To: Dr. Michael W. Ecker 92-02-27 REC, 909 Violet Terrace, Clarks Summit, PA 18411 From: Harry J. Smith, 19628 Via Monte Dr., Saratoga, CA 95070 Enclosure: 1) MS-DOS disk (ButterFy.Bas and .Exe Plus MorseTue.Bas and .Exe) 2) Copy of letter to A. K. Dewdney on Golygons 3) MS-Dos disk (Golygons.Pas, .Exe) Dear Mike, Butterflies: I did not include the .Exe files with my delivery of the ButterFly BASIC programs because I did not have a copy of QuickBASIC, only had QBASIC. I now have QuickBASIC 4.5 so I can send them with this letter. Morse-Thue: I am also sending some QuickBASIC programs based on an article written by Dr. Clifford A. Pickover in Algorithm 3.1 about the Morse-Thue binary sequence. Serial Isogons of 90 Degrees: Dr. A. K. Dewdney had an article about serial isogons of 90 degrees (he called them Golygons) in Scientific American July 1990. It was based on "Serial Isogons of 90 Degrees." by Lee Sallows, Martin Gardner, Richard K. Guy, and Donald Knuth in Mathematics Magazine. Edited by G. L. Alexanderson (in press). The actual article is now available in The Mathematical Association of America's Mathematics Magazine, Vol. 64, No. 5, December 1991. << Correction: The word "golygon" is Lee Sallows invention and not Kee Dewdney's as I stated. Lee corrected me via email on January 20, 2001 >> The Scientific American article challenged the reader to find extensions of Golygons. For example, Kee asked "Can we find prime-sided golygons? The lengths of consecutive sides here increase by the sequence of odd primes: 1, 3, 5, 7, 11, 13 and so on." I found two 16-sided solutions and documented them in a letter to Dewdney in July of 1990 (copy of letter enclosed). I sent a copy of this letter to Dr. Knuth at Stanford University at the same time. Knuth must have kept my name in a data base of people interested in serial isogons of 90 degrees, because last week he sent me a reprint of the article published in December 1991. The final article clears up some details, particularly about how to calculate the exact number of serial isogons of any given order. My extensions to serial isogons allowed the first (smallest) side to be larger than one, but requires consecutive sides to be in serial order. Another extension only required the sides to be consecutive prime numbers, and I found some 12- sided serial prime isogons of 90 degrees. My final extension required the sides to be consecutive primes, all of which are twin primes. The path: 663569N 663571E 663581S 663583W 663587S 663589W 663599N 663601E is the smallest serial twin prime isogon of 90 degrees. Sincerely, Harry

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