
Re: Ariadne's thread
Posted:
Apr 17, 2012 7:04 AM


(intermezzo, part four)
A reader asked me about Pascal's Triangle in early China. The famous triangle was mentioned for the first time "in the works of Chu Shikie (1303), the greatest of the Chinese algebraists of his time" (David Eugene Smith, History of Mathematics). Most ancient books were burned in 213 BC. We know little of the earlier achievements, but can hope to reconstruct some of them by playing with numbers, guided by what little survived the flames. I am sure the early mathematicians played with number patterns. For example the stairway of Yang numbers provides the squares:
1 sum 1 1 3 sum 4 1 3 5 sum 9 1 3 5 7 sum 16 1 3 5 7 9 sum 25
Another stairway provides the powers of 2 that play the key role in the I Ching, nothing else than the triangle that was named for Pascal but known centuries if not millennia before him:
1 sum 1 1 1 sum 2 1 2 1 sum 4 1 3 3 1 sum 8 1 4 6 4 1 sum 16 1 5 0 10 5 1 sum 32 1 6 15 20 15 6 1 sum 64
Also the Egyptian series of the Horus eye can be developed from a stairway (read 'n as 1/n):
1 = '1 1 = '2 '2 1 = '2 '4 '4 1 = '2 '4 '8 '8 1 = '2 '4 '8 '16 '16 1 = '2 '4 '8 '16 '32 '32 1 = '2 '4 '8 '16 '32 '64 '64
An Egyptian month of 30 days multiplied by the series of the Horus eye '2 '4 '8 '16 '32 '64 yields 29 '2 '32 days, one lunation or synodic month of 29 days 12 hours 45 minutes (exact value 29 d 12 h 44 m 2.9 s, average from 1989 AD).
Further conversions of 1 in another stairway:
1 = '1 1 = '1x2 '2 1 = '1x2 '2x3 '3 1 = '1x2 '2x3 '3x4 '4 1 = '1x2 '2x3 '3x4 '4x5 '5 1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6 1 = '1x2 '2x3 '3x4 '4x5 '5x6 '6x7 '7
1 = 1x2 '2x3 '3x4 '4x5 '5x6 '6x7 ...
1x2 '2x3 '5x6 '6x7 '9x10 '10x11 ... = pi/8
Complementary stairways of pi:
8 times '1x3 '16 8 times '1x3 '5x7 '32 8 times '1x3 '5x7 '9x11 '48 8 times '1x3 '5x7 '9x11 '13x15 '64 ...
4 minus 8 times '3x5 '24 4 minus 8 times '3x5 '7x9 '40 4 minus 8 times '3x5 '7x9 '11x13 '56 4 minus 8 times '3x5 '7x9 '11x13 '15x17 '72 ...
Simple yet complex  my formula for understanding the legacy of early times. I dedicate my work to the authors whose books and scrolls went up in flames in 213 BC in China, and later in the Library of Alexandria.

