, ,
__ |\ /| __
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\ \\ |: Y Y :| // /
^. )\ jj | Mi Amore | jj /( .^
"v-"~"-> Leonardo >-"~"-v"
Y \ \ / Y
| o oi \ jo o |
j ~T~ ) ( ~T~ j
\._ '-_.> >._-' _./
| "~" \ / "~" |
l _ ~"-; |\ /| ;-"~ _ l
/ ~"-, \l \ / l/ ,-"~ \
/ -. \/\\/ \//\/ .- \
Y \ Y Y / Y
! I l l I ! art adapted from
/"\ /_ /] ]\ _\ /"\ s1171180@
) .Y ~ (----~ "( (" ~----( ~ Y. ) giaeb.cc.monash.edu.au
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Assume that a pair of rabbits (a) matures at age 2 months and
(b) produces a pair of offspring every month from maturity on..
After x months, how many pairs will there be if none of the
rabbits dies during this time?
month - pairs
------------------------------------------------ P' = 0 mo ---
0 - 1 P'
------------------------------------------------ P" = 1 mo ---
1 - 1 P"
------------------------------------------------ P# = 2 mo ---
2 - 2 P# |_ P' first reproduction
--------------------------------------------------------------
3 - 3 P# | P"
|_ P' second reproduction
--------------------------------------------------------------
4 - 5 P# | P# |_ P' first reproduction
| P"
|_ P' third reproduction
--------------------------------------------------------------
5 - 8 P# | P# | P"
| |_ P'
| P# |_ P'
| P"
|_ P'
--------------------------------------------------------------
6 - 13 P# | P# | P# |_ P'
| | P"
| |_ P'
| P# | P"
| |_ P'
| P# |_ P'
| P"
|_ P'
--------------------------------------------------------------
7 - 21 P# | P# | P# | P"
| | |_ P'
| | P# |_ P'
| | P"
| |_ P'
| P# | P# |_ P'
| | P"
| |_ P'
| P# | P"
| |_ P'
| P# |_ P'
| P"
|_ P
--------------------------------------------------------------
8 - 34 P# | P# | P# | P# |_ P'
| | | P"
| | |_ P'
| | P# | P"
| | |_ P'
| | P# |_ P'
| | P"
| |_ P'
| P# | P# | P"
| | |_ P'
| | P# |_ P'
| | P"
| |_ P'
| P# | P# |_ P'
| | P"
| |_ P'
| P# | P"
| |_ P'
| P# |_ P'
| P"
| P'
next
==============================================================
Each number in the series 1, 1, 2, 3, 5, 8, 13, 21, 34, etc.
is the sum of the previous two numbers. Dividing each number
in the series by the one which follows it produces a ratio
which stabilizes around .618034 --
___FIBONACCI___ ___FIBONACCI___ ___FIBONACCI___
number ratio number ratio number ratio
1 89 .617978 10946 .618034
1 1.000000 144 .618056 17711 .618034
2 .500000 233 .618026 28657 .618034
3 .666667 377 .618037 46368 .618034
5 .600000 610 .618033 75025 .618034
8 .625000 987 .618034 121393 .618034
13 .615385 1597 .618034 196418 .618034
21 .619048 2584 .618034 317811 .618034
34 .617647 4181 .618034 514229 .618034
55 .618182 6765 .618034 832040 .618034
next
==============================================================
"Golden Ratio" = .61803 39887 49894 84820 45868 34365 63811...
can be obtained from the quadratic solution of
g^2 + g = 1
5^(1/2) - 1 5^(1/2) + 1
g = ------------- G = -------------
2 2
1 1 1
g = ------- = ------- .618034 = ----------
1 + g G 1.618034
next
==============================================================
To construct the "Golden Rectangle" RECT:
R_________Q_____T Begin with the square SQRE.
| | | Determine the midpoint M of side ES.
| L|-----|F Set a compass the length MQ and construct
| | | an arc from M to intersect the extension
| | | of ES at C.
|____v____|_____| Extend RQ and construct a perpendicular
E M S C to it at T from C.
The "Golden Ratio" is the ratio of ES to EC, and also SC to TC.
If you construct the line LF you create another square SCFL and another
golden rectangle LFTQ. You can continue to subdivide the rectangles
into smaller squares and rectangles. By drawing a curved line through successively smaller squares, you can construct the "Golden Spiral".
Draw straight lines connecting each of the vertices of a pentagon. You will have constructed two kinds of "Golden Triangle" (fifteen of each): one with the base in golden ratio to the sides, the other with both sides in golden ratio to the base.
The "Golden Ellipse" has one axis in golden ratio to the other.
Dividing a circle into two arcs, one in golden ratio to the other, generates the "Golden Angle" of 137.5 degrees.
These patterns show up in buildings (Parthenon, Pyramids), the angles at which leaves sprout from stems, the shape of pine trees and standard chicken eggs. The navel divides the human body into a golden ratio, as the neck does the upper half and knee the lower. The spiral is the characteristic shape for many things, from spiral galaxies through nautilus shells to the alpha waves emitted from rotating particles.
All this from highly fertile and apparently immortal rabbits!