# Fibonacci’s Sequence

1,1,2,3,5,8,13,21,34…Fibonacci’s sequence is one of the most popular string of numbers that are discussed in class today. And it’s not even Fibonacci’s greatest addition to math! However, the sequence explains many different aspects of life and once you have heard about it, you will not view anything the same again.

Background:

Fibonacci’s sequence is the list of numbers where the next number is a sum of the previous two. Fibonacci invented the sequence as a riddle: a pair of baby rabbits can produce offspring after two months and then every month thereafter the pair produces a pair of rabbits one male and one female (Life 2017). The sequence accounts for each pair of rabbits counting as one and as the numbers grow and more rabbits are being produced that gives us the sequence that we know today. This occurs since the previous two numbers (of pairs of rabbits) will produce offspring.

Spirals: When the sequence is taken and used graphically it can form an even more impressive value: the Fibonacci spiral. This is created by taking squares where the length of one side is the value of each of the numbers in the sequence and then these squares are built off of each other to form larger and larger rectangles built of the Fibonacci squares (Life 2017). The trick to making the rectangle work is to take the new square and line it up with the added length of the previous two squares. To turn this rectangle into a spiral a line is drawn from the inside corner of the innermost one-square and through its opposite corner and the curve continues to travel through opposite corners (Life 2017). This spiral is very famous in the math world because of all of the work that goes into creating it.

Bugs: A similar spiral is created by setting four bugs lose to play an interesting game of tag. The bugs are set in an exact square and they all begin to move towards each other in the same direction at the same rate and as they move towards the bug to their right, they create spirals into the middle of the square (Fibonacci). This pattern can also be extended to different geometries; however, a square provides enough information. The proportion as the bugs move looks very familiar as it is the ratio between a Fibonacci number and its previous iteration (Fibonacci). This comparison is referred to as the Golden Ratio.

The Golden Ratio: The Golden Ratio has a glorified name for a reason. The Golden Ratio is equivalent to 1.618 or  and it can be expressed in many ways as well since it is such a beloved value (Fibonacci). The Golden Ratio is created from Fibonacci’s Sequence based on the ratio of the dimensions of the rectangle (length divided by width) that helps define Fibonacci’s curve. This ratio and rectangle also appear in many famous works of art and throughout history: the Parthenon was built to the dimensions of the Fibonacci rectangle, the Mona Lisa has the essence of the spiral starting on her face, and Composition in Red, Yellow, and Blue also is drawn according to the ratio (Golden). It is said that the ratio is very pleasing to the eye and draws people in, which perhaps is why so many famous pieces of art replicate it.

Plants: Plants follow this pattern of numbers that were all created from a simple riddle! Flowers grow Fibonacci numbers of petals in spirals, pinecones do the same, and even the pineapple follows this pattern which has been deemed efficient for explaining characteristics of plants (Fibonacci). This means that nature even follows this sequence of numbers and makes it out to be the most important discovery that a mathematician has made. The sequence applies to all parts of life and we have not heard anything yet.

Fibonacci’s Sequence is based on the principle that each new element in a sequence is the sum of the previous two. To apply this to music, each new measure is a combination of the two immediately before it. There are two examples of this within the piece separated by a measure of rest. The first version uses quarter notes in the first and second measure and as these notes are added together, they are adjusted to take up less time in the measure so that it is easier to determine when the next element of the sequence begins. The second version begins with eighth notes and complicates much more quickly, even as it follows the same guidelines as the first version. Played by Liv Long