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Topic: RMP # 48
Replies: 70   Last Post: Jul 23, 2010 8:57 AM

 Messages: [ Previous | Next ]
 Franz Gnaedinger Posts: 330 Registered: 4/30/07
Re: RMP # 48
Posted: Oct 6, 2007 5:37 AM

I think it's a popular misunderstanding among math historians
that the Egyptians had no idea of irrational numbers. What
they had was a different idea of such numbers: we can't point
point them out exactly, but we can calculate with a 'cloud' or
'halo' of rational numbers around those elusive numbers ...
The supreme god Ra, one of whose emanations was the sun,
and whose hieroglyph was a small circle with a dot in the center,
had a hundred names (if memory serves), but nobody knew his
real name, not even his own daughter. This idea survives in Islam:
Allah got 99 names but nobody knows the true name ...

Also the Greeks were calculating with rational numbers only,
and we are doing the same: all our values for the square root
of 2, and for pi, are rational, even the value of pi Yasumada
Kanada calculated to over fifty million places - just a rational
number.

The Egyptians accepted this basic fact: we can only handle
rational numbers. And at the same time they went beyond:
we can approximate irrational numbers with algebraic and
geometric methods. - Calculate the square root of 2 using
the trial and error method. You'll inevitably find better
and better approximations that reveal a simple pattern,
the pattern of a number column I reconstructed in 1979
while examining the geometry of Leonardo da Vinci's
_Last Supper_ at Milan. This number column allows to
find ever better values for the square root of 2 without
reaching the true value (the numbers are endlessly
increasing, and soon not even a computer can process
them any more). What about the circle? Picture a square
measuring 10 by 10 royal cubits or 70 by 70 palms.
The diagonals measure practically 99 palms. Draw a circle
around the center of the grid. The radius measures 5 royal
cubits, while the circumference will pass the ends of the
four axes, plus eight points of the grid according to the
'sacred triangle' 3-4-5, all in all twelve points of the grid,
which divide the circumference into as many arcs. Measure
them and you'll find eight arcs measuring pratically 90
fingers each, and four arcs measuring practically 40
fingers each, yielding a circumference of practically 880
fingers. Divide this length by the diameter 10 royal cubits
or 70 plams or 280 fingers and you get 22/7 or 3 1/7 or
3 '7 for pi ... Then you can go a step further. Subdivide
the grid 10 by 10 cubits into a grid of 50 by 50 smaller units.
Now the circumference will pass four more points of the grid
which are defined by the triple 7-24-25 ... Go on to the grids
250 by 250, 1250 by 1250 ... ever smaller units, and the
circumference of the circle will each time pass four more
points of the grid. By connecting the points with straight
lines you get a sequence of polygons that grow ever closer
to the cirlce, and their calculation requires nothing more
than the approximations of the square roots of 2 and 5,
which can be done with the help of analoguous number
colums. The full method including drawings of the polygons
can be found here: http://www.seshat.ch/home/rhind1.htm
I seriously believe that the Egyptian mathematicians of the
Old Kingdom were the first to calculate the circumference,
area, and number of the circle in a systematic way, over
two millennia prior to Archimedes.

Regards, Franz Gnaedinger

Date Subject Author
9/30/07 L. Cooper
10/1/07 Franz Gnaedinger
10/1/07 L. Cooper
10/1/07 Milo Gardner
10/1/07 L. Cooper
10/2/07 Milo Gardner
10/3/07 Milo Gardner
10/3/07 L. Cooper
10/3/07 Franz Gnaedinger
10/4/07 Milo Gardner
10/4/07 Milo Gardner
10/4/07 Milo Gardner
10/4/07 Milo Gardner
10/4/07 Milo Gardner
10/4/07 Milo Gardner
10/4/07 L. Cooper
10/5/07 L. Cooper
10/5/07 Milo Gardner
10/6/07 L. Cooper
10/6/07 Franz Gnaedinger
10/6/07 Milo Gardner
10/6/07 L. Cooper
10/6/07 Franz Gnaedinger
10/6/07 Milo Gardner
10/7/07 Franz Gnaedinger
10/7/07 Milo Gardner
10/7/07 Franz Gnaedinger
10/7/07 Milo Gardner
10/8/07 Franz Gnaedinger
10/8/07 Milo Gardner
10/8/07 Franz Gnaedinger
10/8/07 Milo Gardner
10/9/07 Franz Gnaedinger
10/9/07 L. Cooper
10/10/07 Franz Gnaedinger
10/11/07 Franz Gnaedinger
10/11/07 L. Cooper
10/12/07 Franz Gnaedinger
10/12/07 Franz Gnaedinger
10/12/07 L. Cooper
10/13/07 Franz Gnaedinger
10/13/07 L. Cooper
10/13/07 Franz Gnaedinger
10/15/07 Franz Gnaedinger
10/18/07 Ed Wall
10/19/07 Franz Gnaedinger
10/20/07 Milo Gardner
12/5/07 Milo Gardner
12/14/07 Franz Gnaedinger
12/14/07 Milo Gardner
12/14/07 Milo Gardner
10/11/07 Milo Gardner
10/12/07 L. Cooper
10/12/07 Milo Gardner
10/13/07 L. Cooper
7/11/10 Dioxippus
7/12/10 Milo Gardner
7/20/10 Dioxippus
7/21/10 Milo Gardner
7/21/10 Dioxippus
7/21/10 Milo Gardner
7/23/10 Dioxippus
7/23/10 Milo Gardner
10/6/07 Hossam Aboulfotouh
10/8/07 Milo Gardner
10/11/07 Hossam Aboulfotouh
10/12/07 Milo Gardner
10/13/07 Hossam Aboulfotouh
10/13/07 Hossam Aboulfotouh
10/19/07 Hossam Aboulfotouh
10/27/07 Matt Hugh