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by H. Peter Aleff
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Pentagrams before Pythagoras
Most people familiar with the golden ratio know it is prevalent in the natural world. The golden ratio is expressed in the spirals of seashells gathered from the beach near Cancun resorts to the petals of tropical flowers surrounding a Dominican Republic hotel in the Caribbean. The importance of the golden ratio for artists, mathematicians, and architects from ancient times to the present cannot be denied.
The oldest surviving written evidence for our species' knowledge of the unique "golden" section dates back to Byzantine times, about 888 CE or slightly later1, when the earliest extant copy of the mathematical textbook "The Elements" was written. The contents of this book, in turn, was attributed to the classical Greek mathematician Euclid who wrote around 300 BCE. He discussed this proportion as the "division in extreme and mean ratio".
Before Euclid, the Athenian sculptor and architect Phidias (490 to 432 BCE) gets generally credited with having pioneered the use of this ratio in his design of the Parthenon temple for Athena, the Greek goddess of Wisdom, science, and art. Some scholars deny the presence of this ratio in that temple or any other building of its time, but it is to honor Phidias for its use that modern mathematicians often designate this ratio with the Greek letter f = phi.
Phidias’ knowledge of this proportion, as well as the relevant chapters by Euclid, are said to be based on information from the Greek mathematician Pythagoras (about 580 to about 500 BCE)2. Some disciples of this semi- legendary philosopher and founder of a number- mystic sect reputedly used a pentagram as one of the symbols for his mathematical doctrine, and one of them is said to have displayed this phi- based geometrical figure on his door post as a secret sign of mutual recognition3. The sign and its meaning could remain secret despite that public display because only those initiated into the mysteries of Pythagoras' geometry were able to draw it correctly and to appreciate its deep significance as a symbol for and gateway to those mysteries.
This is where the documented or reported trail of phi and the pentagram stops, at least in the current mainstream histories of science.
Geometrically, the pentagram is an extension of the pentagon which is the same design without the star- arms. Both include in their proportions many instances of the golden ratio, and to draw either figure properly one must first construct that ratio, as illustrated on the "Golden Drawings" page, and as explained below. This construction requires analytical thinking.
For instance, Sir Thomas Heath, the eminent translator of many Greek mathematical and astronomical texts, says about the pentagon in his "Summary of the Pythagorean Mathematical Discoveries":
"... as [the construction of a regular pentagon] depends upon the construction of an isosceles triangle in which each of the base angles is the double of the vertical angle, and this again on the cutting of a line in extreme and mean ratio, we may fairly assume that this was the way in which the construction of the regular pentagon was actually evolved.
It would follow that the solution of problems by analysis was already practiced by the Pythagoreans, notwithstanding that the discovery of the analytical method is attributed by Proclus to Plato. As the particular construction is practically given in Euclid IV:10,11, we may assume that the content of Euclid IV was also partly Pythagorean."4
The mathematical historian Roger Herz- Fischler concurs in his book “A Mathematical History of the Golden Number”:
Herz- Fischler says this postulated program of inscribing polygons into a circle was conceived in Greece. However, the Egyptians had long been scribing polygons around circles, and this requires the same mathematical approach and skills.
Pharaonic stone masons made round columns by starting with polygon facets around the desired circle; then only did they cut away the excess material. According to Dieter Arnold, an Egyptologist who studied the ancient construction methods, this approach can be observed in the unfinished corner torus of Pylon IV at
“... is not yet completely round but polygonal, thus preserving an intermediate step between the rectangular boss and the rounded torus”.
Other columns were intentionally left polygonal, for instance, the two sixteen- sided limestone columns at the entrance of the small temple which Thutmose III built in Abydos. The same geometrical skills would also have been required for the half- and three- quarter- round engaged columns in king Djoser’s Saqqara buildings. Those columns are, moreover, both fluted and tapered, requiring precise guidelines in progressively adjusted sizes.
Whatever method of analytical geometry the Egyptian column cutters may have used for drawing those many regular polygons, it seems that the pentagram expressed the essence of this science not only for the Pythagoreans but also for the temple designers of the Ramesside era, some 750 years before Pythagoras was born.
Geometry was the special turf of Seshat, the divine mistress of temple plans. She presided over the so-called "House of Books", later also called the "House of Life", where the priests and sages maintained and transmitted traditions in all areas of knowledge, from medicine to magic and dream books, and above all the correct performance of rituals which included the design of all temples.
These proto- Universities or library archives appear in the titles of dignitaries from the Fourth Dynasty on. For instance, one of king Khufu’s sons was "Priest of Seshat presiding over the House of Books".
Seshat’s most prominent task was the laying out of sacred buildings, together with the king. Her foundation ritual of "stretching the cord" is sculpted on many temple walls, and it was essential for assuring that the geometry of the building would correctly reflect the structure of heaven and earth which the temple was built to reproduce.
As described in the chapter "Maat soulmate Seshat convicted for possessing pot and undeclared math", a beautiful and well preserved portrait of Seshat among the reliefs in the Luxor temple from around 1250 BCE shows a pentagram at the center of the hemp leaf in her emblem. That pentagram is perched on its stem above her head as if this geometrical figure was already then a symbol for geometry, used there as an extra determinative for the most characteristic art taught by that goddess of geometry, writing, and general learning.
The five-pointed star hieroglyph
This identification of the pentagram with geometry may even go back much farther, all the way to the hieroglyph designer(s) of early
The ancient Egyptians attached great importance to similarities in sound or spelling of otherwise unrelated words. They believed such resemblances were a sign of deep connections between the objects or ideas such words represented, and this ancient principle gives us a glimpse at the associations they seem to have made with this penta- star.
This same star, with a papyrus roll as determinative for abstract ideas, appeared also in the word "seba-eet" for “written teaching, instruction”, whereas the verb "seba" = "to teach, to learn" combined that star with a weapon- wielding arm. (This threatening arm referred presumably to the school master's rod since the word "seba-oo" for "education", based on the same root and with the same determinative, could also mean "punishment".)
Whatever associations this punishing arm may have evoked in the pupils of the scribal schools, the teachings symbolized by the five- pointed star were also associated with doorways because the same "seba" signs as in the "teaching, learning" verb meant "gate" when their "armed- arm" determinative was replaced with that of a houseplan.
The gates this pun on "learning" represented to the learners may have admitted these to the lucrative careers to which their learning opened the way, or to the wisdom which opened their minds. However, although such modern thoughts may also have played a role back then, the star in the word may have alluded above all to other gates which were even more important.
Stars were thought to be the gates of heaven. Another word for them, "ankh-oo", included the "ankh" sign of life and a star plus the plural sign, and it had the same consonants as "ankh-oo" which was also a plural and meant "the living". Consistent with the ancient Egyptian habit of denying death, "the living" was an euphemism for the dead since the eternal afterlife of these was considered more real and more important than its brief prelude here while the future deceased still walked on earth .
Dead pharaos became stars and circled the celestial north pole together with the other "Immortals", that is, the circumpolar stars which never disappear below the horizon. This may initially have been an exclusive privilege for royals, but as commoners gained access to the afterlife, stars came to represent also the souls of the dead in general.
This connection between the star sign and the dead is expressed again unmistakably in the hieroglyph for the "duat" or "afterworld" which was the same five- pointed star but with a circle around it -- pi surrounding phi.
As we saw earlier, a circle was the symbol of the sun and of its divine eternity. The "duat"- sign and its meaning implied therefore that geometry came from and belonged to that parallel but timeless and invisible realm where the gods dwelled, and the "justified" dead who went to join them. That netherworld realm was thus apparently also the world of the numbers and of the geometry from which it took its symbol.
This connection between numbers and stars and gods and the dead who became gods and/or stars makes sense in magical analog thinking because numbers and geometric objects are as timeless and as intangible as those spiritual beings, and as charged with mysterious powers. The Egyptians’ use of numbers in religious and magical contexts suggests that they perceived gods and numbers as related, just as the Mesopotamians did.
Another habit of magical thinking is that a part can stand for or replace the whole. An important part of the ancient Egyptian afterworld were the gates to and within it through which the sun and the newly deceased had to pass on the way to their resurrection. Such gates appear already in the Pyramid Texts and in the Book of the Dead, and twelve serpent- guarded gates, one after each hour of this night voyage, became later such a defining feature for the netherworld that one of the popular guidebooks to it, first attested among the wall paintings in the tomb of king Horemheb (1319 to 1307 BCE), is now called "The Book of Gates" .
The five- pointed star in its circum- circle could therefore also designate the gates to that afterworld, matching the above presence of the star sign in the word for "gate". This usage of the symbol as an opening to the world beyond survived into Medieval and even Renaissance times when magicians typically drew a pentagram on the floor to summon spirits from that world and enclosed it in a circle to protect themselves and their audience from the dangers inherent in such contacts.
See, for instance, the Scene in Goethe's "Faust" where that magician first draws a pentagram on his door threshold before he summons the devil Mephistopheles. This demon can only escape by ordering a rat to gnaw off one of the pentagram's points so that it no longer forms a closed figure.Pythagoras as plagiarist
These uses of the symbol for geometry indicate that the science it represented meant far more to its practitioners than a way to measure fields. They anticipate by more than two millennia the Pythagorean connection between this sign and that science, and also with doors. They suggest therefore that Seshat, or her priests, had much earlier claims on the analytical method than any Greek mathematician. The prior art in the hieroglyph signs and on Narmer’s mace makes Pythagoras move over and abandon his bragging rights as that method’s alleged inventor.
Indeed, many ancient authors tell us that Pythagoras picked up much of his knowledge from others. This purported discoverer of the golden ratio and of its pentagram symbol for the analytical method is notorious for having claimed as his own many discoveries that he had learned abroad. His almost contemporary, the four decades younger Greek philosopher Heraclitus who lived from about 540 to about 480 BCE and probably had access to some of the same sources as Pythagoras, accused him of systematic intellectual theft:
"Pythagoras, son of Mnesarchos, has done more researching than all other people, and by reading together all these writings he pretended with punditry and artful lies that they were his own wisdom. (...) Pythagoras is the leader of the swindlers."
Similarly, though without the offensive terms or intent, all of Pythagoras’ ancient biographers stated that this reputed founder of Western science  owed much of his learning to the traditions of the Near East.
For instance, the Neo- Platonic (and thus also Neo- Pythagorean) philosopher Iamblichus (about 250 to 325 CE) wrote relatively late but is said to have used early sources. According to his account, Pythagoras began his studies with the philosopher and mathematician Thales of Miletus (about 625 to 550 BCE) on the Aegean coast of Asia Minor and then continued them in the Levant :
"... he sailed to Sidon, both because it was his native country, and because it was on his way to Egypt. In Phoenicia he conversed with the prophets who were the descendants of Moschus the physiologist (that is, Moses), and with many others, as well as with the local hierophants [priests who interpret religious rites and mysteries]. He was also initiated into all the mysteries of Byblos and Tyre, and in the sacred function performed in many parts of Syria. (...)
After gaining all he could from the Phoenician mysteries, he found that they had originated from the sacred rites of Egypt, forming as it were an Egyptian colony. (...)
Here in Egypt he frequented all the temples with the greatest diligence and most studious research, (...) acquiring all the wisdom each possessed. He thus passed twenty- two years in the sanctuaries of temples, studying astronomy and geometry, and being initiated in no casual or superficial manner in all the mysteries of the gods.
At length, however, he was taken captive by the soldiers of Cambyses and carried off to Babylon. Here he was overjoyed to be associated with the Magi who instructed him in their venerable knowledge, and in the most perfect worship of the gods. Through their assistance, likewise, he studied and completed arithmetic, music, and all the other sciences. After twelve years, about the fifty- sixth year of his age, he returned to Samos." 
Of course, Iamblichus may have embellished some of the details in his hagiography, such as the readiness of the local priests to initiate this stranger into all their secret teachings.
We must also keep in mind that we have very little or no first- hand information about Pythagoras, and that his very existence can be questioned. He was a legend- encrusted figure, a miracle worker with a golden thigh who could be in two places at the same time. He was also a son of the god Apollo, and so exalted that a river once greeted him by name when he crossed it.
Still, even if this fairy- tale Pythagoras may have never lived, the traditions from antiquity about his teachings are real, and it matters little if some other fellow (whom later writers only happened to call Pythagoras) may have spread the doctrine said to be his.
The solid core of these traditions reflects that the greater part of the mathematical knowledge and discoveries known to the early Classical Greeks, and claimed by or ascribed to Pythagoras, actually came from the cradles of civilization in the Levant, the long- established trading partners and teachers of the then culturally just emerging Greeks.
Pentagons in Solomon's Temple
One of the groups most active in this transmission of ideas were the Phoenicians, acknowledged as such in several Greek myths like that of Cadmus bringing the alphabet to Thebes. In Iamblichus’ account, Pythagoras himself was of Phoenician descent and started his quest by traveling to that homeland of his.
Several centuries before Pythagoras, at a time when Greek mathematics was barely beyond the stage of counting all the legs on a tripod, the Hebrew king Solomon hired Phoenician specialists to build his Temple, and he maintained close contacts with these gifted traders and craftsmen. The designer of his famous Temple in Jerusalem incorporated into that building many examples of the golden ratio, including the same phi- based construction from which the Pythagoreans would later derive their above recognition sign, and he used it in the same location as they would.
The New English Bible translates in 1 Kings 6:21 that at the entrance to the Holy of Holies
If this translation is correct, then it implies that Solomon’s builders were also aware of this analysis- requiring construction, just like their Egyptian neighbors. Moreover, their use of of the pentagon for the cross- section of these door posts matches the way the above Pythagoreans would later affix the pentagram to the door posts to identify their dwellings to other members of their group. The only difference is that in the Temple, the pentagon was not oriented horizontally towards people but turned upwards to heaven since the door it marked was intended for God.
The use of this recognition symbol in the door posts also matches how Jews from at least the Second Temple period on marked their door posts with mezuzahs to identify themselves to God for his protection, and it echoes how their ancestors in Egypt had smeared lamb’s blood on their door posts and lintels (Exodus 12:7) to identify themselves to God when he slew the Egyptians’ firstborn.
All this evidence for pre-Greek knowledge of the golden ratio may be circumstantial, but it is cumulative and makes it appear much more likely than not that Pythagoras picked up the doctrine of the pentagram in Phoenicia or Egypt.
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