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 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:
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
this gives Phi its unusual properties:
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