River Map Ho-T’u and Magic Square Lo-shu of Early
China

© 2012 by Franz Gnaedinger

Illustrations

River Map and Magic Square, from: James Legge, *The I Ching, *Clarendon Press 1899, Dover
Publications New York 1963 (a very fine book that inspired this paper)

The same square and practically the same map are given by David Eugene Smith in *A History of Mathematics*, 1923, Dover
reprint 1953/58 (richly illustrated, one of my favorite sources), only that the
five and five dark circles of the earth below and above the heavenly domino
five are, for the sake of clarity, connected with a rectangle. A modified
reconstruction of the lost diagram is proposed in this paper

The circle might keep a memory of a Neolithic
calendar sanctuary, a circle of poles providing sighting lines for the rising
and setting sun on the solstices (and equinoxes), on the right side the morning
sun of the spring equinox that just rose over the horizon, somewhere on the
northern part of the Ordos Plateau, a couple of millennia ago

Part 1

Mapping the River of Time – a lunisolar calendar
encoded in the Yellow River Map of ancient China that inspired the I Ching or
Book of Changes

The Stone Age way of counting lunations was to
lay out pebbles, 30 29 30 29 30 29 30 … pebbles for 1 2 3 4 5 6 7 … lunations
or synodic months.

15 and 17 lunations counted that way yield 443
and 502 days respectively, together 945 days for 32 lunations, doubled 1890 days
for 64 lunations.

The Göbekli Tepe lunisolar calendar had a month
of 30 days and a basic year of 12 months or 360 days. Add 3 days of midsummer
and 2 days of midwinter and you get a regular year of 365 days. Add 3 midwinter
days instead and you get an occasional leap year of 366 days.

63 periods of 30 days are 1890 days and
correspond to 64 lunations, as found above. One lunation, calculated that way,
lasts 29 days 12 hours 45 minutes, exact value 29 d 12 h 44 m 2.9 s (average
from 1989 AD), mistake less than one minute per lunation, or half a day in a
lifetime.

Now for the River Map Ho-T’u. A dragon-horse
emerged from the Yellow River Ho, bearing a map on its back, the Yang numbers 1
3 5 7 9 given in light circles and the Yin numbers 2 4 6 8 10 given in dark
circles arranged in a square, inner square 1 3 2 4 sum 10, outer square 6 8 7 9
sum 30, individual numbers given as lines of light or dark circles, beginning
with 1 at the bottom of the inner square, in clockwise direction. The center is
held by a domino five in light circles, below and above a line of five dark
circles each, together the Yin number 10.

The number 10 represents a basic year of ten
periods of 36 days, in all 360 days. Add the vertical line of the domino five
for 3 midsummer days and the remaining circles for 2 midwinter days and you get
a regular year of 365 days. If you add instead the horizontal line of the
domino five for 3 midwinter days you obtain an occasional leap year of 366
days.

25 years require 6 leap years (in all 9131
days). The number 6 is given as sum of the vertical line and horizontal line of
the domino five (central circle counted twice). The number 25 is the sum of the
Yang numbers 1 3 5 7 9.

The number 36 of the calendar period is the sum
of the numbers 1 2 3 4 5 6 7 8. The square of 8 equals 64. Outer square of the
River Map 6 8 7 9 sum 30. The product of the Yang numbers 7 and 9 equals 63,
times 30 days 1890 days or 64 lunations or 2 by 2 by 2 by 2 by 2 by 2
lunations, mirrored in the cycle of 64 binary hexagrams in the I Ching.

64 lunations correspond to 63 periods of 30
days, 128 lunations to 126 periods of 30 days or 105 periods of 36 days,
yielding the l/p ratio 126/105. Simpler ratios are 6/5 or 12/10 and 11/9. Generating
additive number sequences of more l/p ratios:

6/5
(plus 11/9) 17/14 28/23
39/32 50/41 61/50 …

Best value: 50 periods of 36 days for 61
lunations.

12/10
(plus 11/9) 23/19 … 78/64 89/73 …

Best value: 73 periods of 36 days for 89 lunations.

89/73
(plus 128/105) 217/178 345/283
473/388 …

Best value: 388 periods of 36 days for 473
lunations, mistake not even six seconds per lunation, a good hour in a
lifetime.

The period of 36 days offers many practical,
good, very good and excellent values for counting lunations over long periods
of time – mapping the river of time, as it were.

(The River Map is a reconstruction. I’d like to
make a suggestion: instead of halving the Yin number 10 one might arrange it as
a circle around the domino five. Outer square 20 by 20, inner square 16 by 16,
diameter of a small circle 1, distance between the centers of two small circles
on a straight line 2, radius of the circle of the basic year 4, diameter 8,
outer diameter 9, inner diameter 7

)

The River Map Ho-T’u was complemented by the
Magic Square Lo-Shu on the back of a tortoise that also emerged from the Yellow
River Ho (David Eugene Smith)

4 9 2

3 5 7

8 1 6

5 heaven
(domino five)

1 north
and winter and water

3 east
and spring and wood

9 south
and summer and fire

7 west
and autumn and metal

2 earth

4 wood

6 metal

8 earth

The Magic Square may be seen as a world map.
Time is present insofar as the cardinal directions go along with the seasons of
the year.

Part 2

Where is the river in the Yellow River Map?

I explained how the Yin number 10 can be
arranged as a circle around the domino five in the center of the map. Now
imagine that circle in motion, turning in clockwise direction, round and round,
making the seasons change, from winter to spring to summer to autumn, from cold
to warm to hot to cool, bringing about changes of every sort – as the waters in
a river rise and sink and rise in perpetual motion.

The number 10 of the river circle is also the
sum of the pairs opposing each other via the heavenly domino five in the Magic
Square

4 9 2

3 5 7

8 1 6

1 and 9, north and south, winter and summer,
water and fire; 3 and 7 east and west, spring and autumn, wood and metal; 4 and
6, wood and metal; 2 and 8, earth. East and west also invoke morning and
evening. Wood and metal are organic and anorganic materials. Earth makes the
plants grow and in turn takes up the fallen leaves. Consider also the colors,
the yellow of dawn and violet of dusk, blue water and orange fire, green leaves
and rusty red iron ore, turning into a glowing red when molten with fire made
of wood … The elements of the Magic Square, paired in opposites, evoke a host
of meanings.

We could say that the eight numbers around the
heavenly domino five in the center of the Magic Square are the frozen

Here again the numbers of the modified
reconstruction of the Ho Map: outer square 20 by 20, inner square 16 by 16, diameter
of a small circle 1, distance of the centers of two adjacent small circles on a
straight line 2, radius of the river circle 4, diameter 8. Those are middle
numbers, relevant for the positions of the centers of the small circles. 8 is
the middle diameter of the river circle. The outer diameter is 9, the inner
diameter 7.

7 8 9 play a role in practical methods of
squaring the circle in the Rhind Mathematical Papyrus of Ancient Egypt. A
circle of diameter 9 and a square of side 8 have practically the same
area.

A small drawing shows a square with
a virtual or imaginary grid of 3 by 3 squares whose corner squares are halved
along diagonals, generating a quasi-circle, an octagon that is close to the
circle inscribed in the square 3 by 3. Also the numbers 1 2 3 4 5 6 7 8 9 of
the Magic Square fill a grid of 3 by 3 squares. Their sum is 45. Halve the
numbers of the corner squares 2 4 6 8 and the sum is reduced to 35. The ratio
45 to 35 equals 9 to 7.

The Egyptians of the Old Kingdom developed a
systematic method for the calculation of the circle on the basis of the Sacred
Triangle 3-4-5. The numbers of this triangle are prominent in the River Map: 3
on the left side of the inner square, 4 on the right side of the inner square,
5 in the center.

The first Egyptian circle of
radius 5 in the square 10 by 10 is defined by the triple

While the calendar concerns time, the squaring
of the circle concerns a change of form, a shift of shape, justifying the
‘River of Time and Change’ that could also have named the circle of hexagrams
in the I Ching. There is a rational core in the Chinese instrument of
divination.

Part 3

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

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

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.

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 9 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.

Postscript

A Neolithic
calendar sanctuary on the Ordos Plateau?

The central part of the modified River Map
could have served as a Neolithic calendar sanctuary on the northern
(northwestern) part of the Ordos Plateau (geographical latitude around 40
degrees, perhaps nine thousand years ago). A circle of ten poles would have
allowed to predict sunrises and sunsets on the solstices (and equinoxes), a
flat horizon and a numerical model of the year provided. The rhomb is given by
the sighting lines of the solstices. (Numbers of the drawing: middle radius 34,
middle diameter 68, outer diameter 72, inner diameter 64, diameter pole 4, long
axis 2 x 55 = 110, short axis 2 x 40 = 80, side of rhomb 13 21 21 13 sum 68.) The morning sun of the spring
equinox just rose over the eastern horizon (tiny yellow circle on the right
side)