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.
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:
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