```                  ,                       ,
__      |\                     /|      __
\ ~-,   | \                   / |   ,-~ /
\  \\  |: 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!