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Lesser = 1 / f .
This is, to my knowledge, a novel hypothesis, and my goal in this paper is to argue its potential for validity. Recall the following facts:
Given these facts and the hypothesis above, we have:
The fact that the ratio of the Greater to the Lesser is f2 and not f is crucial to my interpretation of the Indefinite Dyad. The importance of these relationships will become clear only after a review of some pertinent classical Greek philosophy.
THE PHILOSOPHICAL BACKGROUND
Plato, primarily as a proponent of Pythagorean philosophical doctrines, was very careful with what he did and did not reveal, being under an apparently severe oath of secrecy. Both his writings and lectures are enigmatic, and he only very carefully and subtly provides the clues with which the observer may be capable of uncovering the inner doctrines for themselves. His method in the written dialogues appears to be similar to his reported approach in the Academy, where he would propose the problem to be solved. He would present the problem, puzzle, anomaly, apparent contradiction or incomplete result, intending that the attentive student would abduct an explanatory hypothesis. Thus, the underlying intention was to get the observer (Academy member or dialogue reader) to abduct (hypothesize) an appropriate solution or answer, rather than to accept the dead end, apparent contradiction or incomplete result.
There are several Platonic puzzles and unsolved issues. Some of these arise within the dialogues and others in remarks made by Aristotle and early commentators regarding Plato's doctrines. When several of the key puzzles are viewed in conjunction, they help point in the direction of the required solution. In particular, I will argue that the Timaeus and the Republic together point to the Golden Section. The Timaeus does so by the conspicuous absence of the Golden Section, since Plato provides no appropriate elementary triangle for the construction of the Dodecahedron, often considered the most sublime of the five solids. And in the Republic, Plato subtly and with great economy embeds the Golden Mean in the beautiful ontology of his Divided Line analogy. Together the Timaeus, Republic, and Parmenides 133b ("worst difficulty argument") point to continuous geometric proportion as that which binds together Plato's realms of Being and Becoming. And finally, as we shall see, continuous geometric proportion and the Golden Mean are embedded in Plato's most important ontological principles, the One and the Indefinite Dyad. This should have special ramifications for a whole family of issues surrounding the role of geometry in aesthetics.
Aristotle makes it eminently clear that within the Academy, Plato professed Two Principles, principles that were involved in the construction of the Forms (Universals or Archetypal Numbers), as well as the Sensibles (Particulars) of our Empirical World. The First Principle is generally acknowledged. It is the Good of Plato's Republic, also referred to in the Academy in its more mathematical context as the One. The other Principle was usually referred to as the Indefinite Dyad, and at times as the Greater and the Lesser, Excess and Deficiency, or the More and the Less. Occasionally one would see the Two Principles contrasted in terms of the One as Equality and the Indefinite Dyad as embodying Inequality.
Although there are important references to this Second Principle in the dialogues (especially the Philebus), there is no real clarity as to its meaning and definition. It is an understatement to suggest that Plato was reserved in his references towards it. In fact, when he apparently lectured on the subject of the One and the Indefinite Dyad in his so-called Agrapha Dogmata (Unwritten Lectures) or Lectures On the Good, he continued to veil his presentation in secrecy. Simplicius records, in his Commentarius in Physica 453.25-30:
And as these Two Principles were ontologically prior to and causally involved in the manifestation of both the Forms and Sensible things, it should not be surprising that Plato held them to be of the utmost importance. Thus we learn from Aristotle's pupil and commentator, Alexander, that these Two Principles were "more important than the Ideas" (Commentarius in Metaphysica 88.1) [Barnes 1984: 2440].
Now according to Aristotle and others, what Plato presented to members of the Academy and in public lectures was not always identical to the content of the written dialogues. We learn from Simplicius that:
Thus, there is considerable evidence of Plato avowedly professing that there are the Two Principles of the One and the Indefinite Dyad.
The great mystery has always been, what exactly does Plato mean by the Indefinite Dyad, or as he called it, Excess and Deficiency, or the Greater and the Lesser. Aristotle does tell us:
And of course all Sensible objects of this world are derivative from these original Principles via the Forms or Numbers.
Now in the Timaeus, Plato boldly hints at the deeper revelations to be gained by those who carefully pursue his clues and incomplete analyses. He poses the question:
As Keith Critchlow indicates:
Plato gives us the Ö2 triangle for the construction of the Cube, and the Ö3 triangle for the construction of the Tetrahedron, Octahedron and Icosahedron. But the triangle (or the root numbers embedded in it) necessary for the construction of the Dodecahedron is most conspicuously absent. Regarding the Ö2 and Ö3 primitive triangles, however, Plato states cryptically (and yet very revealingly for the astute student):
The missing triangle for the construction of the Dodecahedron must involve (purely from mathematical considerations) the Golden Section. But why is the Golden Section to be so protected within the Pythagorean tradition? I would like to propose here that for Plato it is its ontological priority over the Forms, the Numbers. It is the discovery that it is embedded in, if not the very basis of, the Principle of the Indefinite Dyad that is so remarkable. And as we shall see, it is this Principle along with the One that is involved in a deeper revelation regarding continuous geometric proportion.
In the Timaeus, Plato states:
Now following the Pythagoreans, Plato places a great deal of emphasis on numbers, ratio (logos), and proportion (analogia). As Aristotle attests in several places, " those who speak of Ideas say the Ideas are Numbers" (Metaphysics 1073a18-20). And again:
In the Republic, Plato presents a series of similes or analogies with the apparent purposes of:
He does this with the Sun Analogy (Republic 502d-509c), the Divided Line (509d-511e), and the Cave (514a-521b). I have discussed these metaphors elsewhere at length [Olsen 1983; 2002], and will be concerned here primarily with how the Divided Line assists in penetrating into the possible nature of the Indefinite Dyad.
THE DIVIDED LINE AND THE "SAYRE CHALLENGE"
Usually when commentators attempt to determine how the line is to be divided, they fail to first fully consider the underlying significance of Plato asking the reader to divide the line unevenly. Now it seems clear to me that if Plato is concerned primarily about continuous geometric proportion, as he appears to assert in Timaeus 31b-32a, then there is one and only one way to divide a line (and it is unevenly) such that you immediately have a continuous geometric proportion, and that is with a division in extreme and mean ratio, or what we now call a Golden Cut: 
Plato then asks us to cut each of those two segments again in the same ratios, namely Golden Cuts. In effect what Plato is asking us to do is to perpetuate the continuous geometric proportion into the four subdivisions of his Divided Line. What he has effectively done through a series of Golden Cuts is to bind his so-called Intelligible and Sensible Worlds (and their subdivisions) together through continuous geometric proportion employing the Golden Section.
Kenneth M. Sayre of the University of Notre Dame, in his 1983 book, posed a very interesting challenge to anyone who would propose a Golden Section solution to Plato's Divided Line. I will call it the "Sayre Challenge." He writes:
I accept the challenge. In fact it will assist us in explicating the underlying significance of the Indefinite Dyad and the One. Let us take a Pentagram, which inherently contains numerous Golden Cuts, and extract one of its lines while retaining its points of intersection.
Thus, we have line ab, which has Golden Cuts at points c and d.
Next take a pair of compasses and, rotating line segment cd (at point c), cut line ab at point e.
One consequence of this construction is what some have referred to as the "anomaly" that line segments dc and ce must be equivalent. Sayre states:
I want to suggest that, to the contrary, the abductive solution to this so-called anomaly helps lead to the most fruitful insights. Plato knew exactly what he was doing. He was very subtly embedding the Indefinite Dyad into his Divided Line, expressing it through continuous geometric proportion.
I propose that the inner two line segments dc and ce should be seen as each representing Unity or One. Let us label the line in a manner consistent with the "Sayre Challenge", with A being the smallest and D the largest. Given that segments B and C are equal (each being 1), it turns out that D will be f, the Greater, and A will be 1/f, the Lesser.
Thus, D : C :: B : A is none other than f : 1 :: 1 : 1/f, i.e., Greater : One :: One : Lesser.
In the Statesman, Plato suggests that:
This agrees with my hypothesized associations. The Greater and the Lesser are to be related not only to one another, Greater : Lesser (a single proportion exhibiting the f2 ratio), but also to the standards that are situated in the mean between the extremes, Greater : One :: One : Lesser (a continued proportion exhibiting the f ratio).
We can now verify the "Sayre Challenge":
(here we have used the facts that 1/f + 1 = f, 1 + f = f2, and f + f2 = f3).
The Divided Line presents Plato's Two Principles, both the One and the Indefinite Dyad. And its accomplishment is that it effectively counters the "worst difficulty" argument of Parmenides 133b, i.e., the argument that there is no connection between the Intelligible Realm (segments D and C) with the Visible Realm (segments B and A). The solution is that the two Realms are bound together through continuous geometric proportion. And not only that, but the powers of the Golden Ratio are carried into the Visible Realm as a kind of enfolded Implicate Patterned Order.
DERIVATION OF THE ROOTS OF 2 AND 3
In other words, the whole number 2 can be generated from the Indefinite dyad. Indeed, recalling that Greater - Lesser = 1, we have Greater + Unity - Lesser = 2.
Now the real secret of Plato's Indefinite Dyad (in addition to generating the whole numbers) is that it may be employed to derive the other crucial roots (Ö2 and Ö3) necessary for the construction of four of the five Platonic solids.
The following construction is the result of carefully combining two insights, one that I had regarding Ö3, and one that Mark Reynolds shared with me regarding Ö2. While contemplating the nature of the Indefinite Dyad, which I had already decided must be the Golden Section and its reciprocal, I had a dream in which I saw the Greater and the Lesser as the legs of a right-angled triangle of which the hypotenuse was Ö3. I jumped out of bed and grabbed a pair of compasses, straightedge and pencil. I did the construction and lo and behold it was true (as we'll verify mathematically below).
Then a year later I had the good fortune of meeting Mark Reynolds. I shared this construction with him, and he in turn showed me how Ö2 was derivable, in a very similar manner, from the Lesser and the square root of the Greater as legs of a right-angled triangle. (In both cases I was attending the KAIROS Summer School studying under Dr. Keith Critchlow and John Michell - Buckfast Abbey in Devon, England in 1997, and Crestone, Colorado in 1998.)
The construction of the Indefinite Dyad Template follows. Though the construction has already proven to harbor many wonderful properties, notice in particular the Quadrilateral DKMH, which acts as a kind of ontological entheogenomatrix. Because of its morphology and seemingly sublime function, I propose to name it the "Golden Chalice of Orion".
(For completeness, we note that KM » 0.708, which is within 0.2% of Ö2/2.) The amazing fact is that the two diagonals of the "Golden Chalice of Orion" are precisely Ö2 and Ö3!
Hence, Ö2 and Ö3 are ultimately derivable from the Greater (DK = f) and the Lesser (DH = 1/f). As a result, the Indefinite Dyad gives rise to the roots that are employed in the construction of the Cube, Tetrahedron, Octahedron and Icosahedron. And of course this Second Principle is directly related to the derivation of the Dodecahedron. As such, the One and Indefinite Dyad were for the Pythagorean Plato the Principles behind all of existence. In the end, I suspect that this was the great Pythagorean secret that Plato could not openly reveal, but only hint at, expecting his attentive followers to abduct the solution. We begin to see why Plato was so careful not to reveal the real nature of the Golden Section and its reciprocal, respectively the Greater and the Lesser.
 I have argued this point exhaustively elsewhere, and would at this stage simply refer the reader to my dissertation [Olsen 1983], especially pp. 45-121. return to text
 It should be noted that there is definite evidence that Plato and his pupils were working with the Golden Section in the Academy. Heath relates, "We are told by Proclus that Eudoxus 'greatly added to the number of the theorems which Plato originated regarding the section, and employed in them the method of analysis [abduction].' It is obvious that the section was some particular section which by the time of Plato had assumed great importance; and the one section of which this can safely be said is that which was called the 'golden section,' namely the division of a straight line in extreme and mean ratio which appears in Eucl. II. 11 and is therefore most probably Pythagorean"[1956: v. 1, 137]. return to text
 Others have proposed the principle of the golden section in the Divided Line (see [Brumbaugh 1954] and [Des Jardins 1976]) or a Fibonacci approximation to it [Dreher 1990]. But these prior attempts do not embody the essential features of the Greater and Lesser (Indefinite Dyad) in relation to the One, as I depict them below. For a critique of the sufficiency of earlier arguments (taken individually) that attempt to establish the golden sectioning of the Divided Line, see [Balashov 1994]. return to text
 We could also simply begin with the line as Unity, and then do the subsequent cuts. The initial golden cut of the line would then give us a longer and a shorter segment, namely, 1/ f and 1/f2, and hence, (1/f) : (1/f2) would be in f ratio. I suspect that the understandable mistake of earlier commentators who were close to uncovering Plato's mystery, and hence the failure to notice the actual relationship between the Greater and the Lesser, follows from the tendency to associate Plato's Greater of the Indefinite Dyad with the "greater" of the two line segments, and the Lesser of the Indefinite Dyad with the "lesser" of the two line segments. With my interpretation, when we divide the line in this way, Unity becomes the "Greater," 1/ f becomes the "Mean," and 1/f2 becomes the "Lesser," relatively speaking. If again, we proceed further with this initial golden cut of a line of Unity, and do our subsequent two golden cuts, we discover that the four resulting line segments are in the following continued proportion: (1/f2) : (1/f3) :: 1/f3 : 1/f4. Here, relatively speaking, 1/f2 "acts" as the Greater, 1/f3 "acts" as the Mean, and 1/f4 "acts" as the Lesser. The relation here of the Greater to the Mean, and the Mean to the Lesser, is again the f ratio. And 1/f2 (the Greater) in relation to 1/f4 (the Lesser in this context) is f2. The advantage of portraying the divisions of the line as I do in this paper, is simply to more clearly reveal the underlying essence of the ratios, as they reflect the One and the Indefinite Dyad, i.e., Greater : 1 :: 1 : Lesser. Otherwise there is a tendency to overlook the crucial f2 relationship between the Greater and Lesser that lies at the heart of this paper. return to text
 Here I am thinking of the Implicate Order that David Bohm, the physicist, suggested is enfolded into the outer Explicate Order of our world. This "patterned order" enfolded into Nature appears to be closely related to the continuous geometric proportion of the One and Indefinite Dyad as expressed, for example, in Fibonacci and Lucas whole number approximations in minerals, plants, animals, microtubules and DNA. See, for example, [Bohm 1980; Dixon 1992; Goodwin 1994; Penrose 1994; Martineau 1995]. return to text
 Neither Mark nor I labor under any false illusion that we have discovered these things; we have simply rediscovered them independently. Perhaps we are uncovering what for many in the past may have been restricted or esoteric knowledge. return to text
 Plato's ontology is based upon his Pythagorean belief that the Divine manifests throughout our world through the Numbers. Thus, he appears to be suggesting that the Principles of the Numbers, namely the Indefinite Dyad in relation to the One, generate or unfold the Divine within all things through this number matrix. During this construction, I would ask the reader to keep in mind a very important comment by Johannes Kepler: "Geometry has two great treasures: one is the theorem of Pythagoras; the other the division of a line into extreme and mean ratio [golden cut]. The first we may compare to a measure of gold; the second we may name a precious jewel" [quoted in Hambidge 1920]. This construction in effect embodies the application of these "two great treasures". return to text
 This is Mark Reynolds's very important contribution that allows this "Platonic Template" to work. return to text
 Notice also the relationship between the sides of
the three squares, AI : AB : CK. They are
in the continuous geometric proportion, f : 1 : 1/f. Thus they
perpetuate the f relationship. In the case of the areas of the three
squares, we have the f2 relationship perpetuated throughout.
Square AIHJ : Square ABCD : Square CKJG
: 1 : 1/f2.
return to text
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