Transcript Fibinocci Numbers and The Golden Section

Fibonacci Numbers and
The Golden Section
Kristi Selkirk
Amber Ballance
Melissa Zale
Thomas J. Hill
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
Who was Fibonacci?
Born: 1170 in (probably) Pisa (now in Italy)
Died: 1250 in (possibly) Pisa (now in Italy)
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Fibonacci’s Four Famous Works
The four works from this period which have come down to us are:
Liber abbaci (1202, 1228)
Practica geometriae (1220/1221)
Flos (1225)
Liber quadratorum (1225)
0 1 1 2
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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
Fibonacci's Mathematical Contributions
Hindu-Arabic Number System (Positional System)
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)
The Fibonacci Series
Fibonacci Number Sequence
Fib(n): 0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987...
Hindu-Arabic Number System
n: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ...
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 …
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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.
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, ...
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Fibonacci Puzzles
Making a bee-line with Fibonacci numbers
0 1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 …
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S
E
E
D
H
E
A
D
S
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 about
the seed head.
Drag me to change angle
<- Zoom ->
Fraction of turn between seeds = 0.950
Angle between successive seeds = 342.164°
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
AC = CD
AD AC
C
AND
D
DB = BA
DA DB
Golden Section In Nature
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Nature Continued…
BIBLIOGRAPHY
Huntley, H.E. The Divine Proportion: A Study in Mathematical Beauty. New York:
Dover Publications, Inc., 1970.
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
http://evolutionoftruth.com/goldensection/
http://www.vashti.net/mceinc/golden.htm
http://www.summum.org/phi.htm
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