# Definition:Golden Mean/Definition 3

## Definition

The **golden mean** is the unique positive real number $\phi$ satisfying:

- $\phi = \dfrac 1 {\phi - 1}$

## Decimal Expansion

The decimal expansion of the golden mean starts:

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

## 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

- 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.