Thursday, May 24, 2018

The mystery of the Great Pyramid

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:
  1. 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.
  2. 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.
  3. 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.
  4. 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:
  1. 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.
  1. 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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