Fibonacci Spiral
©Lorena Loo
©Lorena Loo
I Know What a Spiral Is But What Is a Fibonacci?
Leonardo de Pisa was an Italian mathematician born in Pisa circa 1175 AD. His father, Bonaccio, was a customs inspector on the north coast of Africa which is how Leonardo came to be educated by the Mohammedans of Barbary. It was they who introduced him to the Arabic system of numbers. Leonardo published a book, Liber Abaci (Book of Abacus), which introduced this number system to Europe and earned him the reputation as the greatest mathematician of the Middle Ages.
One of the problems in Liber Abaci concerned calculating the population of rabbits after a certain number of months if you start with just two adult rabbits. Out of this solution came the so-called Fibonacci numbers, named after Leonardo who was known as Fibonacci, a contraction of Filius Bonacci (son of Bonacci).
So now we know the Fibonacci numbers are literally son of Bonacci numbers. But what are these numbers? They form an additive sequence of numbers where the first two numbers are 1 and 1. Each successive term after that is determined by adding the immediately preceeding two numbers in the sequence. For example, the third term or number in the sequence is the sum of term one and two which is 1+1=2. Now we have the third term we can add it to the second term to get the fourth term: 1+2=3. The (n+1)th term in the sequence is then the sum of the nth term and the (n-1)th term. Mathematically this is expressed as T(n+1) = Tn + T(n-1). In the table below, the first 20 Fibonacci numbers are listed in the second column.
Leonardo de Pisa was an Italian mathematician born in Pisa circa 1175 AD. His father, Bonaccio, was a customs inspector on the north coast of Africa which is how Leonardo came to be educated by the Mohammedans of Barbary. It was they who introduced him to the Arabic system of numbers. Leonardo published a book, Liber Abaci (Book of Abacus), which introduced this number system to Europe and earned him the reputation as the greatest mathematician of the Middle Ages.
One of the problems in Liber Abaci concerned calculating the population of rabbits after a certain number of months if you start with just two adult rabbits. Out of this solution came the so-called Fibonacci numbers, named after Leonardo who was known as Fibonacci, a contraction of Filius Bonacci (son of Bonacci).
So now we know the Fibonacci numbers are literally son of Bonacci numbers. But what are these numbers? They form an additive sequence of numbers where the first two numbers are 1 and 1. Each successive term after that is determined by adding the immediately preceeding two numbers in the sequence. For example, the third term or number in the sequence is the sum of term one and two which is 1+1=2. Now we have the third term we can add it to the second term to get the fourth term: 1+2=3. The (n+1)th term in the sequence is then the sum of the nth term and the (n-1)th term. Mathematically this is expressed as T(n+1) = Tn + T(n-1). In the table below, the first 20 Fibonacci numbers are listed in the second column.
The ratio of any two successive Fibonacci numbers approximates the golden ratio. The higher up in the sequence we go, the closer the approximation we obtain to phi if we take the ratio of the larger to smaller number. Compare phi calculated to 9 decimal places (1.618033989 ) with the ratio of successive Fibonacci numbers in the third column (labelled Fibonacci Ratio) of the table above.
Many people erroneously think it is only the Fibonacci numbers which so uniquely approximate phi this way. That is a false perception. The truth is that any additive number sequence defined by the relationship T(n+1) = Tn + T(n-1) also approximates the golden ratio by taking the ratio of any two successive numbers in the sequence. Like the Fibonacci numbers, the higher up in the sequence you go, the closer the ratio approaches phi.
For example the Lucas numbers form an additive sequence with this defining relationship. But the first two numbers in this sequence are 2 and 1 rather than 1 and 1. Alternatively, we can define an additive number sequence called X with this same relationship but with 5 and 200 as its first two numbers. Once again the ratio of any two sucessive numbers in the sequence approaches phi the higher up in the sequence we go. The above table tabulates the results for the Lucas and X numbers for the first 20 numbers.
Square Dance of the Fibonaccis
By forming a series of squares whose sides increase in length according to the Fibonacci numbers, we end up constructing a rectangle which closely approximates the Golden Rectangle (length to width ratio is phi). The animation below demonstrates just such a construction from a series of eight squares. It begins with a 1x1 square and builds up to a 21x21 square. The resulting rectangle is of length to width ratio of 34:21.
Many people erroneously think it is only the Fibonacci numbers which so uniquely approximate phi this way. That is a false perception. The truth is that any additive number sequence defined by the relationship T(n+1) = Tn + T(n-1) also approximates the golden ratio by taking the ratio of any two successive numbers in the sequence. Like the Fibonacci numbers, the higher up in the sequence you go, the closer the ratio approaches phi.
For example the Lucas numbers form an additive sequence with this defining relationship. But the first two numbers in this sequence are 2 and 1 rather than 1 and 1. Alternatively, we can define an additive number sequence called X with this same relationship but with 5 and 200 as its first two numbers. Once again the ratio of any two sucessive numbers in the sequence approaches phi the higher up in the sequence we go. The above table tabulates the results for the Lucas and X numbers for the first 20 numbers.
Square Dance of the Fibonaccis
By forming a series of squares whose sides increase in length according to the Fibonacci numbers, we end up constructing a rectangle which closely approximates the Golden Rectangle (length to width ratio is phi). The animation below demonstrates just such a construction from a series of eight squares. It begins with a 1x1 square and builds up to a 21x21 square. The resulting rectangle is of length to width ratio of 34:21.
© Copyright Lorena Loo
A quarter circle can be constructed in each of the squares to approximate a spiral called a Fibonacci spiral. In my article Ninety Degrees of Separation, I showed how the golden log spiral is formed within a Golden Rectangle and how closely this is approximated by a series of quarter circles. The Fibonacci spiral also very closely approximates the golden log spiral. Or Fibonacci is almost Phi-bonacci.
Text and images (unless otherwise indicated in the credits) are copyrighted
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
© by Lorena Loo
In the instances where images in the public domain have been modified as in the case of geometrizing them, the modified images are no longer public domain but the copyright of the author who made the modifications. Here that means me, Lorena Loo. That is by copyright law.
References:
Ron Knott has an excellent web site covering the Fibonacci numbers and the golden section.
Ron Knott has an excellent web site covering the Fibonacci numbers and the golden section.