Fibonacci Numbers and The Golden Section

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987...

Thomas J. Hill

Kristi Selkirk

Melissa Zale

Amber Ballance

Who was Fibonacci?

Born: 1170 in (probably) Pisa (now in Italy)Died: 1250 in (possibly) Pisa (now in Italy)

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The four works from this period which have come down to us are:

Fibonacci’s Four Famous Works

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987...

Liber quadratorum (1225)

Flos (1225)

Practica geometriae (1220/1221)

Liber abbaci (1202, 1228)

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Fibonacci's Mathematical Contributions

1 2 3 4 5 6 7 8 9 and 0

Roman Numerals

I = 1, V = 5, X = 10, L = 50, C = 100, D = 500 and M = 1000

For instance, 13 would be written as XIII or perhaps IIIX.

2003 would be MMIII or IIIMM. 99 would be LXXXXVIIII and 1998 is MDCCCCLXXXXVIII

For example, XI means 10+1=1 but IX means 1 less than 10 or 9. 8 is still written as VIII (not IIX)

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Hindu-Arabic Number System (Positional System)

The Fibonacci Series

Hindu-Arabic Number System

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Fibonacci Number Sequence

Fib(n): 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987...

0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987…

n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...

Patterns in the Fibonacci Numbers & Cycles in the Fibonacci Numbers

Here are some patterns people have already noticed:

Pattern Number 1

0,1,1,2,3,5,8,13,21,34,55,...

Pattern Number 2

00, 01, 01, 02, 03, 05, 08, 13, ...

For the last three digits, the cycle length is 1,500

For the last four digits,the cycle length is 15,000

For the last five digits the cycle length is 150,000

and so on...

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Fibonacci's Rabbits The original problem that Fibonacci investigated (in the year 1202) was

about how fast rabbits could breed in ideal circumstances.

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The number of pairs of rabbits in the field at the start of each month is

1, 1, 2, 3, 5, 8, 13, 21, 34, ...

Fibonacci Puzzles

Making a bee-line with Fibonacci numbers

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0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987…

S E E D

H E A D S

<- Zoom ->

Drag me to change angle

What's the best angle to have between SEEDS in a seedhead?

The best angle must have NO gaps which waste space and evenly space the seeds aboutthe seed head.

Angle between successive seeds = 342.164°

Fraction of turn between seeds = 0.950

What is the Golden Section (or Phi)?(Also called The Divine Proportion)

Golden Section, in mathematics, a geometric proportion in which a line is divided so that the ratio of the length of the longer line segment to the length of the entire line is equal to the ratio of the length of the shorter line segment to the length of the longer line segment.

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The Golden Section in Architecture The Parthenon and Greek Architecture

Even from the time of the Greeks, a rectangle whose sides are in the "golden proportion"

(1 : 1.618 which is the same as 0.618 : 1)

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Golden Section in Art

A B C D

AC = CD

AD ACAND

DB = BA

DA DB

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Golden Section In Nature

Nature Continued…

BIBLIOGRAPHY

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html

Huntley, H.E. The Divine Proportion: A Study in Mathematical Beauty. New York: Dover Publications, Inc., 1970.

http://www.vashti.net/mceinc/golden.htm

http://www.summum.org/phi.htm

http://evolutionoftruth.com/goldensection/

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