Re: hypothetical Yangshao calendar (early China)

From: Franz Gnaedinger (frgn_at_bluemail.ch)
Date: 03/05/05 

Date: 4 Mar 2005 23:25:22 -0800

I argued that 'my' calendar - a month of 30 days,
a year of 12 months plus 5 and occasionally 6 days,
64 lunations equal 63 months or 1890 days - stems
from Mesopotamia in the 6th millennium BC and spread
from there to Egypt, Persia, India, and China.

Horus, in ancient Egypt, was the Celestial Falcon.
His one eye was the sun, his other eye was the moon.
The moon eye was destroyed by Seth and restored by
Thoth. However, a small part was missing, and that
small part was the gap between 'moon' and month:
'64 of 30 days.

In mathematical terms, the Horus Eye was represented
by a series of unit fractions: '2 '4 '8 '16 '32 '64.
Multiply 30 days by that series and you obtain one
'moon': 30 days x '2 '4 '8 '16 '32 '64 = 29 '2 '32
days (mistake only 58 seconds).

The Horus Eye series can be developed as follows,
and then carried on ad infinitum:

  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 and so on

That principle was well known in the Old Kingdom,
while the mathematician of the Middle Kingdom had,
I believe, a second 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 etc.

The resulting series contains a fascinating subseries:

  '1x2 '2x3 '5x6 '6x7 '19x10 '10x11 ... = pi/4

A transformation of that series approximates pi/4:

  '1x3 ('16)
  '1x3 '5x7 ('32)
  '1x3 '5x7 '9x11 ('48)
  '1x3 '5x7 '9x11 '13x15 ('64)
  '1x3 '5x7 '9x11 '13x15 '17x19 ('80) and so on

A famous equivalent of the first pi-series was found by
the Indian Madhavan (ca. 1340-1425 AD)

  pi/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 ...

while the Indian Nilakantha, who flourished in the 15th
century AD, found more than a dozen pi-series, among
them a fast one:

  pi = square root of twelve, multiplied by the series
       1 - '3x3 + '5x3x3 - '7x3x3x3 + '9x3x3x3x3 ...

For those who complain about eurocentrism in the
humanities: the Indian pi-series are older than the
ones from Europe, where the Madhavan series is named
for Gregory, and the pyramid builders of ancient Egypt
discovered a systematic method for calculating the
circle on the basis of the Sacred Triangle 3-4-5:

  www.seshat.ch/home/egypt2.htm

Preview: The number 64 in China / A second calendar
modifies the first one / More on the second calendar /
Paving the way for the lunisolar calendar of 19 years,
known as Chang cycle in China, and Metonic cycle in
Europe. All messages ready, delivered on a daily basis,
new ideas galore, and you get my work for free.
-
Regards Franz Gnaedinger

> Chinese yue = English moon/month = German Mond/Monat.
>
> A lunation, for example from one to the next full moon,
> lasts 29.53058912 days. A calendar requires natural
> numbers. 29.53... lies between 29 and 30, but a little
> closer to 30, which number has the advantage of being
> divisible by several smaller numbers apart from 1:
> 2, 3, 5, 6, 10, and 15. So it makes sense to replace
> a moon or lunation by a month of 30 days.
>
> 12 months yield 360 days. Add 5 and occasionally 6 days
> and you obtain a year of 365 and sometimes 366 days.
>
> Add continual periods of 30 days and observe the moon
> for years. If you are a good astronomer and make records
> you may come across a fine relation: 64 moons equal 63
> months or 1890 days (mistake only about one hour).
>
> >From this relation you may calculate a moon: 30 days
> multiplied by a factor of 63/64. Now that factor equals
> the famous Egyptian series of the Horus Eye. Hence
>
> 1 moon = 30 days times '2 '4 8 '16 '32
>
> = 29 '2 32 days (mistake only 58 seconds)
>
> Horus was the Celestial Falcon. His one eye was the sun,
> his other eye was the moon. Seth destroyed his moon eye,
> whereupon Thoth restored it. However, a small part was
> missing, and this small part turns out to be a duration,
> namely the gap between moon and month: '64 of 30 days.
>
> Where and when did 'my' calendar originate? I plead for
> Mesopotamia in the 6th millennium BC, from where it may
> have spread to Egypt, Persia, China, and India.
>
> Have a look at a good reproduction of the famous plate
> from the youngest Halaf level of Tell Arpadjije (near
> Ninive) from around 5500 BC. The central rosette counts
> 32 leaves: double the number and you obtain 64. On the
> outmost rim are 36 white crosses: multiply them by a
> long week of 10 days and you obtain 360 days or 12
> months.
>
> Next time: the number 64 in Egypt
> -
> Regards Franz Gnaedinger www.seshat.ch
>
>
> > My hypothetical Horus / Yangshao / Liangzhu calendar
> > is based on a month of 30 days, a year of 12 months
> > plus 5 and occasionally 6 days, and a cycle of 64
> > lunations that equal 63 continual months = 1890 days.
> > One lunation is obtained by multiplying a month of 30
> > days by the famous Egyptian series of the Horus Eye:
> >
> > 1 lunation = 30 days x '2 '4 '8 '16 '32 '64
> >
> > = 29 '2 '32 days (mistake only 58 seconds)
> >
> > The Chinese Bi, from around 3400 till 2000 BC, are flat
> > jade disks, diameters between 10 and 30 cm, with round
> > perforations of a largely variable size in the center.
> > The Bi is considered a symbol of the sky. I came across
> > five Bi, measured them, and found numbers that may refer
> > to crucial numbers of my calendar, especially 30 and 64.
> > So a Bi could actually be a symbol of the sun and moon,
> > the month of 30 days and the cycle of 64 lunations.
> >
> > FIRST BI: radius perforation / radius disk = 1/15
> > ring perforation ring = 14 1+1 14 or 7 1 7
> > diameter disk = 30 or 15; small grid 7 1 7 by 7 1 7
> > rotating square 7 1 by 7 1 = 8 x 8 = 64
> >
> > SECOND BI: radius hole / radius disk = 1/5 or 3/15
> > ring perforation ring = 4 1+1 4 or 12 3+3 12
> > diameter disk = 10 or 30; small grid 4 2 4 by 4 2 4
> > four corner squares of small grid = 4 x 4 x 4 = 64
> >
> > THIRD BI: radius perforation / radius disk = 1/4
> > ring perforation ring = 3 1+1 3
> > diameter 8, square 8 x 8 = 64
> >
> > FOURTH BI: radius perforation / radius hole = 1/3
> > ring hole ring = 2 1+1 2 or 4 2+2 4 or 10 5+5 10
> > diameter = 6 or 12 or 30; medium grid 4 4 4 by 4 4 4
> > four corner squares of medium grid = 4 x 4 x 4 = 64
> >
> > FIFTH BI: radius perforation / radius hole = 7/8
> > ring perforation ring = 8 7+7 8 or 4 7 4
> > diameter disk 30 or 15; small grid 4 7 4 by 4 7 4
> > four corner squares of small grid = 4 x 4 x 4 = 64
> >
> > The idea of a square grid going along with a BI
> > seems justified by the early Chinese sign of heaven:
> > a man with a square for a head.
> >
> > Next time: answering Paul's question
> > -
> > Regards Franz Gnaedinger www.seshat.ch

Relevant Pages