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?