From Math Images
- The spiral curve of the Nautilus sea shell follows the pattern of a spiral drawn in a Fibonacci rectangle, a collection of squares with sides that have the length of Fibonacci numbers.
Basic DescriptionThe Fibonacci sequence is the sequence where the first two numbers are 1s and every later number is the sum of the two previous numbers. So, given two 's as the first two terms, the next terms of the sequence follows as :
The Fibonacci numbers can be discovered in nature, such as the spiral of the Nautilus sea shell, the petals of the flowers, the seed head of a sunflower, and many other parts. The seeds at the head of the sunflower, for instance, are arranged so that one can find a collection of spirals in both clockwise and counterclockwise ways. Different patterns of spirals are formed depending on whether one is looking at a clockwise or counterclockwise way; thus, the number of spirals also differ depending on the counting direction, as shown by Image 1. The two numbers of spirals are always consecutive numbers in the Fibonacci sequence.
Nature prefers this way of arranging seeds because it seems to allow the seeds to be uniformly distributed. For more information about Fibonacci patterns in nature, see Fibonacci Numbers in Nature
The Fibonacci sequence was studied by Leonardo of Pisa, or Fibonacci (1770-1240). In his work Liber Abacci, he introduced a problem involving the growth of the rabbit population. The assumptions were
- there is one pair of baby rabbits placed in an enclosed place on the first day of January
- this pair will grow for one month before reproducing and produce a new pair of baby rabbits on the first day of March
- each new pair will mature for one month and produce a new pair of rabbits on the first day of their third month
- the rabbits never die, so after they mature, the rabbits produce a new pair of baby rabbits every month.
The problem was to find out how many pairs of rabbits there will be after one year.
On January 1st, there is only 1 pair. On February 1st, this baby rabbits matured to be grown up rabbits, but they have not reproduced, so there will only be the original pair present.
Now look at any later month. June is a good example. As you can see in Image 2, all 5 pairs of rabbits that were alive in May continue to be alive in June. Furthermore, there are 3 new pairs of rabbits born in June, one for each pair that was alive in April (and are therefore old enough to reproduce in June).
This means that on June 1st, there are 5 + 3 = 8 pairs of rabbits. This same reasoning can be applied to any month, March or later, so the number of rabbits pairs in any month is the same as the sum of the number of rabbit pairs in the two previous months.
This is exactly the rule that defines the Fibonacci sequence. As you can see in the image, the population by month begins: 1, 1, 2, 3, 5, 8, ..., which is the same as the beginning of the Fibonacci sequence. The population continues to match the Fibonacci sequence no matter how many months out you go.
An interesting fact is that this problem of rabbit population was not intended to explain the Fibonacci numbers. This problem was originally intended to introduce the Hindu-Arabic numerals to Western Europe, where people were still using Roman numerals, and to help people practice addition. It was coincidence that the number of rabbits followed a certain pattern which people later named as the Fibonacci sequence.
Fibonacci Numbers in Nature
A More Mathematical Explanation
Symbolic Definition of Fibonacci SequenceThe Fibonacci sequence is the sequence UNIQ44edd8076c [...]
Symbolic Definition of Fibonacci Sequence
The Fibonacci sequence is the sequence where
The Fibonacci sequence is recursively defined because each term is defined in terms of its two immediately preceding terms.
Identities and Properties
Binet's Formula for Fibonacci Numbers
Fibonacci Numbers and Fractals
- There are currently no teaching materials for this page. Add teaching materials.
Maurer, Stephen B & Ralston, Anthony. (2004) Discrete Algorithmic Mathematics. Massachusetts : A K Peters.
Posamentier, Alfred S & Lehmann Ingmar. (2007) The Fabulous Fibonacci Numbers. New York : Prometheus Books.
Vorb'ev, N. N. (1961) Fibonacci Numbers. New York : Blaisdell Publishing Company.
Hoggatt, Verner E., Jr. (1969) Fibonacci and Lucas Numbers. Boston : Houghton Mifflin Company.
Knott, Ron. (n.d.). The Fibonacci Numbers and Golden Section in Nature. Retrieved from http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html
Wikipedia (Golden Ratio). (n.d.). Golden Ratio. Retrieved from http://en.wikipedia.org/wiki/Golden_ratio.
Fibonacci Numbers in Nature & the Golden Ratio. (n.d.). In World-Mysteries.com. Retrieved from http://www.world-mysteries.com/sci_17.htm
Wikipedia (Mandelbrot Set). (n.d.). Mandelbrot Set. Retrieved from http://en.wikipedia.org/wiki/Mandelbrot_set.
Devaney, Robert L. (2006) Unveiling the Mandelbrot Set. Retrieved from http://plus.maths.org/issue40/features/devaney/.
Weisstein, Eric W. (n.d.). Mandelbrot Set. In MathWorld--A Wolfram Web Resource. Retrieved from http://mathworld.wolfram.com/MandelbrotSet.html.
Future Directions for this Page
Things to add(possible ideas for future)
- Fibonacci numbers and Pascal's triangle
- A helper page for recursively defined sequence
- A section describing the Fibonacci numbers with negative subscripts. this appears in Finite Difference of Fibonacci Numbers section
Things to 'not' add
- A derivation of the exact value of the golden ratio. The derivation is redundant with the information in the golden ratio page.
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