Fibonacci Day

 

INTRODUCTION

November 23rd is celebrated annually as Fibonacci day that honors one of the most influential mathematicians of the Middle Ages ⎼ Leonardo Bonacci. Also known as Leonardo of Pisa, he is popularly known as Leonardo Fibonacci. Fibonacci is a contraction of filo Bonacci, which means the son of Bonaccio.

WHY 23RD NOVEMBER?

November 23rd was chosen as the day for celebrating Fibonacci day, because when the date is written in the mm/dd format (11/23), the digits in the date form a Fibonacci sequence: 1, 1, 2, 3. 

FIBONACCI

A Fibonacci sequence is a series of numbers where a number is the sum of the two numbers before it. For example, 0, 1, 1, 2, 3, 5, 8...is the Fibonacci sequence. Here, 0 + 1 = 1, then 1 + 1 = 2, 1 + 2 = 3, 2 + 3 = 5, 3 + 5 = 8...and so on.

THE FIBONACCI SEQUENCE AND THE GOLDEN RATIO

The Fibonacci sequence and the Golden Ratio are closely related in mathematics, nature, and art. The Fibonacci sequence is a series of numbers where each term is the sum of the previous two terms, starting with 1 and 1. For example, the first 10 terms of the Fibonacci sequence are:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55

The Golden Ratio is an irrational number that is approximately equal to 1.618. 

It is also known as phi (φ) or the divine proportion. The Golden Ratio can be obtained by dividing a line segment into two parts such that the ratio of the whole segment to the larger part is equal to the ratio of the larger part to the smaller part. For example, if a line segment AB is divided into two parts AC and CB such that AB/AC = AC/CB, then AB/AC is equal to the Golden Ratio.

One way to see the connection between the Fibonacci sequence and the Golden Ratio is to look at the ratios of consecutive terms in the Fibonacci sequence. As the terms get larger, these ratios get closer and closer to the Golden Ratio. For example:

• The ratio of the 2nd and 1st terms is 1/1 = 1
• The ratio of the 3rd and 2nd terms is 2/1 = 2
• The ratio of the 4th and 3rd terms is 3/2 = 1.5
• The ratio of the 5th and 4th terms is 5/3 = 1.666…
• The ratio of the 6th and 5th terms is 8/5 = 1.6
• The ratio of the 7th and 6th terms is 13/8 = 1.625
• The ratio of the 8th and 7th terms is 21/13 = 1.615…
• The ratio of the 9th and 8th terms is 34/21 = 1.619…
• The ratio of the 10th and 9th terms is 55/34 = 1.617…

As you can see, these ratios are getting closer and closer to the Golden Ratio as the terms get larger.

Another way to see the connection between the Fibonacci sequence and the Golden Ratio is to use a formula that relates them. There is a formula that can be used to find any term in the Fibonacci sequence using only the Golden Ratio and its inverse (which is also equal to φ - 1 or approximately 0.618). The formula is:

F_n = (φ^n − (−φ)^−n) / √5

where F_n is the nth term in the Fibonacci sequence and n is any positive integer. For example, using this formula, we can find the 10th term in the Fibonacci sequence as follows:

F_10 = (φ^10 − (−φ)^−10) / √5 = (1.618^10 − (−0.618)^−10) / √5 = (17.944 − (−0.017)) / √5 = (17.961) / √5 = (17.961) / (2.236) = (8.034) * (0.447) = (3.591) * (2) = (7.182) * (0.5) = (3.591) * (1) = 55

This formula shows that any term in the Fibonacci sequence can be expressed using powers of the Golden Ratio.

The Fibonacci sequence and the Golden Ratio also appear in many natural phenomena, such as the spiral patterns of sunflower seeds, pinecones, shells, galaxies, and hurricanes; the branching patterns of trees, plants, veins, and nerves; and the proportions of human faces, bodies, and artworks. Some people believe that these patterns reflect a hidden harmony and beauty in nature that can be explained by mathematics.

This rat somehow creates the perfect golden ratio.

The Fibonacci sequence is a series of numbers that starts with 0 and 1, and each subsequent number is the sum of the previous two. For example, 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on. This sequence has many interesting properties and applications in mathematics, art, and nature.

This cat is literally sleeping in the Fibonacci-style.

One of the most fascinating aspects of the Fibonacci sequence is how often it appears in the natural world. Many plants, animals, and phenomena exhibit patterns or shapes that follow the Fibonacci sequence or are related to the golden ratio, which is the limit of the ratio of consecutive Fibonacci numbers as the sequence goes to infinity. The golden ratio is approximately equal to 1.618.

Some examples of the Fibonacci sequence in nature are:

• The spiral pattern of seeds in a sunflower or the scales of a pineapple: These spirals follow the Fibonacci sequence in both directions: clockwise and counterclockwise. For instance, a typical sunflower has 55 spirals in one direction and 89 in the other; both are Fibonacci numbers.
• The shape of a nautilus shell or a snail shell: These shells grow in a logarithmic spiral that approximates the golden ratio. The nautilus shell is often considered a symbol of the Fibonacci sequence and the golden ratio.
• The arrangement of leaves on a stem or branches on a tree: Many plants have leaves that are positioned at an angle that minimizes the overlap with other leaves. This angle is often related to the golden ratio and results in a Fibonacci number of leaves per turn around the stem.
• The petals of a flower or the segments of a fruit: Many flowers have a number of petals that is a Fibonacci number, such as lilies (3 petals), buttercups (5 petals), or daisies (34 petals). Similarly, many fruits have a number of segments that is a Fibonacci number, such as oranges (10 segments), lemons (8 segments), or pineapples (8 or 13 spirals).
• The flight pattern of a falcon or the shape of a hurricane: Some animals and natural phenomena move in a spiral motion that resembles the Fibonacci spiral. For example, a falcon dives toward its prey in a spiral that follows the golden ratio. A hurricane also forms a spiral that approximates the golden ratio.

These are just some of the many examples of how the Fibonacci sequence manifests itself in nature. Scientists and mathematicians have been fascinated by this sequence for centuries and have tried to explain why it is so prevalent and what it means for our understanding of nature and beauty. Some possible reasons are:

• The Fibonacci sequence is efficient and optimal for growth and packing. For example, seeds in a sunflower or leaves on a stem can fill up space without gaps or overlaps by following the Fibonacci sequence.
• The Fibonacci sequence is adaptive and resilient to change. For example, plants that follow the Fibonacci sequence can adjust their growth to different environments and seasons by adding new elements to the sequence.
• The Fibonacci sequence is aesthetically pleasing and harmonious. For example, humans tend to perceive shapes and proportions that follow the golden ratio as more beautiful and balanced than others.

The Fibonacci sequence is one of the most amazing and mysterious patterns in nature. It shows us how mathematics can reveal hidden order and beauty in seemingly chaotic and random phenomena. It also challenges us to explore deeper connections between nature, art, and science.

And this is one of my most favorite mathematics topics 😄 

Information credit: Google