Fibonacci spiral

Parametric equations

The Fibonacci sequence $F_n$ satisfies $F_n=F_{n-1}+F_{n-2}$. Squares of side $F_n$, arranged in a spiral, with a quarter-circle in each square, give the classical construction that approaches a logarithmic spiral.

The recurrence:

Each piece of the spiral is a quarter-circle. The limiting ratio is $\underset{n\to \infty }{\lim }\frac{F_{n+1}}{F_n}=\phi =\frac{1+\sqrt{5}}{2}$.

Interaction

• Number of terms n: the slider shows squares through the nth Fibonacci number
• The outer rectangle stays on the figure. Its side lengths are neighboring Fibonacci numbers, and that ratio approaches $\phi$

What to notice

• Neighboring squares have sides $1,1,2,3,5,\dots$, and each arc of the spiral is one quarter-circle.
• The readout $F_n/F_{n-1}$ moves toward $\phi$. Compare it with the outer rectangle’s aspect ratio.
• The faint blue curve is a logarithmic-spiral reference. The gold arcs are the Fibonacci approximation.

Related

• Geometric view of arithmetic and geometric sequences
• Equiangular spiral
• Basel problem

Further reading

• Fibonacci number (Wikipedia)
• Golden rectangle (Wikipedia)