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Replies: 86   Last Post: Jan 28, 2013 5:19 AM

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 Franz Gnaedinger Posts: 330 Registered: 4/30/07
Posted: Apr 17, 2012 7:04 AM

(intermezzo, part four)

The famous triangle was mentioned for the first time
"in the works of Chu Shi-kie (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

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.

Date Subject Author
11/17/11 Franz Gnaedinger
11/17/11 Milo Gardner
11/18/11 Franz Gnaedinger
11/18/11 Milo Gardner
11/19/11 Franz Gnaedinger
11/19/11 Milo Gardner
11/20/11 Franz Gnaedinger
11/20/11 Milo Gardner
11/20/11 Milo Gardner
11/21/11 Franz Gnaedinger
11/22/11 Franz Gnaedinger
11/22/11 Milo Gardner
11/23/11 Franz Gnaedinger
11/24/11 Franz Gnaedinger
11/24/11 Franz Gnaedinger
11/24/11 Franz Gnaedinger
11/24/11 Milo Gardner
11/25/11 Franz Gnaedinger
11/26/11 Franz Gnaedinger
12/2/11 Franz Gnaedinger
12/2/11 Milo Gardner
12/3/11 Franz Gnaedinger
12/4/11 Franz Gnaedinger
12/4/11 Milo Gardner
12/5/11 Franz Gnaedinger
12/5/11 Milo Gardner
12/7/11 Franz Gnaedinger
12/8/11 Milo Gardner
12/10/11 Franz Gnaedinger
12/12/11 Franz Gnaedinger
12/12/11 Milo Gardner
12/13/11 Franz Gnaedinger
12/13/11 Milo Gardner
12/15/11 Franz Gnaedinger
12/15/11 Milo Gardner
12/15/11 Milo Gardner
12/16/11 Franz Gnaedinger
12/16/11 Milo Gardner
12/18/11 Franz Gnaedinger
12/18/11 Milo Gardner
12/19/11 Franz Gnaedinger
12/20/11 Franz Gnaedinger
12/20/11 Milo Gardner
12/21/11 Franz Gnaedinger
12/22/11 Franz Gnaedinger
12/23/11 Franz Gnaedinger
12/24/11 Franz Gnaedinger
12/29/11 Franz Gnaedinger
1/2/12 Franz Gnaedinger
1/3/12 Milo Gardner
1/4/12 Franz Gnaedinger
11/28/11 Velev, Petyr
1/6/12 Franz Gnaedinger
1/6/12 Milo Gardner
1/9/12 Franz Gnaedinger
1/17/12 Franz Gnaedinger
1/19/12 Franz Gnaedinger
1/19/12 Milo Gardner
1/27/12 Franz Gnaedinger
2/10/12 Franz Gnaedinger
2/28/12 Franz Gnaedinger
3/2/12 Franz Gnaedinger
3/23/12 Franz Gnaedinger
3/24/12 Milo Gardner
4/9/12 Franz Gnaedinger
4/10/12 Franz Gnaedinger
4/13/12 Franz Gnaedinger
4/17/12 Franz Gnaedinger
4/18/12 Franz Gnaedinger
4/18/12 Franz Gnaedinger
5/5/12 Franz Gnaedinger
5/7/12 Franz Gnaedinger
5/7/12 Milo Gardner
5/8/12 Franz Gnaedinger
5/8/12 Milo Gardner
5/8/12 Franz Gnaedinger
5/8/12 Franz Gnaedinger
5/9/12 Franz Gnaedinger
5/10/12 Franz Gnaedinger
8/14/12 Franz Gnaedinger
1/13/13 Franz Gnaedinger
1/19/13 Franz Gnaedinger
1/23/13 Franz Gnaedinger
1/23/13 Franz Gnaedinger
1/24/13 Franz Gnaedinger
1/26/13 Franz Gnaedinger
1/28/13 Franz Gnaedinger