Why is it that both
$\phi$
and
$\tau$
are used to designate the Golden Ratio
$\frac{1+\sqrt5}2?$
Answer
The Golden Ratio or Golden Cut is the number
$$\frac{1+\sqrt{5}}{2}$$
which is usually denoted by phi ($\phi$ or $\varphi$), but also sometimes by tau ($\tau$).
Why $\phi$ : Phidias (Greek: Φειδίας) was a Greek sculptor, painter, and architect. So $\phi$ is the first letter of his name.
The symbol $\phi$ ("phi") was apparently first used by Mark Barr at the beginning of the 20th century in commemoration of the Greek sculptor Phidias (ca. 490-430 BC), who a number of art historians claim made extensive use of the golden ratio in his works (Livio 2002, pp. 5-6).
Why $\tau$ : The golden ratio or golden cut is sometimes named after the greek verb τομή, meaning "to cut", so again the first letter is taken: $\tau$.
Source: The Golden Ratio: The Story of Phi, the World's Most Astonishing Number by Mario Livio; MathWorld
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