Golden Mean

The Golden Mean can be construed as the basis of philosophy and Sacred Geometry, one of the Transcendental Numbers, [*] and is typically derived from Fibonacci Numbers. [* Technically, from a strictly mathematical standpoint -- as has been pointed out by several readers -- the Golden Mean, Phi, is the solution to a polynomial equation and thus not mathematically transcendental. However, if one looks at the other definitions of transcendental, most would agree that it qualifies.]

According to Robert Lawlor [1], “Ancient geometry rests on no a priori axioms or assumptions.  Unlike Euclidian and the more recent geometries, the starting point of ancient geometric thought is not a network of intellectual definitions or abstractions, but instead a meditation upon a metaphysical Unity, followed by an attempt to symbolize visually and to contemplate the pure, formal order which springs forth from this incomprehensible Oneness.  It is the approach to the starting point of the geometric activity which radically separates what we may call the sacred from the mundane or secular geometries.  Ancient geometry begins with One, while modern mathematics and geometry being with Zero.”

Other authors have noted that “Both the ancient Greeks and the ancient Egyptians used the Golden Mean when designing their buildings and monuments.” “Artists as diverse as Leonardo da Vinci and George Seurat used the ratio when constructing their paintings. These artists and architects discovered that by utilizing the ratio 1 : 1.618..., they could create a feeling of order in their works.  Even today, artists are still using this proportion in their works, and scientists, like Roger Penrose are discovering new things about the Golden Mean and its place in science, mathematics, and nature.”

<http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibInArt.html> is an excellent website on the subject, and notes among many other things, the connection with classical music and the Golden Mean.  In a 1996 article in the American Scientist, for example, Mike Kay reported that Mozart’s sonatas were divided into two parts exactly at the Golden Mean point in almost all cases.  Inasmuch as Mozart’s sister had said that Amadeus was always playing with numbers and fascinated by mathematics, it appears that this was either a conscious choice or an intuitive one.  Meanwhile, Derek Haylock noted that in Beethoven’s Fifth Symphony (possibly his most famous one), the famous opening “motto” appears in the first and last bars, but also at the Golden Mean point (0.618) of the way through the symphony, as well as 0.382 of the way (i.e., the Golden Mean squared).  Again, was it by design or accident?  Keep in mind that Bartók, Debussy, Schubert, Bach and Satie may have also deliberately used the Golden Mean in their music.

(6/6/05) Another excellent website is http://goldennumber.net. This one even includes phi to 20,000 decimal places -- just in case you need this for your next term paper. In particular, one might also want to check out Pascal's Triangle. The latter is just too strange for words, incorporating as it does Fibonacci Numbers.

In a much more esoteric vein, Ronald Holt, Director, of the Flower of Life Research has included in <http://www.floweroflife.org/spiral01.htm>, “To The Golden Spiral In All of Us” (dated April 21, 1999), in which he notes, “Sacred geometry is the study of geometric forms and their metaphorical relationships to human evolution as well as a study in fluid evolutionary transitions of mind, emotions, spirit, and consciousness reflected in the succeeding transition from one sacred geometric form (consciousness state) into another.”

Furthermore, “True sacred geometric forms never fixate or stagnate on one single form.  Instead they are actually in constant fluid transcendence and change (evolve or devolve) from one geometric form to another at their own speed or frequency."

(5/31/05) The Golden Spiral, in fact, has been found in Crop Circles -- sort of Mother Nature's "performance art" in wheat and other crop fields -- and thus do sort of grow on you. An excellent description of this design is given by ka-gold's golden spiral.

FfFfFfFfFfFfFfFfFfFfFfFfFfFfFfFfFfFfFfFfFfFfFfFfF

The Golden Mean can be determined via geometry by taking a square with all sides equal to 1, drawing an arc with the center of radius at the midpoint of one side and through the corner of an opposite side, and extending the original side to where it intersects the arc.

The length of the extension will then exactly equal f [and the base’s total length being F.]  From this same geometry, we can calculate F by noting that the diagonal in the square from the midpoint of one side to an opposite corner is equal to the square root of the sum of the squares of the opposite sides (i.e. 1 and 1/2) as per the Pythagorean Theorem.  From this, we calculate the square root of 5/4 (1.11803398875...), and then add 1/2 the side, to obtain 1.61803398875...

The ancient Greeks, who were really into aesthetic geometric appeal, established as one of their primary axioms concerning proportion, to always use the golden mean in dividing a line, i.e. dividing it at a point, C, on the line AB:

A________________C_________B

such that:

AB/AC = AC/BC = 1.6180339875...

For clarity, the Golden Mean can be assumed to be either one of two values, given by [2]:

F = 1.61803 39887 49894 84820 45868 34365 63811 77203 09180...

or

f = 0.61803 39887 49894 84820 45868 34365 63811 77203 09180...

The Golden Mean, the number, is the only number in which, among other things, satisfies the mathematical relationships:

F = 1/F + 1       ;       f  =  1/f - 1

For the esoteric crowd, f can also be represented by the very strange equation:

f = 1 + 1/{1 + 1/[1 + 1/(1 + 1/{1 + 1/[1 + 1/(1 + ...)]})]}

Another way to view the above equation is to think of it as representing rabbits attempting to draw their family tree in a furry, more prolific version of Roots.  [This aside is reference to one original example of the usefulness of Fibonacci Numbers, which mathematically is the easiest manner to obtain the values of the Golden Mean (and to as many powers as one might like) by simply dividing one member of the Fibonacci Series with another.]

In fact, the Golden Mean Mathematics is a whirlwind of fascinating mathematical ideas and curiosities.  Suffice it to say -- for the mathematically-less-inclined -- that the Golden Mean is: 1) intimately tied to the number 5 (and in particular to regular pentagons and five-pointed stars), 2) relates directly to the ideal human body (i.e. the Golden Mean raised to various powers are indicative of the body’s proportions), and 3) has been used extensively in ancient (and modern!) architecture -- among other things.

In the relationship between F, f, and 5 (they even sound alike!), we take a 5-pointed star -- the points of which form an inscribed five-sized, regular pentagon.  By an arbitrary choice of measurement units, the length of a line drawn from one point of the star to an opposite point, can be set equal to f.  This results in the line between two adjacent points (one side of the pentagon) automatically equaling f2.  The line from a point to the interior pentagon is then f3, the side of the interior pentagon is f4, and so forth, ad infinitum.  Then by connecting these points in sequence, we suddenly discover we’ve been thrown a curve: a new geometrical delight known as

We might also mention in passing that the connection of the Golden Mean to the five-pointed star may be why the sacred practice of Wicca seems so preoccupied with this universal symbol -- regardless of whether or not their practitioners fully understand the significance.  (Or why flags of many nations, states, or corporations -- the latter such as Texaco and others -- all enthusiastically use the five pointed star!  Not that these nations, states, etceteras, are practicing Wicca, but...  Well... You understand!)

With respect to the body proportions, this point can be demonstrated by taking the measurements of the average of many people -- with women’s measurements (for some inexplicable reason) approaching the ideal more closely than men -- whereupon we find that the position of the navel (a human’s first channel of nourishment and life) divides the body’s height at precisely the Golden Mean.  Furthermore, if the distance from the brow (top of the eye) to the nose is 1, then the distance from the brow to the crown is F.  Going in the opposite direction, the distance from the nose to the base of the neck is F, the neck to the armpit is F2, the armpit to the navel is F3, the navel to the reach of the fingers is F4, and the distance from the fingertips to the soles is F5.  On a smaller scale, in measuring the length of the bones in the human hand, we find measurements of  1, F, F2, and F3 (the last bone being within the palm of the hand).

The below drawings, courtesy of Michael S. Schneider [3], emphasize these points.

Beginning to get a hand on this?  Michael S. Schneider [3], for example, notes that: “The body’s structure is a mirror of our psyche, a denser expression of the energetic patterns of our soul.  Body and soul somehow partake of the same design.  But in what way can a mathematical ratio permeate our souls?  Through beauty.  A deep part of ourselves recognizes in flowers and dancers the beauty of the mathematical infinite and sees in it the endlessness of our own depths.  Natural beauty resonates with the archetypal nature within us.”

On a decidedly more mundane application of such “natural beauty”, investors in the stock market have used the delights of the Golden Mean for the purpose of making money and, hopefully, large wheelbarrows of it!  As it turns out, F and f are important in the stock market, where the index averages (such as the Dow Jones Industrial Average) typically rise a certain number of points (say a hundred), and then fall back a number of points equal to 0.618 x 100, before again rising to new heights.  The amount of the initial leg of this cycle is not always clear, but the retreat path is much more clearly defined.

Obviously, this cyclical nature also works in the opposite direction as well (a fact which novice investors might want to keep in mind).  It must be noted that there are any number of variations in the amount of the market’s rise and fall, as well as variations in the time periods from minutes to decades.  In effect, there are numerous cycles within cycles within cycles.  Nevertheless, this predictive technique (known as Elliott Wave Theory) has been practiced successfully by numerous stock market analysts, who seem to have an unusually deep appreciation for F Lo Sophia and the money it can effortlessly make them.

There now seems to be a justifiable reason for our ancient ancestors having such a reverence toward the Golden Mean.  Besides The Great Pyramids, their architecture showed it through such examples as the west facade of the Greek Parthenon, which perfectly fits within a golden rectangle (whose dimensions are 1 and F).  The west facade of the Notre Dame Cathedral in Paris is also loaded with Golden Mean ratios, and more recently, the United Nations Building in New York City is designed as three golden rectangles (equivalent to three Parthenons stacked upon one another).  Perhaps we shouldn’t attribute all our knowledge of the Golden Mean to the ancients, as such examples of Sacred Geometry keep cropping up in more modern structures.  At least those designed by more “enlightened” architects.

Why?  Because even artists with absolutely no interest in math (or an aversion to the subject) will inevitably respond to the beauty in the architecture which arises from the mathematics!  This is likely true of everything from “cubism” to Mozart.  For it is the mathematical resonance within the sacred geometry symbolism which touches our soul.

The Golden Mean also touches such intriguing phenomena as The Fifth Element, the Harmony of the Spheres, Connective Physics, Philosophy and the Tree of Life.

Forward to:

___________________________

References:

[1]  Robert Lawlor, Sacred Geometry, Philosophy and Practice, Thames and Hudson, London, 1982.

[2] Handbook of Chemistry and Physics, 56th Edition, CRC Press, 1975-1976.

[3] Schneider, Michael S., A Beginner’s Guide to Constructing the Universe, Harper-Collins Publishers, 1994. [http://www.constructingtheuniverse.com]