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 Did you know . . .?


The Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 . . .) occurs throughout the worlds of nature, art, music, and mathematics!

Portrait of FibonacciEach term in the series is produced by adding together the two previous terms, so that 1 + 1=2, 1 + 2=3, 2 + 3=5, and so on. The sequence takes its name from a famous thirteenth-century European mathematician, Leonard of Pisa (?1170-1250), also called Fibonacci. Fibonacci was one of the first Europeans to use Arabic numbers, whose use he explained in his 1202 Liber abaci.

(Image source: http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Fibonacci.html)

Fibonacci gave this sequence as an answer to the following mathematical puzzle:

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?

The answer is the sequence 1, 1, 2, 3, 5, 8, 13, 21, . . ., as illustrated below:

Illustration of pairs of rabbits at one-month intervals

(Image source: http://www.jimloy.com/algebra/fibo.htm)

Diagram of sunflower seed headFibonacci numbers occur many times in the natural world. Plants tend to have a number of leaves that is a Fibonacci number, and flowers have a Fibonacci number of petals. Seeds in a flower head are often arranged in spiral patterns that are related to Fibonacci numbers (for example, the number of spirals that curve to the left and the number of spirals that curve to the right will be adjacent numbers in the Fibonacci sequence).

(Image source: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#seeds)

Spiral shells also exhibit patterns related to the Fibonacci sequence:

Nautilus shell, cross-section Animated GIF of successive rectangles built on squares with sides that are Fibonacci numbers sprialling outwards Nautilus-like spiral inscribed over spiral of Fibonacci squares

(Source of images: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html#spiral)

Fibonacci numbers are also important in art and music. The ratio between successive Fibonacci numbers approximates an important constant called "the golden mean" or sometimes phi, which is approximately 1.61803. The higher you go in the Fibonacci sequence, the more closely the ratio between two successive numbers in the sequence approximates phi. (By the way phi2=phi + 1!)

A golden rectangle, with line dividing its longer side in proportion 1 : .61803 (golden section)A rectangle whose sides are in the proportion 1 : 1.61803 is supposed to be the most aesthetically perfect rectangle (the "golden rectangle"). The Parthenon in Athens has such a rectangle as its face, and phi is said to have figured in the construction of the Great Pyramids. The "golden section," in which a line is divided into segments of lengths in the ratio 1 : .61803 is supposed to be an aesthetically ideal way to divide a line.

Numerous artists have used the golden section in their works, as well as composers, including (perhaps) Beethoven and Mozart.

(Image source: http://pauillac.inria.fr/algo/bsolve/constant/gold/gold.html)

Illustration of Pascal's triangle with diagonals added to produce Fibonacci sezuenceBy the way, the Fibonacci sequence also shows up in Pascal's triangle, if you add the diagonals!

(Image source: http://forum.swarthmore.edu/dr.math/faq/faq.pascal.triangle.html)

In his own words:

How many pairs of rabbits can be bred from one pair in a year?
A man has one pair of rabbits at a certain place entirely surrounded by a wall. We wish to know how many pairs will be bred from it in one year, if the nature of these rabbits is such that they breed every month one other pair and begin to breed in the second month after their birth. ... Liber abaci (1202)

(Quotation source: http://www-groups.dcs.st-andrews.ac.uk/~history/Quotations/Fibonacci.html)

Fun links:

Unfold the Golden Rectangle: http://www.vashti.net/mceinc/Unfold0.HTM

Using the Fibonacci numbers to calculate pi: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpi.html#piandfib

Some Fibonacci number puzzles: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibpuzzles.html

For more information:

Incredible set of pages on Fibonacci numbers and the golden section by Dr. Ron Knott: http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Jim Loy's pages on the Fibonacci numbers (http://www.jimloy.com/algebra/fibo.htm) and the golden rectangle and golden ratio (http://www.jimloy.com/geometry/golden.htm)

Dr. Math on Fibonacci numbers and the golden mean: http://forum.swarthmore.edu/dr.math/faq/faq.golden.ratio.html

A brief overview: http://plus.maths.org/issue3/fibonacci/

On the golden mean: http://pauillac.inria.fr/algo/bsolve/constant/gold/gold.html

On Fibonacci: http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Fibonacci.html

And: http://cedar.evansville.edu/~ck6/bstud/fibo.html

By Laura Smoller, UALR Department of History.

June 2001.