Outside My Window

This spiral is demanding that I blog about Fibonacci.  He was in the news last Friday because there’s a new book about him, and two weeks ago Chuck Tague said that the Fibonacci sequence describes spirals like these.

So I wondered… What are Fibonacci numbers?  And what do they have to do with spirals?

Fibonacci was a great Italian mathematician of the Middle Ages.  In 1202 he published Liber Abaci (Book of Calculation) in which he introduced the Hindu-Arabic numeral system to Europe and taught them calculations using digits 0-9 with place value.  Until that time, Europe used Roman numerals for commercial bookkeeping.  Imagine how hard that was!  CCXLVIII + MDCCCLXXIX = ?

In Liber Abaci Fibonacci discussed mathematical problems and calculations.  One of his “story problems” was about rabbits:

• Start with 1 male and 1 female rabbit in a field
• They produce 1 male and 1 female rabbit every month from their second month of age onward.
•  The young rabbits mature, pair up, and mate producing 1 male and 1 female per month from the second month of age onward.
• The rabbits never die. 

How many pairs of rabbits will be in the field at the end of one year?

The answer is a mathematical pattern that describes how many pairs of rabbits exist at the end of each month.   Start with 0 and 1 and put them in a row.  Add them together to produce the next number in the sequence.  Put this number at the end of the row and add those last two numbers to get the next one.  Keep doing this forever.  

       0,1
0+1=1  0,1,1
1+1=2  0,1,1,2
1+2=3  0,1,1,2,3
2+3=5  0,1,1,2,3,5
3+5=8  0,1,1,2,3,5,8

Thus the Fibonacci sequence:  0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610,…

How do Fibonacci numbers describe spirals? 

Imagine drawing squares whose edge lengths are the units in the Fibonacci sequence.  Each time you draw a new square, make it touch the ones you drew before.  Because each number in the sequence is larger, the new squares touch the old ones on their long edge.  Eventually you’ll notice that you are drawing squares in a spiral.  …Yes, this is hard to imagine.  Click here for an animation that shows how this works. 

How does this apply in nature?  Amazingly, the number of parts on a plant, the branching arrangements and the spirals of seed heads often follow the Fibonacci sequence.  Here are some real life examples (scroll down on the website).

I hope I haven’t lost you by now!  

I’ve only grazed the surface of Fibonacci in nature so if you’d like to learn more see this educational math website from Surry, UK that has good, simple examples and animations.

I really like numbers.  I get excited by these things.

(photo of a maypops tendril by Chuck Tague)

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