Who was Fibonacci?

In the early 1200’s, an Italian mathematician Leonardo of Pisa (nicknamed Fibonacci) discovered the famous Fibonacci sequence. This sequence falls under the Mathematical domain of number theory and its most famous problem concerned rabbits. The problem read:

Start with a single pair of rabbits. Any pair of rabbits of one generation will produce a pair for the next generation, and then another pair of rabbits for the generation after that. But then they will die. How many rabbits will be produced in the nth generation?

The answer to this well-known problem is the famed Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21…to get each subsequent term, add the sum of the two numbers that precede it. For example, 1+1=2= the third term in the sequence.

A fascinating extension to this sequence is that the Fibonacci numbers turn up in many areas of nature, as will be later discussed. Also, this sequence can be directly related to the Golden Ratio (1+√5/2 ) which results in the Golden Number Phi (1.618033988749…).

An amazing finding concerning the sequence is that the ratio of two consecutive Fibonacci numbers approaches the golden ratio or the golden number Phi. For example:
f(5)=5, f(4)=3
5/3 = 1.6666…
f(6)=8, f(5)=5
f(7)=13, f(6)=8
The values become closer and closer to the golden number as the sequence continues, which is a fascinating discovery.
It is also worthwhile to take note of the fact that if you multiply any term in the series by the value of the golden ratio, the answer that results will be close to the next number in the series. For example, 5 x 1.618=8.090. if higher numbers in the series are multiplied by the golden ratio, the result becomes closer and closers to the next number in the series.

Other than his great mathematical discoveries, there is very little known about Fibonacci’s personal life. However, these discoveries have made a tremendous impact on the mathematical world.

Fibonacci In Nature

There are many areas of nature where the Fibonacci sequence can be found and some of these areas include, flower petals, plants, fruit, the human face, the human hand and animals (i.e. rabbits). The Fibonacci sequence can be found in the growth of living things ranging from simple cells to human beings (World Mysteries).

Flower petals follow the Fibonacci sequence and can identified in the growth of many plants. If you closely examine a flower that still has all of its petals intact, you will notice that the number of petals is a Fibonacci number. For example, a calla lily has one petal, and a daisy can have twenty-one petals, with one and twenty-one being Fibonacci numbers. One of the most common flowers related to the Fibonacci sequence is the above mentioned daisy. Most field daisies consist of thirteen, twenty-one, thirty-four, fifty-five or eighty-nine petals, which are all numbers found in this famed sequence. Other flowers whose number of petals is a Fibonacci number include lilies, euphorbia, trilliums, bloodroots, sunflowers, buttercups, and irises. The number of petals is not a product of chance, but rather a question of efficiency during the growth process of these flowers.

The Fibonacci sequence can again be found in the natural surroundings and is part of the natural growth of many plants that are encountered everyday such as pinecones, fir cones, coniferous trees, and palm trees.

The Fibonacci sequence has also been linked to the human face and hands. Many famous artists, for example, have used golden rectangles in the structure of their artwork. Of course, the concept of “golden rectangles and ratios” was noticed by Leonardo of Pisa (Fibonacci) as outlined in the above history of the Fibonacci sequence. Leonardo da Vinci’s painting of the Mona Lisa is based on the arrangement of the golden rectangle. Many argue that this is the reason behind its beauty.

The Fibonacci sequence can also be found in the structure of the human hand. Each part of the index finger, beginning from the tip down to the wrist, is larger than the preceding section by about the ratio of 1.618 (the golden ratio) (Human Hand).

This fascinating sequence is found in the beauty that is nature. Taking time out to appreciate the intricacies and subtleties of nature is sure to be an eye-opening and enlightening experience.

Fibonacci in Nature: Fun Facts & Visuals

Part 2
Part 3


Fibonacci in the classroom

The Fibonacci sequence can be used in many different subject areas and it can be used to teach patterns in math, music, art, language arts and science. The teaching of patterns begins in the early grades where students are required to identify various forms of patterning and continues into the elementary grades. For example, an elementary mathematics course outcome for the Newfoundland and Labrador curriculum is that students will explore, recognize, represent and apply patterns and relationships both formally and informally. When teaching mathematical patterns, teachers can relate the Fibonacci sequence to patterns that can be seen in real life such as those of flower petals.

The sequence can be taught within the music curriculum using the keys on a piano, which as mentioned above, follow the Fibonacci number pattern. Teachers can demonstrate to the students that the piano follows the Fibonacci sequence by having 5 black keys, 8 white keys and 13 notes which comprise one octave. This sequence can also be identified in various music scales such as the pentatonic, diatonic, and the chromatic scales (Math and Music).

The Fibonacci sequence can also be found in the area of science and more specifically, in the areas of space and physics. When teaching a unit on space, teachers can provide children with interesting facts on how the universe is based on the Golden Ratio, as there are many galaxies in the universe that contain the Golden Ratio in their structure. Also, teachers can use the Fibonacci sequence when teaching a unit on light. For example, when light is held over a surface (i.e. glass), one part of the light passes through the surface, one part is absorbed and one part is reflected. Therefore a multiple reflection occurs which means that there are a number of reflections that the ray of light takes inside the glass before it re-emerges. The number of rays that re-emerge reflect the Fibonacci number sequence.

Within the language arts curriculum there are ways to bring in the concept of the Fibonacci sequence and its numbers. For example, in the area of poetry, several of the rhythms used are Fibonacci numbers, such as iambic pentameter. As well, the number of lines in particular forms of poetry are Fibonacci numbers, such as limericks, which have five lines or Haiku poems which have three.

The Fibonacci sequence can also be used in teaching spatial proportions in art education. Teachers can use spirals, concentric circles and repeated rectangles to show how interesting patterns can be formed using these figures when teaching students about proportional preferences. Also, when teaching a section on portrait drawing, it is important to inform students of the correct facial and body proportions when drawing a human figure. These proportions enable the students to create a drawing that is lifelike and directly proportional to the subject he/she is drawing.

The Fibonacci sequence does not have to be strictly limited to the mathematics curriculum. Incorporating it into other subject areas will allow students to see how the sequence can be relevant to "real life" which makes the information meaningful and relevant for the student. Above all else, this should be a primary goal of any mathematics curriculum.
Read More About The Golden Ratio (Phi) Here


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