# Definition:Golden Mean

## Contents

## 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:

- $\phi \approx 1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$

This sequence is A001622 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## $1 - \phi$, 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**.

$\phi$ tends to appear more in amateur publications.

However, the notation $\phi$ seems 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.

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.

## Sources

- 2008: Ian Stewart:
*Taming the Infinite*... (previous) ... (next): Chapter $2$: The Logic Of Shape