Copyright 2001, by Ernest P. Moyer
Revised February, 2003
There are no conflicts in the placement of the data points in the several graphical plots except for Figures Seven, Eight, and Ten, and their requirements of placement in Figure Eleven. The designer played with the numbers to provide us with the patterns shown.
The degrees of freedom through five pyramids is not exhausted but we do not have evidence on the Flat and the Bent. Hence we cannot make further estimates based on data available. An important task is future completion of the data set.
The mathematical expressions, and the constants and variables used in the pyramid designs, are all typical of what we find in modern science and engineering. I need only open the pages of the Handbook of Chemistry and Physics from the Chemical Rubber Publishing Company to see them staring me in the face. The ancient designer/builder of the Great Pyramids lived in a similar technological environment, or he carried memory from another era and culture. Although we have considerable evidence of advanced technology other than the pyramid designs we do not have tools, surveying equipment, machining equipment, metals capable of producing stone, limestone or otherwise, and so on.
Some may object that we are interpreting data according to modern understanding, while the ancient designer may have performed according to a completely different conceptual framework. This suggestion is refuted by the exterior designs showing Pythagorean triangles, and the half-angle design of the passages, all of which follow current mathematical understanding. Further discussion will show the inadequacy of this suggestion.
1. We see the neat manner in which the designer provided numerical values that would be easily discernible to an investigator of his designs. These are 1/2Pi, 10 Pi, 10/2Pi, ln 10, and so on. This shows his familiarity with mathematical and physical constants as we understand them today.
2. He knew the difference between the common base (10) and natural base (e) number systems. Of course one can be exchanged for the other. Log x = ln x / ln 10, or ln x = log x / log e. We can reasonably infer that he understood logarithmic systems to any base. Certainly, if he were familiar with one he would be familiar with another. If one wished to express the regression lines on the graphs in the alternate system one could easily do so.
3. He was intimately familiar with exponential and logarithmic functions. The manner in which he tied together two exponential decay curves to provide the nearly linear relationship between the two, as illustrated in Figure Five, is an outstanding piece of cleverness.
4. The use of Pi and e are identical to that which we use in modern fields of engineering and physics.
5. He derived equations based on the variable N. This variable is found in such modern developments as mathematical expressions for cyclical or harmonic phenomena, and in infinite mathematical series. His plan demanded a sufficient number of structures to permit graphical display that would expose his design around that variable.
6. He created equations based on r / 2 Pi as a radian measure. The multiplier of 1,000 was necessary to obtain a practical graphical display because the angular measure was so small. One minute of arc is approximately 1/3437 of a radian. By providing slopes on his curves to this mathematical expression he showed us his familiarity.
Thus he was intimately familiar with circle concepts as we understand today. He expressed the relationship of angular distances with radian measure. The mathematical expressions from Figure Eleven are sensible only if he used that form.
7. He recognized the use of incremental values nesting inside physical lengths. The evidence hints at concepts that approach modern calculus. How much he knew of differential mathematics is not certain. He clearly gave us an indication of ability to design to such criteria.
8. He understood the mathematical concept of mean values. Perception of the relationships among side deviations can only be understood when compared against the means.
9. He understood the concept of mathematical absolute numbers. Figures Four and Five use absolute numbers to achieve their results.
10. He used graphical design methods. The results of this study can only be understood if he drew out those relationships through graphical analyses. Simple mathematical calculations would not make the relationships obvious, nor evident, to mental perception not for himself, nor for later investigators.
11. He understood manipulation of design parameters to achieve presentation objectives without violating mathematical degrees of freedom. With seven known structures, four sides, deviations on those sides, and orientation of the individual sides, he had great freedom in his development of mathematical displays. Again, his true genius is evident through such finesse.
Such level of mathematics was not known until modern times.
1. The first treatise on algebra was written by Diophantus of Alexandria in the 3rd century AD. The term derives from the Arabic al-jabr or literally the reunion of broken parts.'' It gained widespread use through the title of a book Ilm al-jabr wal-mukabala the science of restoring what is missing and equating like with like written by the mathematician Abu Jafar Muhammad (active c.800-847). He introduced the writing down of calculations in place of using an abacus.
But graphical analysis of relationships between variables was not available until Rene Descartes (1596-1650), published the first treatise on the geometric representation of algebraic equations in 1637. The ancient Greeks were highly developed in space perception and used figures to represent geometrical relationships but did not analyze them algebraically. They did some analytical work but did not make mathematical associations which would permit graphical display of physical relationships as we know them today.
2. Tables for the use of logarithms were first published in 1614 by John Napier (1550- 1617). They were motivated by the need to alleviate tedious trigonometric calculations for ship positions in the burgeoning maritime traffic of the sixteenth and seventeenth centuries. The tables also were highly useful for calculating compound interest for the financiers of that aggressive commercial age. Although the idea of multiplying numbers through addition of logarithms goes back to Archimedes' study on mathematical series (287 B.C.), with contributions from medieval mathematicians, it was not until the recognition of the ease of substituting addition for multiplication through trigonometric relationships by Wittich (1584) and Clavius (1593) that led to practical developments.
3. The amazing mathematical power of the natural base number e = 2.71828 . . . was not recognized until the late sixteenth century. Although Leonhard Euler (1707-1783) was the first to use the letter e to represent the number, tables of logarithms to the base e were first published by Speidell in 1619.
4. The relationships demonstrated in the several figures are those of applied mathematics. Logarithms, exponential functions, natural base e, and radian measure are useful to physicists and engineers. Natural decay in electrical circuits, mechanical systems, and physical processes follow exponential forms. The solution of differential equations leads to both common and natural base number representations.
5. Our knowledge of ancient Egyptian mathematics is severely limited. Refer to a summary by Marshal Clagett (8). The two major sources are the Moscow Papyrus and the Rhind Mathematical Papyrus (6), both dating a nearly a thousand years after the Great Pyramid Construction Project, circa 1900 B.C. They show only primitive mathematics with elementary operations in arithmetic, calculation of areas and volumes, including for pyramids, and simple progressions. They did not use decimal multiplication and division; they used a procedure of doubling of numbers to find multipliers and divisors nearly the same technique as employed in modern computers. Although this procedure is useful for small numbers it rapidly becomes cumbersome for large numbers.
6. The only other information we have on mathematical knowledge of ancient times is found on the clay tablets of the Old Babylonian empire, circa 1900 to 1600 B.C. Neugebauer and Sachs (7) published translation and analysis which showed more advanced mathematical techniques than displayed in the Moscow and Rhind Papyri. Scholars of Old Babylon used relatively sophisticated algebraic expressions for geometric figures including circles, segments of circles, diagonals of squares, right-angled triangles, (with Pythagorean solutions), irregular triangles, trapezoids, irregular solids and, most importantly, exponents and logarithms. Problems for exponents included powers between 2 and 10. Problems for logarithms were given to base 2 and 16.
In all examples known to us the Babylonians illustrated sophisticated problem solutions but just short of achieving theoretical expressions. We have no evidence of their mathematical evolution or how they achieved their knowledge.
7. The Egyptian papyri and the Babylonian clay tablets have an important feature which has not been voiced. They both show learning by rote. The methods of problem illustration are how to. This is the way you multiply numbers; this is the way you calculate areas; and so on. Nowhere do we have evidence of how the formulae were developed or the theories deduced. As a consequence we believe they arrived at those methods through trial and error, that there was no theoretical development. For this reason many of us are unwilling to accept the more sophisticated examples for their obvious sophistication. The solution of a truncated pyramid in the Moscow papyrus is regarded with skepticism, although many such examples are found on the Babylonian clay tablets. Then again the use of the Pythagorean relationship 1500 years before Pythagoras for complex Pythagorean solutions is received with even more difficulty. The evidence is contrary to our notions of the slow evolution of mathematical knowledge. The real problem is that advanced mathematics was known in the far distant past, and incorporated in the pyramid designs, but then was lost to the world. Such phenomenon could occur only if the society which possessed such knowledge was also dying.
8. If it took better than 400 or a 1,000 or 2,000 years to develop modern mathematics how many centuries did it take ancient Egyptians, members of the most conservative society known in history, to develop such mathematics? Why is there no other record of it? If it is the product of a technical culture where is the evidence for such culture? And why was it lost?
How did the designer know all this sophisticated mathematics without a social base? How could he have developed it entirely on his own, when we took millennia to achieve the same results? We cannot reconcile such sophisticated knowledge without a social base for its nurture which then became lost to the world.
This has profound implications. He had to be fully expectant that his project would be brought to completion with the number of structures he planned. As we saw from the analysis this demanded the complete gamut from Meydum to the Giza 1 Case. He had to be sure the social resources were available to produce such massive works. But he also had to be sure that the management of such a vast enterprise would achieve its goals to the refinement we saw. This either meant that he was personally present during the entire program, or that he taught others who possessed similar determined purpose. If the kings merely believed the pyramids were each intended for their individual burials then they would be short of the knowledge and reasoning behind the project. That each was intended individually for their personal honor and fame is the view of history. Ancient records, written centuries afterwards, support such belief. But if the designer/builder had a motive hidden from the eyes of the kings, they would not understand his plan or methods. This means they could not have been part of the project except that the designer/builder depended upon their support for the social resources. This explains why Seneferu would have three structures to his credit, and why Khafre would claim a pyramid that was built before the pyramid assigned to his father. He was left with the prospect of making claim to something not originally assigned to him. Again, this is why modern Egyptologists would make that chronological assignment after Khufu.
The recognition of a stupendous enterprise, with social and royal support, under hidden design, and with private resources of knowledge as so often ascribed to ancient priesthoods, serves to explain this unbelievable phenomenon.
There seems to be no other reasonable approach to the Master Puzzle of the ancient past.
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