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.
For the geometric mean (b) of two numbers (a) and (c),
b is the square root of a times c.
b = Ö ( a * c )
4 = Ö ( 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
If we instead let the length of line B equal 1,
this gives Phi its unusual properties:
B = Ö ( A * C
) AND B + C
= A
1 = Ö ( Ø * 1/Ø
) AND 1 + 1/Ø = Ø
1 = Ö ( 1.618...
* 1/1.618... )
AND
1 + 1 / 1.618... = 1.618...
Note also that:
1 / 1.618... =
0.618... = 1.618...  1
1 / Ø =
0.618... = Ø  1 
