# Golden Section

54 55 aPPenDix iii The inDefiniTe DyaD Plato, as a Pythagorean, was under an inviolable oath not to reveal the deeper truths of the mystical Pythagorean mathematical order. Like Pythagoras, he spent considerable time in Egypt studying the mathematical mysteries with the priesthood there and in his writings intentionally conceals deeper truths under a shroud of mystery. As a teacher and writer, Plato practiced Socratic midwifery, presenting anomalous puzzles, problems, and incomplete solutions, along with significant hints, both within the Academy and in his dialogues. Readers needed to abduct retroduct or hypothesize a solution to the anomalous situation. Academy members were presented with problems such as the doubling of the cube, or giving an account of the heavens that would accommodate and explain the apparent irregular motions of the planets, thus saving the appearances. In the dialogues, Plato carefully selects several interrelated problems that are very subtly posed. Taken together they point to the great mystery of the Golden Section and its Reciprocal, none other than the Greater and Lesser of the Indefinite Dyad. It is clear from Aristotle and other members of the Academy that, in unwritten lectures, Plato more openly revealed the deep truths of how the One or Good of the dialogues combined with the Greater and Lesser of the Indefinite Dyad to produce the Hierarchy of Intelligible Forms, Mathematicals, and Sensible Particulars. In the Metaphysics 987b 1922, Aristotle writes Since the Forms are the causes of all other things their elements were the elements of all things. As matter, the Great and Small were Principles as substance, the One for from the Great and Small, by participation in the One come the Forms, the Numbers. And yet even to the Academy members, the presentation was necessarily enigmatic. As Simplicius records in his Commentary on Aristotles Physics 187a12 Plato maintained that the One and the Dyad were the First Principles, of Sensible Things as well. He placed the Indefinite Dyad also among the Objects of Thought and said it was Unlimited, and he made the Great and the Small First Principles and said they were Unlimited, in his lectures On the Good Aristotle, Heraclides, Hestiaeus, and other associates of Plato attended these and wrote them down in the enigmatic style in which they were delivered. In the Parmenides 133b, Plato presents the worst difficulty argument how can the immaterial Intelligible and material Sensible worlds have any contact or interaction In the Timaeus 31b32a, Plato makes it clear that continuous geometric proportion is the best of all bonds. This involves there being an intermediate geometric mean relationship. He then gives us the socalled Lambda relationships of 1, 2, 4, 8 and 1, 3, 9, 27 35b36b. In the Republic 509d, Plato asks us to divide the line unevenly, representing the Intelligible and Sensible Worlds. In effect he is telling us to generate a continuous geometric proportion between the whole and the parts with the simplest of all cuts, the Golden Cut. When we apply the same ratio, the Golden Cut, to the two segments, we now get the most interesting geometric proportion between the parts, F1 11F. Unity becomes the geometric mean between F and 1F. Therefore, the geometric relationship is F 1 1F. I argue that this is the Greater Unity Unity Lesser, or Greater Unity Lesser. Thus, the worst difficulty argument is answered through continuous geometric proportion. The Intelligible and Visible worlds are interlocked, interlaced, fused together through the magic of the Golden Mean in its Greater and Lesser relationships with Unity. Furthermore, Plato in the Timaeus conspicuously omits the triangle required to construct the dodecahedron, even though he declares that this regular solid represents the Cosmos itself. This triangle would of course necessarily require him to overtly recognise the Golden Section. Yet he does provide the 2 triangle for the construction of the cube, and the 3 triangle for the construction of the tetrahedron, octahedron and icosahedron. He also notes that the 3 triangle is used to compose a 3rd triangle, the equilateral triangle. Platos nephew Speusippus, who headed the Academy after Platos death, wrote in On Pythagorean II I Numbers a fragment of which still survives that the equilateral triangle with all sides equal represents Unity or One, the triangle with two sides equal represents Two, and the 3 triangle with three unequal sides represents Three. Plato clearly calls the 2 and 3 triangles the most beautiful triangles. However, he then hints to the astute reader that these then we assume to be the original elements of fire and other bodies, but the principles which are prior to these God only knows, and he of men who is a friend of God. Timaeus 53de Thus, Plato suggests there may actually be principles prior to these triangles. In fact, I would suggest, the principles may be discoverable in the missing 4th triangle, the one necessary to construct the dodecahedron. Plato then goes on to say anyone who can point out a more beautiful form than ours for the construction of these bodies shall carry off the palm, not as an enemy, but as a friend. Now, the one which we maintain is the most beautiful of all the many triangles is that of which the double forms a 3rd triangle which is equilateral. The reason of this would be too long to tell he who disproves what we are saying, and shows that we are mistaken, may claim a friendly victory. Timaeus 54ab It is particularly relevant that Plato names a 3rd triangle here, the equilateral triangle, that Speusippus has clearly identified as representing the One. The 2 triangle represents the Two, and the 3 triangle represents the Three. However, in Pythagorean language, one swears by the fourfold Tetraktys. And very suggestively, the Timaeus opens with the cryptic statement One, two, three, but where is the fourth, my dear Timaeus, of those who were yesterday my guests and are to be my entertainment today One of the guests, the philosopher, has been taken ill. He is conspicuously absent, as Plato intends much more by these opening remarks. The fourth is now the 4th triangle which is missing. It is the triangle that would build the dodecahedron, but it would require revealing the Golden Section. And not only that, it might reveal the very principles behind the Cosmos itself, namely, the One and the Indefinite Dyad of the Greater F and the Lesser 1F see The Golden Chalice, Fig. 5, p. 4041, where 2 and 3 are derivable precisely from the Greater and the Lesser. The clincher comes in Alexanders Commentary on the Metaphysics, where he retains from Aristotle a very poignant observation regarding Plato Thinking to prove that the Equal and Unequal other names for the One and Indefinite Dyad are first Principles of all things, both of things that exist in their own right and of opposites he assigned equality to the monad, and inequality to excess and defect for inequality involves two things, a great and a small, which are excessive and defective. This is why he called it an Indefinite Dyad because neither the excessive nor the exceeded is, as such, definite. But when limited by the One the Indefinite Dyad, he says, becomes the Numerical Dyad. And we know that when the One is added to the difference between the Greater and Lesser, it equals, not approximately, but exactly, Two. Thus, Greater Lesser Unity Two F 1F 1 2. Finally, it is also the way in which the Golden Section and its Reciprocal, together with Unity, set the standard of Proportion through the Geometric Mean relationship that is relevant. This pertains not only to Truth, the Reality of the Unfoldment of the Cosmos, but also to Beauty and Goodness. Thus in the Statesman 284a1e8, Plato writes It is in this way, when they preserve the standard of the Mean that all their works are Good and Beautiful. The Greater and the Less are to be measured in relation, not only to one another i.e., GL F 2, but also to the establishment of the standard of the Mean i.e., G1 F, and 1L F. This other comprises that which measures them in relation to the moderate, the fitting, the opportune, the needful, and all the other standards that are situated in the Mean between the Extremes. We now begin to see the extension to Aesthetics Beauty and Ethics Goodness. Aristotle proposed, not surprisingly, the notion of the Golden Mean of Moderation between the extremes of too much and too little. For example, courage is preferable to the extremes of foolhardiness and cowardice. Thus, the Indefinite Dyad in relation to Unity provides the basis for Truth, Beauty and Goodness. Johannes Kepler simultaneously concealed and revealed the nature of the great mystery, uttering the simple words Geometry has two great treasures one is the theorem of Pythagoras the other, the division of a line into extreme and mean ratio. The first we may compare to a measure of gold the second we may name a precious jewel. this is an abbreviated version of a paper first appearing in Nexus Journal of Architecture and Mathematics, vol.4, no.1. www.nexusjournal.com IV III