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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: May 9, 2012 2:35 AM

River Map HoT'u and Magic Square Lo-Shu
of Early China (revised and illustrated version)

4) Unfolding Cosmos

Where did the Yang line and Yin 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
the 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 90, 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 diameter of the unit circle.
Begin with 4/1 and add repeatedly 3 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 9 and a square of side 8 have practically
the same area, implicit pi-value 256/81.
http://www.seshat.ch/home/china05.GIF

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 of a square
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 double 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 value for the square root of pi 576/325,
an excellent value from the sequence

16/10 (plus 16/9) 32/19 48/28 ... 576/325 592/334 (296/167)

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.

(to be continued)

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