# Definition:Golden Mean/Historical Note

## Historical Note on Golden Mean

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.

## Sources

- 1953: H.S.M. Coxeter:
*The Golden Section, Phyllotaxis, and Wythoft's Game*(*Scripta Math.***Vol. 19**: pp. 135 – 143) - 1961: Martin Gardner:
*The Second Scientific American Book of Mathematical Puzzles and Diversions* - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$ - 1992: George Markowsky:
*Misconceptions about the Golden Ratio*(*College Math. J.***Vol. 23**: pp. 2 – 19) www.jstor.org/stable/2686193 - 1995: Peter Schreiber:
*A Supplement to J. Shallit's Paper “Origins of the Analysis of the Euclidean Algorithm”*(*Hist. Math.***Vol. 22**: pp. 422 – 424) - 1997: Donald E. Knuth:
*The Art of Computer Programming: Volume 1: Fundamental Algorithms*(3rd ed.) ... (previous) ... (next): $\S 1.2.8$: Fibonacci Numbers - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $1 \cdotp 61803 \, 39887 \, 49894 \, 84820 \, 45868 \, 34365 \, 63811 \, 77203 \, 09179 \, 80576 \ldots$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**golden section** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**golden section**