nature's word : musings on sacred geometry

Introduction to Phi / The Golden Proportion

Defining Proportion:

It is easy to become distracted by the actual numerical values that the various numbers of sacred geometry are attributed to, such as 3.142… for Pi, 1.732… for the square root of three, and 1.414… for the square root of two, and thereby place importance on those numbers as absolutes themselves. However, we are reminded by the fact that these numbers are all irrational (they have an infinite number of integers after the decimal point with no apparent pattern discernable in that infinite string) that they in fact have no absolute value what so ever, and as such we cannot fixate on the specific values of these numbers. Instead, we must keep in mind that these numbers represent proportions – in other words, they are relative values that are delineated by the geometric figures that they appear in. As an example, the diagonal of a perfect square measures the square root of two (1.414…) only because we can assume that that the edge length of any perfect square equals 1, and it is the relation (or proportion) between the two lines that gives the square root of two its value. To state it simply, if we were to present a line that measures 1.414 inches, it would mean nothing at all to a sacred geometer – unless of course it was presented alongside a line that measures 1 inch, in which case it would come close to visually approximating the square root of two proportion.

Before we deal with the number Phi, or 1:1.618…, we must be sure that the ideas of proportion are clear in our minds. Proportion, as mentioned above, is of central importance when dealing with any sacred geometric number, but the idea of proportion takes on a particularly pertinent role when working with the number Phi. As such, let’s discuss our understanding of proportion before we get into the details of this most astonishing number.

In geometry, a proportion is a relation between various numbers. As a straightforward example, 1:2::6:12, which is verbosely stated as “one relates to two in the same way that six relates to twelve.” Geometrically speaking, the statement 1:2::6:12 is describing two sets of two lines. The fact that this is a true proportion tells us that the two sets of lines are sharing a similar relationship with one another – namely that in both sets, the second line is twice as long as the first. If we were to look at the two sets of lines without being given any specific line length values, it would be possible to say that both sets have equal proportionate values. In other words, the actual lengths of the lines do not have to be assigned in order to understand that they share the same proportionate values.

As such, the most important piece of information that we can glean from our proportion is not the numbers 1, 2, 6, or 12, which could in fact be any values that fall in line with our geometric figures. Instead our central piece of information is the fact that these numbers are tied together by the relational known as doubling, i.e., in both sets the second line is twice the length of the first. If we were to assume that each dot along the lines in our image at left represented two units instead of one, our proportion would read 2:4::12:24. The actual measured values of the lines would change, but the proportionate value between the two sets would remain exactly the same.

In this way, it is the number that relates the line lengths to one another that stands in the spotlight when considering a proportion, with the specific line lengths themselves being of secondary importance. Once again, this is similar to our earlier mentioned example of the square root of two and the square – it is not the actual measure of either edge length or diagonal that is of prime importance, but instead is the relationship of the two line lengths to one another that delineates our understanding of the figure.

Striving as ever to relate the ideas of geometry to our actual experiences of life, we must find personal meaning within the confines of the language of proportion. In fact, it is not much of a stretch to do so if we consider one word that has already been used several times in this introduction to describe proportion – that word is “relation.” Separate geometric entities are being tied together into a relationship by a proportion, with the line lengths representing the entities in a geometric proportion. But if we step out of the bounds of geometry, then we no longer have to use numerical values within our proportion. Instead we could use proportion to describe any four variables that relate to one another in a similar manner – either numerically or otherwise. To represent our proportion in this new, broader sense, we can simply replace the specific numbers with letter variables, i.e., A:B::C:D. With geometry, of course, it is a much more simple issue to prove whether or not the four variables involved in a proportion do in fact share a common relationship (either the two sets of numbers are related to one another by the same number or they are not), but this does not mean that we cannot use proportion to compare common relationships amongst any variables that we choose to use, be they number values or not. And who says that we need mathematical proof for describing relationships between things, anyway?

In the case of our previously mentioned proportion (A:B::C:D), we have four separate variables being tied together by one relational. This type of proportion has been called a “discontinuous” proportion, due to the fact that we are relating two completely separate sets of variables to one another, with A and B representing one set and C and D representing the other. Numerically speaking, many sets of variables can be tied together in such a way – in fact, an infinite number of sets can be related to one another through a discontinuous proportion. As an example, if we know for certain that A equals 3 and B equals 9, then we also know for certain that our relational is a tripling of the first variable to obtain the value of the second. But can we say for certain what values C and D must have? In point of fact, there are an infinite number of possible values that C and D could have. Granted those numbers would have to relate to one another by tripling, but that one restriction could apply to any two sets of numbers from 1/3:1 (and smaller) to 1,000,000,000,000:3,000,000,000,000 (and beyond). Thus a discontinuous proportion is not a very rare or comparably special type of relationship to form between variables.

If we wish to form what is known as a “continuous” proportion, we must link two sets of variables together with not only the relational that describes their interaction, but also with a common variable between the two sets, i.e., A:B::B:C. Here we find that one element (namely B in our example) is one half of both sets of variables that are being related to one another within the proportion. As the name implies, there is a “continuous” relationship represented, from A to B and from B directly to C. The number of possibilities here has been greatly reduced from our earlier discontinuous proportion, and a more solid understanding of how A relates to C can be taken as well. To return to our numerical example above, if we have a continuous proportion and we know that A and B equal 1 and 3 respectively, we can say with all certainty that the value of C must be 9.

Thus a continuous proportion is in some ways a far more desirable proportion to work with, as the relationship that it is defining is far more specific than that of a discontinuous proportion – all variables involved are relating to one another through a common variable, as opposed to relating to one another strictly through the proportion’s relational.

Even in the case of continuous proportions, there are still absolutely vast numbers of possibilities for the numerical values that could be assigned to the variables A, B, and C. By removing D from the initial discontinuous proportion, we have certainly narrowed down the possibilities and created a more specific statement involving the variables in question – yet we still cannot state any value for certain unless we define at least two of the three variables in the proportion. The question must arise then if it is somehow possible to remove one more variable from the proportion, and by doing so create a proportion that expresses an extremely specific relationship. But how could it be possible to relate only two variables when two sets of two variables are required in order to form a true proportion? By simply stating A:B::A:B or A:B::B:A, we are not stating anything at all in a proportional sense: “A is to B just as it is to itself.” Though the statement is true, the relationship is not being defined as it relates to anything besides itself, and thus no proportional statement is being formed.

Various ancient cultures asked the question if there was some way to form a true proportion by using only two variables, and they found that there is indeed a way to form such a specific and exact proportion. The proportion is known, as some may have guessed or already know, as the Golden Proportion, or the number Phi. There is one way – and only one way – to create a proportion wherein there are only two variables: A:B::B:(A+B). To ensure our understanding, let’s take a look at a graphic example of this most rare of proportions:

As the image illustrates, the Golden Proportion defines a proportion wherein the smaller line segment relates to the longer line segment in the same way that the longer relates to the sum of both shorter and longer segments. If we assume that the longer segment is equal to 1, then the entire line must measure 1.618…, or if we assume the total length (A+B) to equal 1, then the line that divides the totality (often labeled the “Golden Division”) falls at 0.618… If the proportion is broken down algebraically, it will show that the division must always fall at the specific value 0.618…, or 1:1.618…, which is the value known as Phi.

Well, it is very interesting that it is actually possible to create a proportion such that it only contains two variables, and that there is only one possible way to do such a thing numerically, and that the number happens to measure the irrational value of 1.618… But what is the meaning of this number?

Phi and the Human Experience:

To explain the meaning of Phi, we have to go back to our earlier idea that proportion does not have to be restricted to numerical values, and can instead be used to express relationships between non-numerical variables. For our example, we shall use proportion to describe a relationship that everyone can understand – the relationship between themselves and the world around them. To keep it simple, we will call the conscious awareness that experiences the phenomena of the world “the observer” and the phenomena that the observer experiences “the observed.”

Let’s start with a discontinuous proportion: A:B::C:D. To help us understand the idea of this discontinuous proportion, let us sdefine what each of these four terms can be taken to represent: B equals the observer right now and A is some specific phenomena that is being observed in the present, whereas C and D would have to represent something that has happened to someone else the our observer (B) is aware of. Although this sounds odd, it is in fact not that rare of an occurance, if we translate this into the statement, “The observer is relating to the current event in the same way that he/she heard that someone else related to an event in their life.” In other words, the observer is relating to what is happening to them by comparing it to something they have heard about happening to another person. It sounds a bit odd when spelled out in this manner, but this is exactly the way in which most children learn to relate to their environment – by remebering what they have heard and been taught by their parents.

So, to step a bit closer to direct relation with the observed, we can relate via a continuous proportion: A:B::B:C, or “The observer relates to the observed in the same way that the observer related to an observation of their own in the past.”

Continuous Proportion:
A : B :: B : C
A = a current observation
B = the observer in at the present time
C = a past observation

The key factor that has changed, of course, is that it is the act of relating to their own past experience that helps them to understand what is happening to them in the present. This is an extremely common type of relation, and in fact describes quite well the type of relationship that most people have with their environments on a daily basis. To state the relationship through the language of proportion: “The observer relates to the observed just as the observer related to a similar observations in the past.” In other words, as soon as something happens in our environment, we cycle back through our past experiences to find a comparable event, and then we react to the new phenomena in a similar fashion to the way that we related to the past event. All in all, it isn’t a particularly bad way to relate to one’s experiences, except for the fact that there is not a very direct relationship occurring observer and the present phenomena.

We can say that both discontinuous and continuous relations are not very direct relations because a type of prejudice (literally, to pre-judge) is intrinsic with both types of relation – basing one’s understanding of what is occurring now on what one already believes from past experience is prejudice in its most simplified form. In a discontinuous proportion relationship, we are acting on the prejudices formed by other people that have been passed on to us, and in a continuous relationship we are acting on the prejudices that we have formed from our own experience.

So, even in a continuous proportion relationship, we are relating everything as it is happening to our past experiences, and as such we are pre-judging it, never allowing any room for new growth and new relation. Worst of all is if we consider that this type of pre-judgment occurs before we even give the phenomena enough time to manifest, and as such we are not really pre-judging it for what it is, but pre-judging if for what we believe it to be, thus making any type of true and direct relation to what is actually occurring extremely difficult.

Those who are familiar with eastern philosophy (particularly Buddhism) will know that the process of pre-judging that is being described here is nothing other than that part of our consciousness known as ego. Because the understanding of ego and its deleterious side effects are of particular importance to this discussion, we will give a quick summary for those who are not familiar with these ideas.

Simply put, ego is the product of a dualistic understanding of one’s relationship with their environment (please see Lesson Two for more details on dualistic understanding). In other words, the observer views him- or herself as intrinsically separate from the observed. As we learned in Lesson One, however, all parts of the universe are in fact united at the base level, and as such ego is constantly under threat of being destroyed by unity. Quite literally, the dualistic mindset that tells the observer that he or she is separate from the world around them is afraid of being consumed and dissolved back into oneness with the universe. At the very early developmental stages the war between ego and the environment is much more pronounced, but as ego grows in strength the battle becomes less and less aggravated due to the fact that as ego has grown to dominatethe situation. Once ego is firmly established (and thus the dualistic mindset is deeply ingrained on the observer’s awareness) it influences the process of observation by instantly relating all phenomena to past experiences of similar phenomena – in other words, it restricts all understanding of reality to its own personal system of categorization and organization, which it has developed from past experiences of similar phenomena. All new phenomena are judged as either desirable or undesirable, threatening or non-threatening, and then dealt with accordingly. By restricting understanding in this manner, ego manages to keep itself central to all phenomena that it observes, literally basing all understanding of the outside world on how that world relates to the separate and individual observer that it has convinced itself exists.

The unfortunate major side effect of the process described here is the fact that the observer becomes less and less a part of the world around them, and more and more wrapped up in their own particular understanding of the world. Naturally, any phenomena that manifests in the world that would break down the ego’s fundamental system of categorization and organization is pushed away and denied, because any threat to the ego’s system of understanding represents a threat to its very existence. The act of continually pushing away those phenomena that challenge the ego’s system of belief results in a literal “ignore-ance” of the world around the observer. Who can honestly state that it could be healthy to ignore the actual events of one’s life in favor of seeing only those events that make one feel more comfortable with what one wants to believe?

The destructive effects of the ego on the observer are well known to many spiritual disciplines, and several have developed practices, such as yoga, meditation, tai chi, and others, to attempt to lesson the control of ego over the observer. The desired effect is to bring the observer back into a direct relationship with the observed, to put them back in touch with the reality of their world, and thereby to reintroduce the observer’s conscious participation with the unified aspect of the universe. Some would say that letting go of one’s ego to re-submerge into the flow of unity from whence we came is the most noble of all goals for a human being to set for themselves.

Sacred geometry, and in fact a huge portion of nature itself, backs up this idea of egolessness as a prime goal of humankind. It does so through Phi, the Golden Proportion. In the examples provided above, both the discontinuous and continuous relations describe situations in which the observer is relating to the observed by past observations – either past observations of other people’s (discontinuous relation) or observations of their own (continuous relation). In both cases, it is the process of relating through ego – through relating to new phenomena via one’s pre-conceived beliefs – that is being described.

With the Golden Proportion, however, secondary relation is removed. A:B::B:(A+B), where A equals the observer and B equals the observed. In other words, the observer relates to the observed just as the observer relates to the observer and the observed conjoined, i.e., added together to form a unity.

The Golden Proportion Relationship:
A : B :: B : (A+B)
A = the current observation
B = the observer at the present time

The implication is that all prejudice has been removed – not simply prejudice in the modern definition of having a pre-set view on race, gender, class, etc., but the very act of taking one’s mind out of the present in order to refer to anything except for what is happening right now is no longer occuring. Despite the fact that the observer and the observed (A and B) are seperate entities, they are so closely internit with one another in this type of relationship that they have been tied together once again (A+B) into a unified state, all the while paradoxically retaining their individual status.

This is only one of the major interpretations of the Golden Proportion. The example of human consciousness as observer has been used here simply because it is a situation that all of us can relate to. As we shall see, there are others, although all relate to the same central idea of diversification of form within unity relating back to unity. Let’s turn now to discuss a slightly broader example, so as to understand how Phi can be applied to ideas other than human transcendence.

Phi as an expression of “Ji Ji Mu Ge”:

It has been stated several times in the various Lessons that the original state of the universe is a unified one, and at some point that unity divides to create form and movement. But there has also been an insistence throughout this body of literature that the original state of unity is still exists, despite the fact that it has divided into an infinite number of forms. Various explanations have been given to explain this paradox of “ji ji mu ge,” or “Thing and thing, no division,” but it is with Phi that we find the most apt expression in geometry of the paradox of “one-yet-many, many-yet-one.”

Let’s take the most simplified geometric expression of Unity – a single straight line – and use it as an experiment for creating simultaneous division and unity. If we were to take our Unity and divide it, we might be tempted to simply divide the line in half:

For many this would make a very poetic and just situation, wherein two equal halves exist, neither of which dominates the other. In many ways, this is true – except for one major problem: we now have two halves, with neither containing a proportionate relation to the original unity. Our two segments have the following proportion: 1:2::1:2. As discussed earlier, the proportion is not a true proportion, because nothing has been stated at all. By this we mean that no relationship between two sets of variables has been established, because only one set of variables is relating to itself (1:2), which doesn’t make a relational – it makes a simple ratio. All we have now is two halves, both of which relate only to their division (1:2, i.e., one relates to two – period). Thus we have divided unity, but in so doing lost our relation to unity.

From here we would have to start dividing unity using fractions or decimals other than 1/2, or 0.50. Let’s try an example, say dividing unity at the 1/4 to 3/4 mark: 0.25:1 :: 0.75:3. Our segments can no longer both be related to the original unity, because if 0.25 is to 1, then in a true proportion, 0.75 has to be multiplied by 4 (the relational as defined by 0.25 to 1), giving us 3. Thus in this example we cannot relate both segments of unity back to unity – only one, which automatically forces the other out of relation to unity.

The same is true for any division of one – there is no way to create a true proportion that will relate both sections back to unity again. Except, of course, the Golden Division.

A:B::B:(A+B), or 0.382:0.618 :: 0.618:1

The two halves are in the same proportion to each other as the greater segment is to the sum of both. There is nothing other than the two segments and the unity that encompasses them in the proportion, and thus we have divided unity into two segments, the both of which can (and do) directly relate back to the original unity in a true proportional sense – both mathematically and figuratively. We have divided unity into multiplicity in such a way that the multiplicity continually relates back to the unity that contains it, and thus an aspect of “ji ji mu ge” has been explained. We shall shortly see the many ways in which the Phi division does indeed lead to infinite multiplicity within unity.

To summarize, geometrically speaking there is only one way to create a proportion that contains only two variables. That proportion, when algebraically reduced, happens to equal the irrational number known as Phi – 0.618… or 1:1.618… Phi “coincidentally” happens to give us an apt expression for the transcendence of humankind, and also gives us the ability to understand how it is possible that the unified universe can be divided into myriad forms, yet continuously relate back to the ever-present unity of all things.

Now wait until you see how nature uses the number…

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