Pythagoras, Plato and the Golden Ratio The Golden Ratio and the Pentagram in the philosophy of the Pythagoreans By: Ken L. Wheeler Pythagoras, Plato and the Golden Ratio The Golden Ratio and the Pentagram in the philosophy of the Pythagoreans Author: Ken L. Wheeler 1st Edition 2005 All Rights Reserved. Copyright © Oct. 2005 DARKSTAR PUBLICATIONS Printed on acid free paper for archiaval purposes. No part of this book may be reproduced or transmitted in any form or by any means, graphic, electronic, or mechanical, including photocopying, recording, taping or by any information storage or retrieval system, without the permission in writing from Darkstar Publications For information, please contact: Darkstar Publications, 533 Plainview Rd., Lexington, Ky 40517 Published by Darkstar Publications ISBN 0-9712541-5-X $15.00 U.S. Printed in the United States of America Introduction “The Pythagorean Philolaus says that the sun is a vitrescent body which receives the light reflected by the fire of the Cosmos, and sends it back to us, after having filtered them, light and heat; so that you might say that there are two suns, the body of the fire which is in the heaven, and the igneous light which emanates there from, and reflects itself in a kind of a mirror. Perhaps we might consider as a third light that which, from the mirror in which it reflects, and falls back on us in dispersed rays.” [FRAGMENTS OF PHILOLAUS From Boeckh 14. (Stob.Ecl.l:25:3:p.530)] Pythagoras, Plato, and Plotinus after them were well aware, by their intense study of geometry and nature, the Golden section. The purpose of this very small but pithy book is to examine some key passages of Plato and the Phi incommensurability of the Pythagorean pentagram and unveil a heretofore unknown ‘secret’ rediscovered which links the Golden ratio, Pythagoras, the ontological preeminence of the pentagram, and the two most important passages in the Republic of Plato all together. I will show that the Emanationist philosophy/religion of the Platonists employed the Golden ratio into both a “Divided Line” analogy and the “Cave” symbolism to show the variant degrees of proportion and ratio between the visible (aisthetos) and intelligible (noetos) kosmos in a divine Logos (proportion) of which the Monad is Arche. Also I hope to demonstrate that the “Pythagorean oath”, the tetraktys was shown only inside a special ‘Pythagorean triangle’, the 1-1-Phi triangle which is not only the “triple intertwined triangle” of the pentagram, but also the 1-1-Phi-1/Phi analogy of Plato’s Cave and Divided line symbolism for the totality of all things. The Logos (proportion) which comprises the visible and intelligible, I hope to show, formed the foundational doctrine of the Pythagoreans and Platonists who despised all who were “ignorant of geometry” to partake of their instruction, for “As above, so below”, the Platonists before and after understood the “third thing which unified” was a Logos of the Monad, geometrically and arithmetically represented by a power of Phi, which itself is the logos of the necessary emanation of the Monad. Phi is the ratio and relationship of the Monad to its increasingly phenomenal self-image in emanation. “But employing the objects of which the shadows below are resemblances in their turn as images”[Rep. 509]; thru the many ages of nature study the Pythagoreans understood that all nature exhibits a Logos (proportion) to a Principle (1), and this very special ratio was Phi (1 to 1.618033), and so being the case, the 2nd and 3rd Hypostases (the Nous and the Psyche tou pantos [world soul]), though noetic, MUST exhibit this same proportion. After who knows how long a time, the symbolism and analogies I mention below will spell out that which the Emanationists settled upon as the visible models for their knowledge of the ordered kosmos into powers and or proportions of Phi. One very special triangle, its significance uncovered for the first time here in this book, serves as the model for the tetraktys, the pentagram, the Divided line, and the Cave analogies, and its geometers contain the seven sections of Platonic totality. The 1-1-Phi ‘Pythagorean triangle’, the 108-36-36 degrees isosceles triangle. At no time does this book affiliate its discussion with the bastardization of the pentagram in meaning and usage, or the theology of numbers with those groups as commonly denoted in the practice of the white-lighter new-age movement, but only to the root meaning and importance as taught by the Pythagoreans alone, of which the author is certain, is the only point of relevance or importance. That the Pythagoreans and Platonists were obsessive in mind and possessed by talk and models of harmony and proportions as pertains the noetic and visible kosmos is in no doubt, but what is of utmost importance is that Phi comprised the most important “oaths/secrets” of the Pythagoreans and which is the basis of the 5th Platonic solid which “represented the kosmos”. Phi is the ubiquitous logos of the principle to things create in which the mind finds beauty in its ratio and which the entire kosmos is painted in. We know now that even DNA has a ratio of its width to full circle revolution of 1:Phi, as well as spiral galaxies and all things in between at all levels of the microcosmic to the macrocosmic, the great Greek minds of ages past were no less ignorant of these facts even though they didn’t possess the finer instrumentality of our modern world. Everything from rocks and plant growth, human and animal physical proportions, to aesthetics in art, and, as was reported, that Pythagoras discovered that the harmony of musical scales followed to the letter a Phi ratio. This grand discovery led the Pythagoreans and Platonists, who knew that from the smallest to the grandest the Golden ratio Phi was represented, with which they created models and analogies for Emanation, and to declare: “(We) are thinking not of these (geometers), but of the ideals which they resemble”. For further specifics on Phi as its pertains to phenomena, art, and proportions in living things, there are several books worthy of purchase. The effort here however is to unravel the heretofore unmentioned ontological and philosophical principles which varying powers of Phi represented in the models, symbols, and analogies of the Pythagoreans and Platonists, and lastly, for the first time anywhere in print, to have provided a brief overview of what exactly the philosophical meaning of Phi represents, both on its own, and in the mind of the Platonists who never tired of cogitating upon ratios, harmonies, proportions and the Phi principle which underlies the very logos of the Absolute, the Divine Monad. “As there still remains one compound (the 5th Platonic solid, the dodecahedron) the fifth, which the Divine used it (as model/image) for the whole, broidering it with designs (images/forms)” [Plato, Timaeus, 55C] “Pythagoras, seeing that there are five solid figures, of which the sphere of the universe was represented by the dodecahedron.” [Aetius, Placita, ii, 6.5 (Thomas, I,217)] 1. The Divided Line of Plato “The vivific fountain of souls is comprehended under two intellects” [Thomas Taylor, Chaldean Oracles #91] “The triad measures and bounds all things” [Thomas Taylor, Chaldean Oracles #76] [The Divided Line; (Plato’s Republic 509d-511e)] Figure 2 You have to imagine, then, that there are two ruling powers, and that one of them is set over the intellectual world (The One and Being-On [Phi]), the other over the visible (the ‘shadows’, the Mimesis of Nous and that of matter). I do not say heaven, lest you should fancy that I am playing upon the name. May I suppose that you are making this distinction of the visible ([1-Unity, also 1/Phi phenomenally], or Unity-Noetic ‘forms’, 1/Phi Memesis [shape] and 1/Phi [matter] ‘shadow’ = The Lesser section) and of the intelligible (The One and Phi [1/phi, 1/phi, and Psyche .381967] = The Greater section) fixed in your mind? I have. Now take a line (Totality, i.e. = 1, also = 4.236067, or 1/.236067 [primordial Dyadism of Unity of the Monad], Pan, Pentagram) which has been cut into two unequal parts (One and Phi ratio), and divide each of them again in the exact same proportion (which gives (1) Phi [1.618033]), (2) One, (3) One, and (4) 1/Phi [.618033]), and suppose the two main divisions, one to the visible (Lesser = Noetic forms [1-Unity], and ‘shapes’ [1/Phi] and matter-‘shadows’ [1/Phi]) and the other to the intelligible (Greater = Phi and the One), and then compare the subdivisions in respect of their clearness and want of clearness, and you will find that the first section [#3 and #4] (Noetic forms [1-Unity-1/Phi], mimesis [1/Phi], and matter, 1/Phi [.618033]) in the sphere of the visible consists of (noetic) images (Kosmos Noetos, Nous-Eidos). And by images I mean, in the first place, shadows (shapes-Hyle- matter 1/Phi [.618033]) and in the second place, reflections (noetic reflections-Mimesis, Kosmos Aisthetos), in water and in solid, smooth and polished bodies and the like: Do you understand? Yes, I understand. Imagine, now, the other section [#1 and #2] (The Greater section, Being-Phi [1.618033] and Hen-the Absolute-1), of which this (the lesser section, the Noetic forms and ‘shadows and shapes’ i.e. matter) is only the resemblance (a mimesis of the 1st hypostasis and Being-On), to include the animals (animate life, all empirically ensouled Being-Zoe-Phi) which we see, and everything that grows or is made.