Again we start with fig. 17b, which we now know to be the Golden Rectangle. The Golden Rectangle is used to generate a diagonal (of a smaller Golden Rectangle) which is then applied to the circumference of the circle. This length is then used to mark-off the sides of the polygon. These same points also provide the apexes for an inscribed star-pentagram.(fig. 29)
Each figure can be used to generate the other in an endless pattern, thus invoking the ineffable yet again. (fig. 30)
If we return to Iamblichus' quote about the man who blasphemed for revealing Pythagoras' teaching and set aside "symmetry and asymmetry," we are left with three other revelations: the construction of the dodecahedron, irrationality and incommensurability. The dodecahedron is comprised of twenty sides where each side is the shape of a pentagon. Therefore, as demonstrated, all three issues as well as the geometric mean, are closely related to each other and to endlessly generating geometric forms which invoke the ineffable.
But what is the ineffable? The ineffable is infinity. The early Pythagoreans had discovered infinity. The problem is that infinity does not admit of ratio. This discovery was kept secret till someone revealed it. Iamblichus, does not provide the person's name in this context, and he provides an incorrect name in a later context. The name is provided in other sources as Hippasus.
It would seem that in about 420 BC, or a little earlier, towards 450 BC, there was a schism in the sect, between the acousmatics and mathematicians. The turmoil was probably due to the forced exile of the Pythagoreans from southern Italy because of political rivalries. The 'listeners' wished to uphold the very word of their teacher. The mathematicians wished to explore the process initiated by Pythagoras. At the same time, there was probably another conflict which fell within the same lines between those who wished to teach for a living and those who believed the teaching sacred and reserved for the initiated. This conflict is documented in Iamblichus by a letter from Lysis to Hippasus. Iamblichus misnames Hippasus, Hipparchos. Lysis on fleeing from Italy moved to Thebes. Hippasus later perished at sea.
The letter states:
"You say we should philosophize in public, for whoever comes along. Pythagoras said not, and so you learnt, Hipparchos, in all seriousness. But you did not keep the teaching safe. You had the taste of the Sicilian high living, man, though you should have got the better of it. If you change, I shall rejoice; if not, you are dead. It is right, they say, to keep in memory his commands on divine and human matters, and not to share the goods of wisdom with people whose souls are not remotely purified. It is not right to hand out to chance-met persons what was achieved with so much effort and toil, nor yet expound to the uninitiated the mysteries of the Two Goddesses of Euless - those who do either are equally wrong and impious.
"Think how long a time we spent cleansing the stains which were ingrained in our breasts, until, with the passage of time, we were able to receive his words. As dyers cleanse and treat with a mordant the parts of the garment which need to be dyed, so that the dye will be fast and will never fade or be lost in the wash, so that wonderful man prepared the souls of those who had fallen in love with wisdom, so that he should not be disappointed in one of those he hoped would become good men. He did not purvey false words of the snares which most sophists, working for no good purpose, entrap young men: he knew about divine and human affairs. But those others make his teaching a pretext and do terrible things, hurting young men in the wrong way and of set purpose."
The reason I have brought the reader so far on these matters is because firstly, the Golden Section or Divine Proportion was of particular concern, as was the idea of proportion, to Leonardo Da Vinci. As we shall see, the Golden Section and the tetraktys are key elements of The Last Supper. Furthermore, the problem of infinity is also re-encountered below, as is Lysis' letter.
Infinity is a spectrum with two extremes. One extreme tends to the infetissimal, the other to an unimaginable magnitude. Looking at fig. 17a., the infetissimal tends to the left, infinite magnitude, or infinity to the right. With time, the problem of infetissimals would be tackled by the Greeks. By Aristotle's time, a hundred years or so later, it is a question of open debate as evidenced by Aristotle's own writings. But the question is kept within strict limits. Infinite divisibility is viewed as a potentiality and not an actuality. 'Potentiality' restricts the phenomena to an imagined space. As for the question of infinite magnitude, it is denied.
The problem raised by the irrational line would also, once revealed, be later explored. By the time of Socrates and Plato, at least four mathematicians; Archytas, Theodorus, Theaetetus and Eudoxus, had transformed the impasse into a theory of irrational lines by confining such problems to the realm of geometrical line segments. Most of this work would reappear in a highly formalized fashion in Euclid.
Which brings us to the great work itself. As probably apparent, the Pythagoreans' greatest impact lay in a geometrical foundation which eventually led to the Thirteen Books of the Elements by Euclid.
Euclid (300 BC), a mathematician, is believed to have elaborated the first formalized system of intellectual human thought. This system would greatly influence a parallel tradition of Greek philosophy and would subsequently shape the course of various Western sciences. Euclid organized his principal work, The Thirteen Books of The Elements, around a kind of mathematical alphabet, starting with three basic definitions and then progressing from proposition to proposition, theorem to theorem, element by element. Earlier mathematicians had attempted such works, such as Hippocrates (450 to 430 BC) and Leon (circa 400 BC). But because of the particular rigor of its organization, the Elements superseded the others and was the only work of its kind to survive.
With Euclid begins the ascendancy of the Alexandrian School (300 BC - 150 BC), which would come to include such pivotal mathematicians such as Aristarchus, Archimedes, Erastosthenes, Appolonius, Hipparchus and Ptolemy. Only two facts seem to be known about Euclid. The first is that he was born after Plato's pupils, yet he predated Archimedes, who died in 212 BC. The second is that he taught in Alexandria.