golden ratio

Golden Ratio

The golden ratio $\varphi$ is the positive solution to:

$${\varphi }^2=\varphi +1$$ $$\varphi =\frac{1+\sqrt{5}}{2}\approx \mathrm{1.618033988...}$$

The unique number where adding 1 gives the same result as squaring. Dividing by $\varphi$ is equivalent to subtracting 1.

The most irrational number – Continued Fraction Representation

$$\varphi =1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\ddots }}}}$$

This is hardest to approximate with rationals. Every term in its continued fraction is 1, making it the slowest possible convergence.

This is why $\varphi$ appears as the worst-case ratio in many algorithms and why fibonacci numbers (successive convergents of this fraction) give the slowest growth for the euclidian algorithm’s complexity.

Fibonacci Connection

$$\varphi =\underset{n\to \infty }{\lim }\frac{F_{n+1}}{F_n}$$ $${\varphi }^n=F_n\varphi +F_{n-1}$$

Powers of $\varphi$

$${\varphi }^n=F_n\varphi +F_{n-1}$$

Every power of $\varphi$ can be expressed as a linear combination of $\varphi$ and 1 with fibonacci coefficients.

E.g.: ${\varphi }^2=\varphi +1$, ${\varphi }^3=2\varphi +1$, ${\varphi }^4=3\varphi +2$, ${\varphi }^5=5\varphi +3$

The Fibonacci numbers emerge naturally from repeatedly multiplying by $\varphi$.

Golden Ratio Convergence

The ratio of consecutive Fibonacci numbers converges to the golden ratio:

$$\underset{n\to \infty }{\lim }\frac{F_{n+1}}{F_n}=\varphi$$

This follows from the recurrence relation: if $\frac{F_{n+1}}{F_n}\to L$, then $\frac{F_{n+2}}{F_{n+1}}=\frac{F_{n+1}+F_n}{F_{n+1}}=1+\frac{F_n}{F_{n+1}}\to 1+\frac{1}{L}$.
At the limit, $L=1+\frac{1}{L}$, giving $L^2=L+1$, whose positive solution is $\varphi =\frac{1+\sqrt{5}}{2}$.

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Closed-Form Solution (Binet's Formula) – Fibonacci numbers grow exponentially with base $\varphi$.

$$F_n=\frac{{\varphi }^n-{\psi }^n}{\sqrt{5}}$$

where $\varphi =\frac{1+\sqrt{5}}{2}\approx 1.618$ (golden ratio) and $\psi =\frac{1-\sqrt{5}}{2}\approx -0.618$
See also: Eigenvector puzzle.

Since $|\psi |<1$, the term ${\psi }^n$ vanishes as $n$ grows, so for large $n$:

$$F_n\approx \frac{{\varphi }^n}{\sqrt{5}}$$

The golden ratio $\varphi$ is the smallest possible base for which a sequence satisfying $F_n=F_{n-1}+F_{n-2}$ can grow exponentially . Any slower growth would eventually turn negative or decay to zero.

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The golden ratio and its reciprocal differ by exactly 1.

$$\frac{1}{\varphi }=\varphi -1\approx \mathrm{0.618...}$$

Geometric Interpretation

A golden rectangle (ratio $\varphi :1$) can be split into a square and a smaller golden rectangle. This self-similar subdivision continues infinitely, generating a logarithmic spiral found in nature.