Geometric and Golden 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

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

So 2 is to 4 as 4 is to 8.

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:

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

B = √ ( A * C )  AND  B + C = A

1 = √ ( Phi * 1/Phi ) AND 1 + 1/Phi = Phi

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 / Phi    =    0.618…    =    Phi  – 1