Definition:Golden Mean
Contents |
Definition
The golden mean is the unique positive real number $\phi$ satisfying
- $\phi = \dfrac 1 {\phi - 1}$
It is also known as the golden ratio or golden section.
Equivalently, $\phi$ is the real number
- $\phi = \dfrac{1 + \sqrt 5} 2$
This follows from the Quadratic Formula.
Its approximate value is:
- $\phi \approx 1.61803\ 39887 \ldots$ This sequence is A001622 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).
Note also that:
- $1 - \phi = - \dfrac 1 \phi$
which follows directly from taking reciprocals of the definition.
This number $1 - \phi$ is often denoted $\phi'$ or $\hat \phi$:
- $\phi' = \dfrac {1 - \sqrt 5} 2 \approx -0.61803\ 39887 \ldots$
Geometrical Interpretation
Let $\Box ADEB$ be a square.
Let $\Box ADFC$ be a rectangle such that:
- $AC : AD = AD : BC$
where $AC : AD$ denotes the ratio of $AC$ to $AD$.
Then if you remove $\Box ADEB$ from $\Box ADFC$, the sides of the remaining rectangle have the same ratio as the sides of the original one.
Thus if $AC = \phi$ and $AD = 1$ we see that this reduces to:
- $\phi : 1 = 1 : \phi - 1$
Continued Fraction Expansion
The golden mean has the simplest possible continued fraction expansion, namely $[1, 1, 1, 1, \ldots]$. That is:
- $\phi = 1 + \cfrac 1 {1 + \cfrac 1 {1 + \cfrac 1 {\ddots}}}$
As demonstrated here, the convergents to $\phi$ are given by the ratios of consecutive Fibonacci numbers.
Historical Note
The symbol $\phi$ originates from the Greek artist Phidias who was said to have used it as a basis for calculating proportions in his sculpture. It is said to produce the most pleasing proportions, and as a consequence many artists have used this ratio in their works.
Euclid called it the extreme and mean ratio:
- A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the less.
(The Elements: Book VI: Definition $3$)
That is:
- $A + B : A = A : B$
Also see
