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The Fibonacci numbers are Nature's numbering system.  They appear everywhere in Nature, from the leaf arrangement in plants, to the pattern of the florets of a flower and the scales of a pineapple.  The Fibonacci numbers are therefore applicable to the growth of every living thing, including a single cell, a grain of wheat, a hive of bees, and even all of mankind.

Fibonacci is still recognized today as the greatest European mathematician of the Middle Ages.  He was born in the 1170’s and died in the 1240’s and there is now a statue commemorating him located at the Leaning Tower end of the cemetery next to the Cathedral in Pisa.  His full name was Leonardo of Pisa or Leonardo Pisano in Italian.  He called himself Fibonacci which was short for Filius Bonacci, standing for son of Bonacci which was his father’s name.  He was one of the first people to introduce the Hindu-Arabic number system into Europe, the system we now use today, based on ten digits, a decimal point and a symbol for zero: 1 2 3 4 5 6 7 8 9 and 0.

Being a mathematician, he attempted to solve a problem which he had raised:

How big would a rabbit colony be each month if rabbits gave birth to a pair of young every month and started breeding at two months of age?

The solution he discovered was to add the rabbits from the previous month with the young from the current month.  This gives a sequence (the Fibonacci sequence) in which "each number is the sum of the two preceding numbers".

Thus, the sequence progresses: 1,1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 and so on.  This is an example of a recursive sequence, obeying the simple rule that to calculate the next term one simply sums the preceding two.

Plants do not know about this sequence - they just grow in the most efficient ways.However, many plants show the Fibonacci numbers in the arrangement of the leaves around the stem.  Some pine cones and fir cones also show the numbers, as do daisies and sunflowers.  Why do these arrangements occur in nature? In the case of leaf arrangement it may be related to maximizing the space for each leaf, or the average amount of light falling on each one.


So nature isn't trying to use the Fibonacci numbers, they are appearing as a by-product of a deeper physical process. This is why the spirals can be imperfect.  The plant is responding to physical constraints, not to a mathematical rule.

 The Fibonacci Spirals and the Nautilus

The Fibonacci Spiral is created by using the Fibonacci numbers 1,1,2,3,5,8,13,21





Stage 1 - Draw

a square 1 unit x 1 unit

Stage 2 - Draw a

second square 1 unit x

1 unit (the sum of the

previous 2, i.e. 0+1=1)

Stage 3 - Draw a third

square 2 units x 2 units

(the sum of the previous

2, i.e. 1+1=2)




Stage 4 - Draw a third square 3 units x 3 units (the sum of the previous 2, i.e. 2+1=3)


Stage 5 - Draw a fourth square 5 units x 5 units (the sum of the previous 2, i.e. 3+2=5)




Stage 6 & 7 – Continue the process for two further stages to create

a final square 13 units x 13 units)


Notice that the rectangles which result at each stage are all roughly the same shape, that is, that the ratio of length to width seems to “settle down” as we build the pattern outward.


We can then draw a spiral by putting together quarter circles, one in each new square. This is a known as The Fibonacci Spiral. 

The curve is strangely reminiscent of the shells of Nautilus and snails. This is not surprising, as the curve tends to a logarithmic spiral as it expands