Golden Mean Mathematics

The Golden Mean, one of the Transcendental Numbers, is fundamental to Sacred Geometry, astronomy (e.g. Harmony of the Spheres and A Book of Coincidence), architecture (e.g. The Great Pyramids), human and other physiologies, the stock market, and most everything else.  It can be easily derived from the Fibonacci Numbers, and mathematically, is nothing less than a tour de force.

For clarity, the Golden Mean is defined as either one of two values, given by [1]:  

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 relationship:

F = 1/F + 1 =  f + 1 =  1/f  

One can show that the only numbers which satisfy these types of the reciprocal nature between f and F, is done by solving the quadratic equation of F²- F - 1 = 0.  I.e.

F = (1/2) [ 1 ± Ö 5 ]

 f can also be calculated by use of the very strange equation:  

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

Or more commonly, derived from a Fibonacci Numbers, by taking the ratio of two adjacent numbers in the series.  The individual Fibonacci numbers, themselves, can be calculated from the equation:

F(n) = (2/Ö5) {- [-2/(1-Ö5)]ⁿ / [1 - Ö5] + [-2/(1+Ö5)]ⁿ / [1 + Ö5]}  

where, obviously, the number 5 (or Ö5) figures prominently.  Is there something magical about 5 or the square root of 5?  Of course!  Otherwise, why bother to ask?  Duh.  Later.

For the moment, we will think in terms of deriving f by dividing each number in the Fibonacci Series, by the immediately following number; while, for F, by dividing each number by the immediately preceding number.  For example, part of the Fibonacci Series includes the following numbers in sequence: 10,946; 17,711; 28,657; 46,368; 75,025; 121,393; 196,418; 317,811; 514,229...  We can obtain the following values for F and f (good to ten decimal places) by noting that:

F = 514,229/317,811 = 1.6180339887..., and f = 317,811/514,229 = 0.6180339887...

But what happens if we divide one number in the series by a second non-adjacent number -- one separated from the first number in the series by other numbers in the series?  I.e.:

F(1) = 514,229/196,418    F(2) = 514,229/121,393    f(3) = 75,025/514,229  ...  

When we do this, we encounter some very interesting results.  F(1), for example, equals 2.6180339887.  But as it turns out, this is just the value for F²!  In fact, as we divide successive numbers in order to find different values of F(n), we encounter a trend.  Or maybe a fad.  Or maybe there’s something really profound going on here.  In any case, with more calculations, we quickly find for any integer value of n:  

Fⁿ  =  F(n-1)              and            fⁿ   =  f(n-1)

This is an impressive result!  Not only does the ratio of adjacent numbers in the Fibonacci Series yield increasingly accurate values of F and f with increasing numbers in the sequence, but we can also calculate all positive integer powers of F and f by using numbers separated by other numbers in the series by one less than the power desired. In this manner, by dividing numbers in the series by a variety of other numbers in the same sequence, we obtain the results shown in Table 1.    

 Furthermore, from this table we can generate a host of mathematical equations relating and inter-relating the various Fⁿ and fⁿ.  We can even obtain “crossing relationships” between Fⁿ and fⁿ, including Fⁿ ± fⁿ.  These are given in Table 2, and can be summarized in non-mathematical terms (as well as by the clever formulas).           

For example, considering each of the first two columns of numbers in Table 1, we can conclude that: 1) adjacent numbers add or subtract to the next number, 2) any two numbers multiply or divide to another number in the sequence by a prescribed formula, 3) the sum of all of the numbers in the f column add to F, 4) the sum of all the reciprocals of the numbers in the F column add to 1/f, 5) the cross product of two numbers in each column equals 1, and 6) the ratio of different powers of F or f equals a power of f or F.  

The intriguing part is F (or f) and all of its harmonics continually turn into themselves.  And if you think that the powers of the Golden Mean (all those fⁿ we’ve encountered above) are only a mathematical curiosity, then consider the ideal human body which not only appears structured to the Golden Mean -- but even displays proportions of the Golden Mean raised to various integer powers.

Using Table 3 we can also consider the results of adding or subtracting fⁿ and F^n. -- the last two columns on the right.  The most obvious aspect is the crossing series of first Fⁿ - fⁿ, then fⁿ⁺¹ + Fⁿ⁺¹, and so forth.  The first of these “crossing series” yields 1, 3, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, 843, 1364...  As it turns out, this is can be thought of as a Modified Fibonacci Series, the only difference being a variation in the starting point -- in this case, 1 and 3.  Nevertheless this sequence also yields as the series limit of the quotients of adjacent numbers, f and F, and non-adjacent numbers, fⁿ and F^n..

The opposite crossing sequence yields yet another Fibonacci Series where the starting points are 0 and 2.2360679... The real kicker is that these numbers are the exact square roots (in symbol form, Ö) of a series of whole numbers, given by: 5, 20, 45, 125, 320, 845, 2205, 5780, 15125, 39605, 103680, 271445, 710645, 1860500...  This series of numbers can also be thought of as a Ö5 sequence, where the numbers become:

Ö5 x [0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89...]

Or just the original Fibonacci Sequence multiplied by Ö5.  This process also brings home to us the very important fact that:

Ö5 = F + f  

The relationship of 5 and the Golden Mean turns out to be absolutely crucial to any understanding of F Lo Sophia.  Accordingly, the reader is advised to memorize the above equation before continuing.  (We’ll wait.  But don’t take too long.)

Now that the relationship between F, f, and 5 is clear (and hopefully memorized), we can refer to 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.  Furthermore, by connecting these points in sequence, we suddenly encounter a new geometrical delight, this one a curve known as The Golden Spiral.  And while we did not intend to throw the reader any curves at this point, this Golden Spiral thing is important.

For further involvement, follow the Yellow Brick Road (which begins as a Golden Spiral) to Connective Physics, The Fifth Element, Mathematical Theory, and yet More Math.  Or, go back to:

Sacred Mathematics         Sacred Geometry         Golden Mean

link over to:

<http://mathworld.wolfram.com/GoldenRatio.html>.

Or throw caution to the winds, and go to:

The Golden Spiral        Philosophy         Geometry of Alphabets

________________________

References:

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

[2]  Blatner, David, The Joy of Pi, Walker Publishing, Inc. USA, 1997.

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