Defining the Divine Proportion
It is known by many names.
Many references will begin telling you it is a number equal to
\[1.618...\]
But this number is incomplete and meaningless.
A proportion is a relationship between two values. Such as 1:2 or a:b or a/b
a geometric me
To understand the true meaning of the Divine Proportion, we first must
understand the principle of a geometric mean. A geometric mean is a
relationship of three values \(a, b, c\) where
\[\large
\frac{a}{b} = \frac{b}{c}\]
if can also be understood as
\[\large
ac = b ^ 2\]
so a simple example would be
\[\large
\frac{1}{2} = \frac{2}{4}\]
But what if we were to define \(c = a + b\) so our geometric mean is
expressed with only two values:
\[\large
\frac{a}{b} = \frac{b}{a+b}\]
We can begin to find a solution for this by multiplying \(a\) and \(b\)
by a common value \(\lambda\)
\[\begin{split}\large
a * \lambda &= 1\\
\large
b * \lambda &= \phi\end{split}\]
This allows us to translate our geometric mean to an expression with one
variable, \(\color{#C90}{\phi}\)
\[\large
\frac{1}{\color{#C90}{\phi}} = \frac{\color{#C90}{\phi}}{1+\color{#C90}{\phi}}\]
if we do a little algebraic manipulation, we see that our mean can be expressed as a quadratic expression.
First, multiply each side by \(\color{#C90}{\phi}\)
\[\begin{split}\large
{\color{#C90}{\phi}} * \frac{1}{\color{#C90}{\phi}} &=
\large
{\color{#C90}{\phi}} * \frac{\color{#C90}{\phi}}{1+\color{#C90}{\phi}} \\
\\
\large
{\color{grey}{\cancel{\phi}}} * \frac{1}{\color{grey}{\cancel{\phi}}} &=
\large
{\color{#C90}{\phi}} * \frac{\color{#C90}{\phi}}{1+\color{#C90}{\phi}} \\
\\
\large
1 &=
\large
\frac{{\color{#C90}{\phi}}^2}{1 + \color{#C90}{\phi}}\end{split}\]
Then, multiply each side by \((1 + {\color{#C90}{\phi}} )\)
\[\begin{split}\large
( 1 + {\color{#C90}{\phi}} ) * 1 &=
\large
( 1 + {\color{#C90}{\phi}} ) * \frac{{\color{#C90}{\phi}} ^ 2}{1 + \color{#C90}{\phi}} \\
\\
\large
( 1 + {\color{#C90}{\phi}} ) * 1 &=
\large
{\color{grey}{\cancel{(1 + \phi )}}}
* \frac{{\color{#C90}{\phi}} ^ 2}{\color{grey}{\cancel{1 + \phi}}} \\
\\
\large
1 + {\color{#C90}{\phi}} &=
\large
{\color{#C90}{\phi}} ^ 2 \\\end{split}\]
Then flip the terms and subtract \((1 + {\color{#C90}{\phi}} )\) from both sides
\[\begin{split}\large
{\color{#C90}{\phi}} ^ 2
&=
\large
1 + {\color{#C90}{\phi}} \\
\\
\large
{\color{#C90}{\phi}} ^ 2 - {\color{#C90}{\phi}} - 1 &=
\large
0\end{split}\]
Now we have a quadratic expression in the form of:
\[\large
{\color{red}{a}}x^2 + {\color{green}{b}}x + {\color{blue}{c}} = 0\]
And can use the quadratic formula to solve for \(\color{#C90}{\phi}\)
\[\large
\frac{-{\color{green}{b}} \pm \sqrt{{\color{green}{b}}^2-4{\color{red}{a}}{\color{blue}{c}}}}{2{\color{red}{a}}}\]
Substitute the coefficients and plug into the formula:
\[\begin{split}\large
{\color{red}{a}} &= 1 \\
\large
{\color{green}{b}} &= -1 \\
\large
{\color{blue}{c}} &= -1 \\
\\
\large
{\color{#C90}{\phi}} &=
\large
\frac{-{\color{green}{(-1)}}
\pm \sqrt{{\color{green}{(-1)}}^2
- 4{\color{red}{(1)}}{\color{blue}{(-1)}}}}{2{\color{red}{(1)}}} \\
\\
\large
{\color{#C90}{\phi}} &=
\large
\frac{1 \pm \sqrt{5}}{2}\end{split}\]
So, \(\color{gold}{\phi}\) equates to two values:
\[\begin{split}\large
{\color{#C90}{\phi}} &=
\large
\frac{1 + \sqrt{5}}{2} \approx 1.618033989...\\
\\
\large
{\color{#C90}{\varphi}} &=
\large
\frac{1 - \sqrt{5}}{2} \approx -.618033989...\\\end{split}\]
The expression can also be rendered using just 5’s
\[\large
.5 \times 5^.5 + .5\]
Two values are in The Divine Proportion when the ratio of the lesser value
over the greater value is equal to the greater value over the sum of the lesser
and greater value.
In other words, when a segment is sectioned into the Divine Proportion, the
parts are in a harmonic relationship to the whole. Setting up a Harmonic
Rhythm.
The very nature of the Golden Ratio is harmonic resonance