Golygon Letter to Dr. Michael W. Ecker

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