# Mathematical Beauty

It is said that beauty is in the eye of the beholder. As a person is more than the sum of its parts, physical beauty is just one aspect of the total beauty. Physical beauty, however, has some sort of mathematical standard. Why is it that we find movie stars beautiful? What is that standard that defines beauty? That standard is hidden in the sequence of numbers 0, 1, 1, 2, 3, 5, 8…

The careful eye will realize that, beginning on the second term, the terms in this sequence is equal to the sum of the previous two numbers. For example, the second term is just 0+1 = 1, the third term is just 1+2 = 3, and so on. This sequence is called the fibonacci sequence after Leonardo of Pisa, also known as Fibonacci. We can model this sequence as a recurrence relation. If we let $F_n$ be the nth Fibonacci number, then by definition it is equal to the sum of the previous 2 Fibonacci numbers $F_{n-1}$ and $F_{n-2}$. Mathematically we write this as

$\displaystyle F_n = F_{n-1} + F_{n-2}$

Looking at the recurrence relation above, we realize that it is an instance of a Homogeneous Linear Recurrence Relation With Constant Coefficients. The good news is we know how to solve this kind of recurrence relation. The characteristic equation is

$\displaystyle r^2 - r -1 = 0$

Using the quadratic formula, the roots are

$\displaystyle r = \frac{ -(-1) \pm \sqrt{ (-1)^2 - 4\cdot 1 \cdot (-1)}}{2\cdot 1}$
$\displaystyle = \frac{1 \pm \sqrt{5}}{2}$

If we define

$\displaystyle \phi = \frac{1 + \sqrt{5}}{2}$

then we can write negative root as

$\displaystyle \frac{1 - \sqrt{5}}{2} = 1-\phi$

From the previous post, we know that we can express the solution of $F_n$ as a linear combination of $\phi^n$ and $(1-\phi)^n$. Therefore, the solution of the Fibonacci recurrence relation is just

$\displaystyle F_n = \alpha_1 \phi^n + \alpha_2 (1-\phi)^n$

where $\alpha_1$ and $\alpha_2$ are constants. Since $F_0 = 0$ and $F_1 = 1$, we can solve for $\alpha_1$ and $\alpha_2$:

$\displaystyle F_0 = 0 = \alpha_1 \phi^0 + \alpha_2 (1-\phi)^0 = \alpha_1 + \alpha_2$
$\displaystyle F_1 = 1 = \alpha_1 \phi^1 + \alpha_2 (1-\phi)^1 = \alpha_1 \phi+ \alpha_2 (1-\phi)$

From the first equation, we solve for $\alpha_1$:

$\displaystyle \alpha_1 = - \alpha_2$

Substituting this into the second equation and solving for $\alpha_2$ we get:

$\displaystyle -\alpha_2 \phi + \alpha_2 - \alpha_2 \phi = 1$
$\displaystyle = -2\alpha_2\phi + \alpha_2 = 1$
$\displaystyle = \alpha_2 (-2 \phi +1) = 1$

Observe that

$\displaystyle 1-2\phi = 1 - 2\frac{1+\sqrt{5}}{2} = 1 - (1 + \sqrt{5}) = -\sqrt{5}$

This means that

$\displaystyle -\sqrt{5} \cdot \alpha_2 = 1$
$\displaystyle \alpha_2 = - \frac{1}{\sqrt{5}}$

and

$\displaystyle \alpha_1 = \frac{1}{\sqrt{5}}$

Therefore, the solution to the Fibonacci recurrence relation is

$\displaystyle F_n = \frac{\phi^n - (1-\phi)^n}{\sqrt{5}}$

Beauty and Fibonacci numbers

After all that tedious computations, so what? What does Fibonacci have anything to do with beauty? If you take the successive ratio of the fibonacci numbers

$\displaystyle \lim_{n \rightarrow \infty} \frac{F_{n+1}}{F_n} = \phi$

Here is a list of the first few fibonacci numbers starting at 1 and the corresponding ratios:

1        1       1        1.000000
2        2       1        2.000000
3        3       2        1.500000
4        5       3        1.666667
5        8       5        1.600000
6       13       8        1.625000
7       21      13        1.615385
8       34      21        1.619048
9       55      34        1.617647
10      89      55        1.618182
11     144      89        1.617978
12     233     144        1.618056
13     377     233        1.618026
14     610     377        1.618037
15     987     610        1.618033
16    1597     987        1.618034
17    2584    1597        1.618034
18    4181    2584        1.618034
19    6765    4181        1.618034
20   10946    6765        1.618034

The first column in the table above is just a line number. The second column is $F_{n+1}$, the third column is $F_n$ and the last column is $F_{n+1}/F_n$. You can see that the ratio approaches value of $\phi=1.618034$.

The constant $\phi$ is called the Golden Ratio by the Greeks. Any structure that follows the golden ratio is structurally beautiful to the eye. Below is an image of a rectangle with labeled sides.

The rectangle is called a Golden Rectangle if

$\displaystyle \frac{a+b}{a} = \frac{a}{b} = \phi$.

The Golden Ratio can be found in many aesthetic works. Leonardo Da Vinci used this in his Vitruvian Man. This is probably why the fibonacci sequence was featured in the beginning of the movie (book) Da Vinci Code.