Definition:Golden Mean
Definition
Definition 1
Let a line segment $AB$ be divided at $C$ such that:
- $AB : AC = AC : BC$
Then the golden mean $\phi$ is defined as:
- $\phi := \dfrac {AB} {AC}$
Definition 2
The golden mean is the unique positive real number $\phi$ satisfying:
- $\phi = \dfrac {1 + \sqrt 5} 2$
Definition 3
The golden mean is the unique positive real number $\phi$ satisfying:
- $\phi = \dfrac 1 {\phi - 1}$
Euclid's Definition
- 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 $\text{VI}$: Definition $3$)
That is:
- $A + B : A = A : B$
Decimal Expansion
The decimal expansion of the golden mean starts:
- $\phi \approx 1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$
$1 - \phi$, or $\hat \phi$
The number:
- $1 - \phi$
is often denoted $\hat \phi$.
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:
- $\phi : 1 = 1 : \phi - 1$
where $\phi$ is the golden mean.
Also known as
The golden mean is also known as the golden ratio or golden section.
Euclid called it the extreme and mean ratio.
The Renaissance artists called it the divine proportion.
The notation for the golden mean is not universally standardised.
In much professional literature, $\tau$ (tau) is used, for the Greek tome for cut.
$\phi$ (phi), which has generally tended to appear more in amateur publications, appears to be becoming more widely used.
Also see
- Limit of Ratio of Consecutive Fibonacci Numbers where the convergents to $\phi$ are shown to be the ratios of consecutive Fibonacci numbers.
- Results about the golden mean can be found here.
Historical Note
The Egyptians knew about the golden mean. It was referred to in the Rhind Papyrus as sacred.
The heights of the Great Pyramids of Gizeh are almost exactly $\phi$ of half the lengths of their bases.
It is believed that the Ancient Greeks used $\phi$ in their architecture, but there is no extant documentary evidence of this.
Surprisingly, they had no short term for this concept, merely referring to it as the section.
However, it was known to the Pythagoreans, who called it the extreme and mean ratio.
The Renaissance artists exploited it and called it the Divine Proportion.
Fra Luca Pacioli discussed it in his book De Divina Proportione.
The first occcurrence of the term sectio aurea ("golden section") was probably by Leonardo da Vinci.
Mark Barr coined the use of the uppercase Greek letter $\Phi$ (phi) for the golden mean, originating from the Greek artist Phidias, who was said to have used it as a basis for calculating proportions in his sculpture.
Its companion value $\dfrac 1 \Phi = \Phi - 1$ was given the lowercase version $\phi$ or $\varphi$.
However, this convention is far from universal, and the larger value $1 \cdot 618 \ldots$ is usually denoted $\phi$.
It is said to produce the most pleasing proportions, and as a consequence many artists have used this ratio in their works.
A famous (or infamous, depending on how much reading you have done around the subject) article by George Markowsky attempts to debunks a number of myths surrounding the number.