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 be the nth Fibonacci number, then by definition it is equal to the sum of the previous 2 Fibonacci numbers and . Mathematically we write this as
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
Using the quadratic formula, the roots are
If we define
then we can write negative root as
From the previous post, we know that we can express the solution of as a linear combination of and . Therefore, the solution of the Fibonacci recurrence relation is just
where and are constants. Since and , we can solve for and :
From the first equation, we solve for :
Substituting this into the second equation and solving for we get:
This means that
Therefore, the solution to the Fibonacci recurrence relation is
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
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 , the third column is and the last column is . You can see that the ratio approaches value of .
The constant 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
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.