Fibonacci: The Numbers of Nature

Many scientists suspected that the universe is structured via numbers, providing mind-blowing evidence and strong theories that make the reader change the way they look at the surrounding environment, and one of these thinkers was Fibonacci, an Italian mathematician that found a number sequence that is seen a lot in nature and it shows how maths is truly everywhere around us.

The Origins Of The Fibonacci Sequence

Leanardo Pisano Bogollo, Fibonacci being his nickname, first introduced the sequence of numbers known as the Fibonacci sequence in his book Liber Abaci in 1202. Although Fibonacci’s Liber Abaci contains the earliest known description of the sequence outside of India, the sequence had been described by Indian mathematicians as early as the sixth century and was used frequently in metrical sciences. Fibonacci is the one who introduced these numbers to Europe in his book. His sequence was derived by mathematicians from a problem that is found in his book:

“A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begets a new pair which from the second month on becomes productive?”

Fibonacci was also a member of an important Italian trading family during the 12th and 13th century. Being part of a trading family, mathematics was an integral part of the business. Fibonacci travelled throughout the Middle East and India and was captivated by the mathematical ideas from his travels. His book was a discourse on the mathematical methods in commerce that Fibonacci observed during his travels. And he was the one who promoted the Hindu-Arabic numerals to Europe (0,1,2,3,4…) alongside the Fibonacci sequence. 

Where is the Fibonacci sequence found?        

Other than being a cool brain teaser; the sequence has been rediscovered in different branches of mathematics, computer science applications, nature, and statistics. 

The Fibonacci sequence

So what is this miraculous number sequence? To find the sequence, we start with the first two numbers (1,1) and the third is the sum of the two before (2), and the process is repeated so on to infinity, as shown below.

The Fibonacci sequence

1,1,2,3,5,8,13,21,34,55,89,144,233,377…

1+1=2         13+21=34

1+2=3         21+34=55

2+3=5         34+55=89

3+5=8         55+89=144

5+8=13       89+144=233

8+13=21     144+233=377

The golden ratio and its relation to the sequence

If you are into math you have probably heard of the golden ratio. If you haven’t, the golden ratio(represented by the Greek letter phi φ) is an irrational number that is approximately equal to 1.618033…. It is named golden for the several times it is found in math and nature. As an example,  it’s found in plants. The rotational angle from leaf to leaf in a repeating spiral can be represented by a fraction of a full rotation around the stem. Alternate distichous leaves will have an angle of 1/2 of a full rotation. In beech and hazel, the angle is 1/3, in oak and apricot it is 2/5, in sunflowers, poplar, and pear, it is 3/8, and in willow and almond, the angle is 5/13. The numerator and denominator normally consist of a Fibonacci number and its second successor.

But are these numbers coincidental? No, they aren’t when it comes to plants with lots of leaves as they have adapted to the surrounding environment to find the ideal way of absorbing sunlight, to prevent a leaf from blocking all of the sunlight from reaching the leaf directly below, the solution is to have an angle, a golden angle that would guarantee that none of the leaves get no sunlight at all, and this angle is one φ of a circle(360/1.618033…) away from the other, for φ can be considered as the most irrational number, it is almost a guarantee that no leaf will completely overlap another.

The group of numbers below shows the quotient of a Fibonacci number divided by the one before- notice anything weird?

1/1=1, 2/1=2, 3/2=1.5, 5/3=1.666666…, 8/5=1.6, 13/8=1.625, 21/13=1.615384615…, 89/55=1.618181818…, 377/233=1.6180257…

The quotient of a number from the sequence and the one before will get closer to the golden ratio when bigger numbers are used.

The golden rectangle

The Fibonacci sequence is often visualised in a graph. Each of the squares illustrates the area of the next number in the sequence. The Fibonacci spiral is then drawn inside the squares by connecting the corners of the boxes.

The squares fit together perfectly because the ratio between the numbers in the Fibonacci sequence is very close to the golden ratio, which is approximately 1.618034. The larger the numbers in the Fibonacci sequence, the closer the ratio is to the golden ratio. The spiral and resulting rectangle are also known as the Golden Rectangle.

Where is Fibonacci seen in nature?

What makes this sequence so famous is the countless times it is seen in nature. Try counting the petals of a flower and you will notice that no matter the type, the number of petals will always be a number in the sequence. Not just that, try counting the spirals found on a pine cone, pineapple, or even sunflowers, you will find the same thing.

The sequence can be observed in the array of sunflower seeds and other plants, and the shape of galaxies and hurricanes as the Fibonacci spiral that is formed by the golden rectangle, the sequence appears in nature because it represents structures and sequences that model physical reality when the underlying mechanism that puts components together to form a spiral they naturally conform to that numeric sequence.

Fibonacci in art and music

Many people presume that Leonardo Da Vinci used the Fibonacci spiral as a measure to draw his intricate painting of the Mona Lisa, it may be just a coincidence,  but that’s not the point, it is just an example of how the sequence can be observed in real life.  

An octave on the piano consists of 13 notes. Eight are white keys and five are black keys, the white keys represent full notes in music, and 8 keys are what it takes for an octave to repeat, so if counting starts on the note C the eighth note is also C but on a higher pitch, and in between the whole notes, half notes are represented by black keys on a piano, notice how there are only 5 black notes, that’s because notes E# and B# are missing (Music Matters explains the reason in this video). In a scale, the dominant note is the 5th note, which is also the 8th note of all 13 notes that make up the octave, 5th and 3rd notes create the basic foundation of all chords.

Nowadays, artists recognised that the Fibonacci Spiral is an expression of an aesthetically pleasing principle – the Rule of Thirds. This is used in the composition of a picture; by balancing the features of the image by thirds, rather than strictly centring them, a more pleasing flow to the picture is achieved.

Why are Fibonacci numbers significant in computer science?

Fibonacci numbers have become a popular introduction for computer science students and noticed in nature in many shapes and forms. For these reasons, many of us are familiar with them.

They are within computer science elsewhere; in surprisingly efficient data structures and algorithms based upon the sequence. Two main examples come to mind:

Fibonacci heaps have better-amortised running time than binomial heaps.

Fibonacci search which shares O(log N) running time with a binary search on an ordered array. 

“Is there some special property of these numbers that gives them an advantage over other numerical sequences? Is it a spatial quality? What other possible applications could they have?” 

This group of developers concluded that these numbers are unique because of their mathematical properties.

1.They grow exponentially fast.

2. Any number can be written as the sum of unique Fibonacci numbers- this is Zeckendorf’s Theorem (though it also states that such a sum doesn’t include any two consecutive Fibonacci numbers).

3.The Fibonacci numbers are efficiently computable.

4.They’re pedagogically useful: Teaching recursion is tricky, and the Fibonacci series is a great way to introduce it.

Fibonacci is only a fraction of how truly beautiful math is. In school we spend most of our time solving for x or y, but we must consider asking why.  It is not just calculation but also application and inspiration for many across the globe.