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The Phi Nest on the Golden Number
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Means


What do we mean by mean?

Math isn't tough, but it can be mean.  The term "mean" in mathematics simply reflects a specific relationship of one number as the middle point of two extremes.

Arithmetic means

The arithmetic mean of 2 and 6 is 4, as 4 is equally distant between the two in addition:

2 + 2 = 4
and
4 + 2 = 6

Arithmetic mean

For the arithmetic mean (b) of two numbers (a) and (c):

b = ( a + c ) / 2

4 = ( 2 + 6 ) / 2

The arithmetic mean is thus the simple average between two numbers. 

Geometric means

The geometric mean is similar, but based on a common multiplier that relates the mean to the other two numbers. As an example, the geometric mean of 2 and 8 is 4, as 4 is equally distant between the two in multiplication:

2 * 2 = 4
and
4 * 2 = 8

Geometric Mean

So 2 is to 4 as 4 is to 8.

For the geometric mean (b) of two numbers (a) and (c),
b is the square root of a times c.

b = ( a * c )

4 = ( 2 * 8 )

The Golden Mean

The Golden Mean is a very specific geometric mean.  In the geometric mean above, we see the following lengths of line segments on the number line:

Yellow line = 2
Blue line = 4
White line = 8

Here, 2 x 2 = 4 and 4 x 2 = 8, but 2 + 4 = 6, not 8.  The Golden Mean imposes the additional requirement that the two segments that define the mean also add to the length of the entire line segment:

Golden Mean

This occurs only at one point, which as you can see above is just a little less than 5/8ths, or 0.625.  The actual point of the Golden Mean is at 0.6180339887..., where:

A is to B as B is to C
AND
B + C = A

Golden Mean

If we instead let the length of line B equal 1,
this gives Phi its unusual properties:

B = ( A * C )  AND  B + C = A

1 = ( * 1/ ) AND 1 + 1/ =

1 =  ( 1.618... * 1/1.618... )

AND

1 + 1 / 1.618... = 1.618...

Note also that:

1 / 1.618...    =    0.618...    =    1.618... - 1

1 /     =    0.618...    =      - 1

 

Phi - The Golden Number
A source to some of Net's "phi-nest" information on the
Golden Section / Mean / Proportion / Ratio / Number,
Divine Proportion, Fibonacci Series and Phi (1.6180339887...)

The Evolution of Truth, 1999-2001

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