Sumerian Plimpton 322
There is
good background information on the Plimpton 322 Sumerian mud tablet in
Wikipedia under Plimpton322. The modern
decimal numbers from that file were imported into an Excel worksheet and
appropriate analysis was done. It is
obvious that the information on the tablet was copied from something else which
had a great deal more calculations associated with it.
Also in
Wikipedia under Pythagorean Triples the Euclid equation for finding random
triples is provided and a proof follows.
However, this approach does not provide a specific ratio of short leg to
long leg. The primary sorting routine used on the left column of the tablet is
the square of the ratio of short to long legs.
Also at that
same location is a plot of all Pythagorean Triples which looks to be tens of
thousands at least. The lines radiating
are the result of a prime triple like 3-4-5 triangles that are multiples many
thousands of time. Below is a plot of the ratios in the left column of the
tablet, namely the ratio of short to long and then squared.
A simple computer program was written which examined all possible
Pythagorean Triples from the smallest to those greater than the
12709-13500-18541 provided on the fourth row of the tablet. The top line triple
of 119-120-169 is in the first position because the short to long ratio is a
maximum for all triples between the limits. Because the low triple is so low,
it reoccurs 109 times as integer multiples of the initial triple, the last one
in this range having a hypotenuse of 109 x 169 = 18421 which is 120 less than
the 18541.
Most of the Pythagorean Triples on Plimpton 322 tablet are prime triples. That means they are not divisable by an
integer to reach a smaller triple. An example of a non-prime would be the 45
-60-75 triangle that can be divided by 15 to become a 3-4-5 triangle. It is not
obvious that the large triangles are primes but all were checked in a computer
program. To check all these triangles required 5 to 10 minutes on a fast
computer which is making millions of calculations in seconds. Doing this without a computer would not be
possible even with dozens of people over their entire lifetime. Perhaps most
people think Plimpton 322 is just a mathematical curiosity, but the massive
effort says no.
One might be tempted to think that the Sumerians had charts made over
centuries for such calculations. Keep in
mind that a calculation with say a=12,709 and b=13,500` would result in numbers
with 9 digits. That many digits would
require at least one inch to write and would require a chart some 1,000 feet in
each direction. Since that would not be
possible one might think it would be broke into pieces, but that would take an
equal amount of numbers just to create a filing system that could be practical.
In short, the Plimpton 322 tablet information was not developed by the person
making the tablet.
When Napoleon went to the Egyptian Giza Pyramids in the eighteenth
century, he took with him several savant syndrome people for making
calculations. It may be possible that some really capable person made the
Plimpton 322 Pythagorean triple calculations, but seems really unlikely as that
type of person was generally not inclined to take on a useful task without
direction and nobody else would even know to try it. Even a savant like Daniel
Tammet (brainman) who verbalized in two books about his methods, still required
a minute or more for some calculations in his head. It would take 5 years at
that rate to do the calculations for just one large triple.
The colored columns above are what appear on the tablet only the row
column appears on the far right of the tablet. The tablet ratio of the short to
the long legs is quite precise. However,
not precise enough for the computer program.
It is amazing how many “near triple” triangles there are where a “near
integer” is only off by less than 0.001 unit. Note that the sum of the short
legs provides a number easily converted to something very close the digits of
the sun to earth average center distance. (149,598,022) in the 2000 Epoch
calculations by the Astronomical Almanac Supplement. One might be tempted to
think this just an accident, but it repeats in other locations.
In another portion of the Excel chart shown below, nearly the same number
is produced as a ratio of the dimensionless number of perimeter squared divided
by area represented by the product of the perpendicular legs.
If the design is intended to call attention to someplace inside the sun,
the numbers 149,592,333 above and 149,593,106.7 would work fine. These numbers
are not that far off from an inner sphere slightly smaller than the earth and
therefore well inside the solar radius of 695,700 km.
The far right column above is the ratio of the diagonal (hypotenuse)
divided by the short leg. The average ratio shown at the bottom in red is very
close to the Coulomb constant digits of 1.602176. If somebody wanted to suggest we think about
electrical matters, this might be a good way to do it.
Another Excel column below used the product of the two right angled legs
and then divided by the hypotenuse. The sum divided by 15 gives the average
(decimal shifted three) at the bottom which is the mass of hydrogen on an
average basis. These published values change slightly ever few years and has
been going up in recent years probably due to better accounting of the hydrogen
isotop ratios.
A few years ago the mass of the earth digits was given at the exact
reciprocal of 5.9742, both in kilograms. This might be a good way to call
attention to earth and basic chemistry of hydrogen.
The graph below shows there is far more order to the design than is
indicated in the overall initial charts above.
Clearly there are groups of triangles that relate to each other. This is
not an issue of measurements or units of measure; this is hard core mathematics
that apparently is of some very high importance to whoever focused on the model
design.
Notice that the blue line touching the top right corner of the six red
marked columns is parallel to the four green marked columns. The two curved
functions indicate that there are multiple relationships going on; perhaps many
more than found for this report.
The precision of the chart above is better explained with the actual precise
numbers in the chart below where the color coding shows the ratios that are of
the very similar equation. One must keep in mind that these are not measurements,
but numbers developed by shear mathematics.
This design is of immense complexity.
Ratio of angle ArcTan (long/short)
One should take note that the Sumerian written language used something
shaped in a triangular shape to make some of the oldest script. A couple millenia later Sumerian script used
a triangular stylus to make the head of the marks. Perhaps there is more than a
casual interest in triangles. Certainly,
whoever created this model had a passion for Pythagorean Triples.
It seems possible that the message could be pointing to “an electric sun”
instead of the more widely accepted theory of a nuclear sun. It will require
perhaps years for mankind to figure out what all might be said in this simple
mud tablet.
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©Copyright 2017
Jim Branson, retired engineering manager, knowhow at ctcweb dot net
