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by H. Peter Aleff |

| Numerals and constants | ||||||||||||||||||||

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phi and pi.
Some people claim these same constants were also embedded in the proportions of the Great Pyramid and other ancient Egyptian monuments, but many mainstream scholars assert those ratios got there by accident, without the builders' knowledge or intent. These academics will probably say the same thing about the numbers on this mace because they believe all early Egyptians were still certifiably unaware of these constants, more than two and two and a half millennia before the glorious Greeks would discover them. However, and notwithstanding that gaggle of gurus, there is nothing inherently impossible about some ancient Egyptian number researchers from Narmer’s time having known these constants with fairly close approximations. By then, people had been exploring numbers for a very long time, and our species' knack for mathematical discoveries is not limited to recent generations, as you can examine in the chapters from "Early math" to "Contributions from Egyptian mathematics". When Narmer united Egypt, the ancestors of his subjects had practiced field geometry and arithmetic for many hundreds if not thousands of years on a daily basis, and the methods for computing both pi and phi are quite simple. Nailing down both these constants to any desired degree of accuracy requires some patience but no mathematical tools more advanced than the consistently mis-attributed “theorem of Pythagoras” about the area equality of the two smaller squares over the sides of a right- angled triangle with the larger one. The absence of this theorem and of these constants from the few written mathematical records that survived in Egypt is no counter- argument because all that remains from three thousand years of mathematical activities by millions of people are exactly eight mostly fragmentary records. This random sampling is too small to allow any informed judgment about how far any of those field- measuring and number- exploring ancestors may have advanced their inquiries into the number world, or how much knowledge they might have transmitted to their pharaonic heirs. To appreciate just how meager said eight fragments are, and how the secrecy surrounding the priestly arts would furthermore have prevented some mathematical knowledge from being widely circulated in the first place, see the chapters from “Our sources on ancient Egyptian math” to "Hermetic secrecy". Since we cannot rule out that the early Egyptian geometers might have known more than what they left us in those eight surviving scraps of their mathematical writings, it seems permissible to explore, as a thought experiment, how they might have perceived these astonishing constants that ruled the endless and self- repeating circle and the ever self- same regeneration of a mysterious ratio which later mathematicians would admiringly call “golden” and “divine”. Although no inscription tells us directly what meanings the Egyptians gave to pi or phi, or whether they knew these at all, the timeless mathematic properties of both constants suggest fairly obvious symbolic associations. We can reasonably reconstruct those probable associations in the context of ancient magical thinking where superficial similarities often linked things and ideas and allowed them to symbolize each other. If any ancient Egyptians happened to know these amazing constants, they would not have thought of them as abstract and faceless numbers, the way we do today, but they would have assigned symbolic meanings to them. They lived in an intensely religious culture which attached symbolic meanings to everything, from beings and objects to actions and ideas and even certain integers. As we saw earlier, the number world existed in parallel with the tangible world since both had been created together and were thoroughly interwoven, and the ancient Egyptians identified or associated numbers with the powers of the divine realm, even representing one numeral with the picture of a god. Like this god whose group held up the sky, numbers must therefore have formed an ideal bridge between the tangible world on earth and the divine one in the sky, particularly since the basic unit stroke for the integer “one” was also the hieroglyphic sign for “real” and for “person”. This symbolic shorthand connected numbers to the people who used them and who seem to have seen them as a permanent abstract counterpart to the ever- changing and perishable “real” world in which they lived. Just as the hieroglyphs of the written words were considered alive and had their own magical powers as well as those of the objects they depicted, so were the numbers living entities, invisible but all- pervading and clearly mightier than the natural processes and heavenly bodies whose cycles they ruled. Even stronger than the readily accessible whole numbers and fractions must have been the mysterious constants hidden between them. We call these constants transcendental because we cannot express them as finite fractions or roots of whole numbers but only with non- repeating expressions that continue to infinity. The ancients may not have known or cared about our definition of transcendental, but they must have noticed something special about these constants because they appear to have given them a similar status of connection to infinity. This is fairly natural for pi since the endless circle is a self- evident representation of that infinity and has been used as its symbol in many cultures, including the ancient Egyptian images of the Shen-Ring and the Ouroboros snake that bites its own tail. In addition, the ways in which the ancient number group composers used this and other constants suggest that that these belonged to a special category of numbers and were among the more potent secrets to be learned from Thoth who combined, not coincidentally, the functions as master of numbers and also of magic. Close cross- connections between whole numbers and simple functions of constants, like those in the ratios and quasi- equations you will find in this book, would therefore have appeared as bridges between the worlds of the humans and of the gods. This parallels the situation in ancient And returning to Egypt, we can now reconstruct and then plug in the likely ancient Egyptian meanings attached to phi and pi: | |||||||||||||||||||||

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