A proportion is a measure of equivalence between two ratios, when the ratio of one element a to a second element b is compared with that between a third element c and a fourth, d. This is usually written: a:b::c:d, e.g., 2:4::7:14, which means that the ratio of 2 to 4 is equivalent to the ratio of 7 to 14. This represents a deeper level of appreciation than the simple difference expressed by a ratio, and the ancient Greeks thought of such a proportion as analogy. The Pythagoreans called the type of thinking involved a discontinuous proportion. This example consists of four terms, but you can see that it may be extended indefinitely.
There is a third kind of proportion in which one of the elements is common to both ratios being compared. For any three consecutive terms in such a progression, it may be said that the ratio of the first to the second is equal to the ratio of the second to the third, and so on. Thus 4:8 = 8:16. The Greeks called this a continuous proportion. As it consists of only three terms, we have to think more deeply about it because its meaning is not immediately obvious to our senses and we ourselves have to 'work out' what the mean term b should be. We are forced to think of the proportion as division of a unity.
Thinking further along this line, we come to yet another proportion which consists of only two terms. It occurs when the smaller term is to the larger as the larger is to the smaller plus the larger. It may be written: a:b::b:(a + b). There are many examples of this type of proportion in which the third term, (a + b), is greater than 1.
In mathematics and the arts, two quantities are in the golden ratio if the ratio between the sum of those quantities and the larger one is the same as the ratio between the larger one and the smaller. The golden ratio is a mathematical constant, and is approximately 1.6180339887.
Many artists and architects have proportioned their works to approximate the golden ratio — especially in the form of the "golden rectangle", in which the ratio of the longer side to the shorter is the golden ratio — believing this proportion to be aesthetically pleasing. The same idea is implicit in the design of "wide-screen" televisions, computer monitors, etc.
The golden rectangle has the unique property that if a square is cut off one end, the remaining rectangle is also golden, and this process can be repeated indefinitely as shown in the diagram.
In Sacred Geometry, Robert Lawlor writes:
It is important to mention that φ represents a coinciding of the processes of addition and multiplication. Addition is the most common process of growth, whether it be of the cells in our body, of wealth, of knowledge, or of experience; it is a deliberate, logically expanding development. Multiplication is really a special form of addition, an accelerated form: 4 X 4 is really 4 + 4 + 4 + 4. But in this acceleration there is the intervention of an extraordinary moment of transformation: what was a linear accumulation suddenly becomes a square, a surface, a plane. There has been a leap of growth.
In the plant, the simple additive growth occurring in the stem suddenly exolodes into a fruit or a flower, or a seed gradually swells from absorbing moisture and germinates. In studies, one's additive accumulation of skills or data suddenly blossoms into a genuine understanding....
There are three significant circumstances in which the ancient researchers of this principle found this simultaneous coincidence of additive and multiplicative processes. Each of these gives the sense of a combination of material and supra-material growth. They are the square, musical harmony, and the proportion φ.
[The reader may find it helpful to refer to Tertium Organum for a discussion of the significance of dimensions. — Ed.]
The Golden Proportion or Golden ratio is a special case of a geometric ratio.
Draw a square making its side, b, one unit of length. By means of a vertical line, divide the square into two equal rectangles, the lengths of whose opposite sides will be 1 unit and 1/2 unit respectively. In the right half of the square, draw a diagonal as shown. By the Theorem of Pythagoras, the length of the diagonal will be √5/2. Rotate this diagonal until it lies along the bottom of the square and extends beyond it by a distance designated by a. The total length of this line, c, will be be (√5/2 + 1/2) units.
It will then be found that the ratio of a to b is equal to the ratio of b to (a + b), or a:b=b:c.
Taking the special case when c = 1, we have a:b::b:1.
Then, a = 1 — b; b = 1 — a; and a + b = 1.
Hence a/b = b/1; b2 = a; and b = √a
Translated into ordinary language, this means that the root of a is equal to the root of b2, so that a is related to b as a root is to a square. Robert Lawlor remarks that:
This completes the mathematical metaphor for the Trinity: 'Three that are two that are One'. It is the ultimate reduction of proportional thought to the causal singularity.
If we ... use proportion as a model for the perceptual activity based on the recognition of differences, we have in this unique Golden Proportion within Unity a case where the perceived difference (that which we experience as an object) plus the perceiver of that object are symbolised as contained within a sustained awareness of an all-encompassing Unity, a:b::b:1. This perceptual state corresponds to the goal of dynamic meditation....
Why, it may be asked, cannot Unity simply divide into two equal parts? Why not have a proportion of one term, a:a? The answer is simply that with equality there is no difference, and without difference there is no perceptual universe for, as the Upanishad says, 'Whether we know it or not, all things take on their existence from that which perceives them'. In a static, equational statement, one part nullifies the other. An asymmetrical division is needed in order to create the dynamics necessary for progression and extension from the Unity.
Fibonacci considered what might happen if a newborn pair of rabbits of opposite sex were enclosed in a walled garden. If rabbits mature after one month and each mature pair were to breed in the second month to produce a new pair in the third month, how many rabbits would there be in the garden after one year?
In the first month, there is one juvenile pair. In the second month, this pair matures and in the third month produces another pair, so we have one mature pair and one juvenile pair. In the fourth month, we have the original pair, a newly mature pair, and a new juvenile pair from the original pair, making three pairs in all. In the fifth month, two new pairs are born (one from each mature pair) and we have three mature pairs for a total of five pairs.
Assuming that that the process continues and that none of the rabbits dies during the year, we can count up the pairs each month as follows: 1 + 1 = 2; 2 + 1 = 3; 3 + 2 = 5; ....; and if we continue we shall find that each new number is the sum of the preceding two numbers.
Thus after 12 months, we shall obtain the "time series": 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, i.e. we should be able to count 233 pairs of rabbits in the garden after one year. This is an example of exponential growth, and if it were allowed to continue much longer, the garden would be full of rabbits and there would be nothing for them to eat but each other.
If you now get your calculator to work out the square of φ, you will get something very close to 2.618033989, and if you calculate the reciprocal of φ, you will get 0.618033989. In other words, to square the Golden Mean, you need only "add one", and to find the reciprocal of the Golden Mean, you need only "subtract one".
Written algebraically, φ2 = φ + 1, and 1/φ = φ — 1.
Please refer back to the golden rectangle for a geometrical explanation of this phenomenon.
The cube of phi, φ3, is a volume arrived at by simultaneously adding and multiplying.
1/φ + 1 = 1 X φ
The volumetric expression of phi, φ3, becomes a new unity, for here the abstract principle of φ achieves expression as a unity on the physical level of volume, the cube. In an ancient Egyptian inscription, Thoth says:
I am One which transforms into Two (polarity)
The progression then occurs as though we were to continue to consider the One as without definition, up until the moment it becomes a tangible manifest unit, the cube; as we've just seen, ø3 = 1. And if the transformative power of redemption is fixed to the material cross of addition, +, then the moment of resurrection comes when this principle allows the cross to fall + X, and an exponential growth occurs, an incomprehensible non-sequential leap to another level of being.
1 + φ = φ X φ = φ + 1
φ + φ2 = φ3 = φ X φ X φ = φ X φ2
I am Two which transforms into Four (surface, 22 = 4)
I am Four which transforms into Eight (volume, 23 = 8)
After all of this, I am One.
Yet as we take note of factual observations such as those referred to above and ponder their possible cosmic significance, we may gradually, or perhaps even suddenly, gain an intuitive grasp of their reliable constancy in the midst of perpetual change. We may then acquire some ability to make practical use of some of the apparently irrational principles which seem to underlie the phenomena of the visible and tangible world.