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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 13, 2012 4:28 AM

(intermezzo, third part)

Unfolding Cosmos

Where did the Yang line and Ying line come from?
A Chinese author by the name of Ku Shi said their
origin was a circle. The circle divided and produced
the whole line of the Yang and the divided line of
the Yin, which lines, in turn, produced all things ...
The initial circle may then have been the empty
circle Wu of the primeval cosmos before the
division into the Yang of heaven and Yin of earth.

Let me imagine a myth of creation as it might have
been told in the Paleolithic or Mesolithic or Neolithic
settlements on the Ordos plateau in the wide northern
curve of the Yellow River.

In the begin was the empty circle Wu. Then the circle
divided. The upper half became the Yang, manifest in
heaven, and the lower half became Yin, manifest in earth.
Hereupon the Tree of Life grew out of the earth, marking
the center of the world - here, in the center of earth,
under the center of heaven - and dividing the world into
north and south, east and west, appearing as a cross
or domino five when imagined from above, and as a pair
of lines when imagined from the side, a whole upper line
and a divided lower line, the center open for the Tree of Life.

Yin and Yang produced all things, and when we play with
the numbers and forms of the (modified) River Map and
the Magic Square we see a mathematical cosmos unfold.

1 1 2, 2 3 4, 5 7 10, 12 17 24, 29 41 58, 70 99 140 ...
these are numbers of the square and octagon.

1 1 3, 2 4 6, 1 2 3, 3 5 9, 8 14 24, 4 7 12, 11 19 33, 30 52 45,
15 26 45 ... these are numbers of the equilateral triangle,
hexagon and cube.

1 1 5, 2 6 10, 1 3 5, 4 8 20, 2 4 10 1 2 5, 3 7 15, 10 22 50,
5 11 25, 16 36 80, 8 18 40, 4 9 20 ... these are numbers
of the double square.

3 4 5, the numbers of the Sacred Triangle, prominent in
the River Map, start a sequence of ever rounder polygons
whose peripheries can be calculated with the numbers
of the square and double square.

4 1 1 are the numbers of the periphery, the horizontal
and vertical axis of the unit square. 3 1 1 are the numbers
of the simplified circumference, horizontal and vertical
diameter of the unit circle, reflected in the Yang number 9
and Yin number 3 3, also in the Yang number 36 and
Yin number 12 12 of the I Ching. The 64 hexagrams
count 192 Yang lines and 192 Yin lines, yielding 11,520,
"the number of all things" (I Ching, Appendix III 53).

4 1 are the periphery and axis of the unit square, 3 1
the simplified circumference and the diameter of the
unit circle. Begin with 4/1 and add repeatedly 30 to 4,
and 1 to 1 :

4/1 (plus 3/1) 7/2 10/3 13/4 16/5 19/6 22/7 25/8 28/9

3/1 (plus 22/7) 25/8 ... 311/99 ... 377/120

9/3 (plus 19/6) 28/9 ... 256/81

The grid 3 x 3 has the area 9. If you halve the corner
squares along diagonals, in such a way that you get
an octagon, you have a good first approximation of the
circle inscribed in the square, area of the quasi-circle 7,
implicit pi-value 28/9. A circle of diameter 8 and a square
of side 8 have practically the same area, implicit pi-value
256/81.

2 3 2 sum 7 - make a rod of that length. 4 3 4 sum 11 -
make a second rod of this length. If the diameter of a circle
measures one short rod, the circumference measures
two long rods. If the radius of another circle measures
one short rod, the area equals two short rods by one
long rod. If the side of a square measures ten short rods,
the diagonal measures nine long rods, and if the side
measures nine long rods the diagonal measures twenty
short ones. Three short rods are the golden minor of
five long rods, in numbers 21 and 55, from the Fibonacci
sequence 1 1 2 3 5 8 13 21 34 55 89 144 ..., a sequence
present in the above numbers of the square, as are the
complementary Lucas numbers 1 3 4 7 11 18 29 47 76
123 199 322 ...

11,520 is the number of all things according to Appendix
III 53 of the I Ching. Why that specific number? One
reason was given above, the other reason is a challenging
problem of an amazingly simple answer. Imagine a circle
of the circumference 11,520. How long is the periphery
of the square of the same area? 13,000. Implicit pi-value
576/325, an excellent value from the sequence

16/10 (plus 16/9) 32/19 48/28 ... 576/375

We observe a mathematical cosmos unfold from the River
Map Ho-T'u and the Magic Square Lo-Shu and evolve in time,
on the way to the I Ching and the important third appendix.

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