triangle. If a circle is drawn through the centers of two circles, concentric with the third circle, then it can be proved that the ratio of areas of the concentric circles is 16:3, corresponding to the musical note F''. The two circles taken as satellites give the note C. The primes (') indicate the
number of octaves above the first octave.
For an equilateral triangle, the ratio of areas of the circumscribed and the inscribed circles (4:1) gives C''. The annulus between the circles when divided by the area of the inscribed circle gives G'.
For a square, the ratio of the areas of circumscribed and inscribed circles (2:1) gives C' and the annulus gives C. Four applications of this theorem give a reduction of 16, and the notes become C'''' and B'''.
For a regular hexagon, the ratio of areas of circumscribed and inscribed circles (4:3) gives F.
Theorems II-IV are a special set within the family of regular polygons---Only the triangle, square, and hexagon will give a diatonic ratio from the circumscribed and inscribed circles. Theorem V is a general theorem from which I-IV can be derived.
OK, did you get that? HA! For me these theorems have been quite maddening and ponderous, driving me insane trying to figure them out, but don't let that stop you. If you think you have figured it out at some point, write and diagram it all out (making sure you can prove theorem V by Euclidean geometry), with your
name, return address and any comments to Circular Times.
I will forward all results to Prof. Hawkins. Have fun!
Dr. Colette M. Dowell
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Send comments, suggestions and questions to: Dr. Colette M. Dowell at: Luminescence@verizon.net