The Fibonacci numbers appear in many other areas of mathematics, such as the sums of the "shallow diagonals" in Pascal's Triangle:
${F}_{n}equals;\sum _{kequals;0}^{\u230a\frac{n-1}{2}\u230b}\left(\genfrac{}{}{0ex}{}{n-k-1}{k}\right)$.
The Fibonacci numbers are also common in nature, relating to a wide variety of biological phenomena. These phenomena include: the arrangement of leaves on a stem, the flowering of an artichoke, the growth of seeds in a sunflower, the spirals of shells, and the family trees of honeybees. The sequence was originally described by Fibonacci as depicting the growth of a rabbit population, but this example makes some very unrealistic assumptions such as the rabbits never die and each mating pair must produce one new pair (one male and one female) every month from the second month on.
This sequence is closely related to the Golden Ratio, 4, which has the value $\mathrm{\ϕ}equals;\frac{1plus;\sqrt{5}}{2}equals;1.6180339887..period;$ Two quantities are said to be in the Golden Ratio if the ratio of their sum to the larger quantity is equal to the ratio of the larger quantity to the smaller one. Algebraically, this is written as: $\frac{a\+b}{a}equals;\frac{a}{b}equals;\mathrm{varphi;}$ where $agt;b$. We find that the ratios of successive terms of the Fibonacci sequence, namely, $\frac{2}{1}\,\frac{3}{2}comma;\frac{5}{3}comma;\frac{8}{5}comma;\frac{13}{8}comma;..period;$ provide the closest rational approximations of the Golden Ratio for a given size of numerator and denominator. In fact, they approach φ as we move further along in the sequence.
Because of this relationship with the Golden Ratio, the Fibonacci numbers can also be used to approximate a Golden Spiral: a logarithmic spiral with the polar equation $r\=a\cdot {\ⅇ}^{b\mathrm{\θ}}$ and a special growth factor of $bequals;\mathrm{ln}\left(\mathrm{varphi;}\right)\cdot \left(\frac{2}{\mathrm{pi;}}\right)$.
By creating a tiling of squares whose side lengths are consecutive Fibonacci numbers and inscribing quarter-circle arcs between the opposite corners of each square, we can illustrate the Fibonacci Spiral, which very closely approximates the Golden Spiral given by $r\={\ⅇ}^{b\mathrm{\theta}}$.