Fibonacci Sequence and Numbers

The Fibonacci Sequence

• At the end of this module, you should be able to discuss the contribution of Fibonacci in mathematics, explain how the Fibonacci sequence came about and find and relate the Fibonacci number with things and happenings in the environment.

Who is Fibonacci?

• Leonardo Pisano or Leonardo of Pisa was born in Pisa, Italy, in 1170.
• He popularized the Hindu-Arabic numeral system or decimal system in Europe and he advocated the use of the digits 0 to 9 and of the place values.
• He is popularly known as Fibonacci, a shortened word for the Latin term "filius Bonacci," which means "son of Bonacci" because his father was Guglielmo Bonaccio.
• Fibonacci discovered the FIBONACCI SEQUENCE, while looking at how generations of rabbits breed.

Fibonacci Sequence in Nature

• Flowers follow the numbers in the Fibonacci sequence about the number of petals they possess naturally.
• Calla lily: 1 petal
• Euphorbia milii: 2 petals
• Trillium: 3 petals
• Hibiscus: 5 petals
• Cosmos: 8 petals

Number of sections in fruits and the Fibonacci sequence

• The number of sections in certain fruits aligns with the Fibonacci sequence, such as:
• Apple: 5 sections
• Star fruit: 5 sections
• Banana: 3 sections

The Fibonacci Spiral

• Using the first ten numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55) in the Fibonacci sequence as dimensions of adjacent squares creates a Fibonacci spiral.
• When arranged in a certain way, the Fibonacci sequence creates a special spiral pattern.

How to create a Fibonacci spiral:

• Begin by drawing two 1 x 1 squares.
• Draw a 2 x 2 square using the sides of the rectangle formed by the first two squares, followed by a 3 x 3 square.
• The sides of the squares will be Fibonacci numbers and this continues by drawing squares using the Fibonacci numbers as sides of the squares.

In creating a spiral-like shape

• Draw curves in the squares beginning with the 1 x 1 squares.
• The Fibonacci spiral can be observed in photographs of storms, specifically hurricanes and tornadoes.
• Sunflower seeds are another example where the Fibonacci spiral can be seen.
• Fibonacci numbers are present on the spirals that appear on the skin of pineapples and pine cones.
• The number of spirals found in them belongs to the Fibonacci numbers.
• There are some parts in the body where the Fibonacci spiral is also evident.

The Golden Ratio

• At the end of this module, you should be able to identify the presence of the Golden Ratio in nature, architecture, and artworks and the human body, as well as enumerate the practical application of this concept in your life.

General Facts

• The symbol of the golden ratio is the Greek letter "phi" – Ф (uppercase letter) or q (lowercase letter), named after the Greek sculptor Phidias.
• The Golden Ratio is an irrational number approximately equal to 1.618, and is also known as Divine Ratio or Divine Proportion. φ = √5+1/2 and q = 2(Sin 54°).

The Golden Ratio Derived from the Fibonacci Sequence

• There are ways to derive the approximate value of the Golden Ratio, such as:
• By using the golden sections, a and b of a line segment where a/b = a+b/a and a/b is considered as the golden ratio.
• From the so-called "continued fraction."
• An easier way to derive the value of the Golden Ratio is by using the Fibonacci sequence, where the bigger Fibonacci numbers are used as a ratio, the closer one gets to the approximate value of φ (1.61803398874989484820...).

Application

• Many artists and architects apply the Golden Ratio in their artworks and creative designs, believing that their works would be more pleasing and beautiful.
• Leonardo Da Vinci used the Golden Rectangle and the Golden Ratio in his painting "Mona Lisa."
• The Golden Rectangle is a rectangle whose sides are in the proportion of the Golden Ratio, observed in notable architectural structures dating back to ancient times as well as art.
• Temples like the Parthenon in Greece are believed to have the Golden Ratio in them.
• The Fibonacci spiral is also known as the Golden Spiral.
• Proportions of the human body, such as the face, follows the Divine Proportion, and the closer the body parts' proportion is to the Golden Ratio, the more aesthetic and beautiful the body is.

The Indispensability of Mathematics

• At the end of this module, you will be able to compare your previous concepts of mathematics with what you have learned in this module and identify at least five instances in your life that mathematics has helped.

Mathematics is indispensable

• Mathematics helps organize patterns and regulations in the world.
• Animals have patterns for camouflage
• Markets and grocery stores categorize items and products.

Mathematics helps predict the behavior of nature and phenomena in the world

• Researchers observe nature and phenomena and try to make a mathematical model that works for their observation and makes sense.
• Mathematics helps control nature and occurrence in the world for our own ends through mathematical modeling, which helps see inputs and their outcomes.

Mathematics has numerous applications in the world

• In the entire history of education, mathematics has held its leading position among all other school subjects.
• Mathematics is the only subject that can be used in all world cultures to produce an educated man.