The Great Pyramid of Giza, also called Pyramid
of Cheops or Pyramid of Jufu, was built to be the tomb of the pharaoh Jufu
(called by the Greeks Cheops), of the fourth dynasty, the high point of the
Ancient Egyptian Empire. The reign of Jufu is usually dated in the 26th century
before Christ, over 4500 years ago.
The current height of the Great Pyramid is
138.8 meters, but the pyramid is truncated, having lost its top. It is easy to
calculate that its original height was about 8 meters higher: 146.7 meters, or
280 Egyptian cubits. The base of the pyramid is a square with a side of 230.34
meters, or 440 Egyptian cubits.
Observe a curious point: the semi-perimeter of the pyramid
(twice the side of the base) is equal to 880 cubits. If we divide it by
the height of the pyramid, we get the following:
(880/220) = (22/7) = 3.142857...
This is a fairly good approximation of the
value of p, supposedly unknown until many
centuries later, when it was discovered by Archimedes. In fact, it is much
better than that of the Rhind papyrus, which we know was later by about seven
hundred years, an approximation I mentioned in the
previous post. What can we deduce from it? There are at least four
possibilities:- The Egyptian builders
of the fourth dynasty knew the approximation of p as the fraction 22/7, discovered two millennia
later by Archimedes. When the Old Empire collapsed and gave way to the
First Intermediate Period, a dark age, they would have lost that
mathematical knowledge, so that the builders of the subsequent Middle
Kingdom just knew a worse approximation of p.
- It is pure chance. In the second pyramid of
Giza, that of the pharaoh Khafra, the quotient of the same lengths is
equal to 3 (or was, when the pyramid was complete). In the third pyramid,
the smallest one, that of Pharaoh Mykerinos or Menkaure, the quotient is
greater, between 3.2 and 3.3. That this value is 3.14 for the largest
pyramid would be the effect of chance.
- Extra-terrestrial
beings taught the Egyptians how to build pyramids. Making that quotient equal to
an approximation of p was a symbolic message, so
that later civilizations like ours could suspect that they had been on the
Earth. Oddly enough, this explanation has been proposed in earnest and has
led to plenty of esoteric and absurd ideas about the Great Pyramid.
- Some other reason,
different from the previous three.
The first explanation is not very credible. Technical
knowledge is not usually lost so easily, unless it is quite complex. As
I explained in another
article in this blog, during the European Middle Ages (which some call the
Dark Age), not only the technical advances of the Romans were not lost, but
others were added, such as water and wind mills, the iron plow, the harness and
the stirrup, the chimney, the buttress, and the mechanical clock. It seems
unlikely that a piece of information as simple as fraction 22/7 would be
forgotten by architects during millennia.
The second explanation does not seem reasonable. It would be, if in the Great
Pyramid the quotient between the semi-perimeter and the height would have been
3.1, for instance; but exactly 22/7 seems too coincidental.
What can we say about the third explanation?
Among other things, that these aliens are too simple. If they
wanted to bequeath us a good approximation of p, why settle for 22/7? Why not use a
much better one? If they were smart enough to make interstellar travel, they
would know the value of pi with several trillion digits, like us. Why leave it
at just three?
Given this situation, most researchers have opted for
the fourth alternative (some other explanation). Yes, but which one?
Two have been proposed, which seem reasonable:
- a) A Japanese team that studied
in Egypt the methods of construction of the Great Pyramid proposed that the Egyptians would use wheels to measure long
horizontal distances. Rather than counting cubits, they
would count the number of turns of a wheel one cubit in diameter. If they opted
for a height of the pyramid equal to 280 cubits and the length of each
side of the base equal to half this figure (140 turns of a wheel with a
diameter of one cubit), this length would be equal to 439.82 cubits, i.e. 440
cubits. The value of p and its approximation by the
fraction 22/7 would have been there automatically, without the Egyptians
noticing.
- b) The
other explanation is based on the angle of the pyramid. To
measure distances shorter than the cubit, the Egyptians used the finger. There
were 28 fingers in a cubit. To measure the inclination of a ramp, they
specified the number of fingers in the base to go up a cubit. Since they
only used whole numbers, the number of possible inclinations between 45º
and 90º was equal to 28: from one finger per cubit, to 28 fingers per cubit
(45º). It happens that the inclination of the Great Pyramid is 22 fingers
per cubit, so the ratio between double the side of the base (the
semi-perimeter) and the height turns out to be (88/28) = (22/7). Again the
value 22/7 appears automatically in the results.
Assume explanation 4a) is the right one: the use of
wheels with a diameter of one cubit. If this is not what happened, it could
have happened, and that is what matters in this reasoning. By using this method
of measurement, the value of p would have automatically entered in the dimensions
of the pyramid, even though the Egyptians of that time might not even know about
the existence of p. In such a case, can we assert that p is just
a creation of human mind?
Perhaps this mental experiment indicates that the value of p was not invented, but
discovered, that it existed before any human being noticed its existence.
The same post in Spanish
Thematic Thread on Mathematics: Previous Next
Manuel Alfonseca
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