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Golden Rectangle


									                                              Golden Rectangle

     AC    AB
If              then AB is the golden mean, the golden ratio or the golden proportion
     AB    BC

                                                                     CD            DE

                                                                     AB  AE

                                                                     AC       AB
                                                                     AB       BC


                          1                                         1  5
                                      or     2    1  0   
                1          1                                          2

Infinite square roots:
              1       1      1     1 

        2  1  1            1       1
        2  1  
              1  5
                              1     1     1 

Infinite Continued fractions:
                            1                        1
           1                               1
                   1 
                              1                      
                         1 
                              1 
                                        “Constantly Mean”
                                         Paul S. Bruckman
                                     The Fibonacci Quarterly (1977)

                                 The Golden Mean is quite absurd
                                 It's not your ordinary surd
                                 If you invert it (this is fun!)
                                 You'll get itself, reduced by one
                                 But if increased by unity
                                 This yields its square, take it from me

                                 Expressed as a continued fraction
                                 It's one, one, one,..., until distraction
                                 In short, the simplest of such kind
                                 (Doesn't this really blow your mind?)

Recursion formula:
              Fn  1  Fn  1  Fn        where F0  1 F1  1

Binet’s formula (1834):

                      1    1  5 n  1  5 

              Fn                        
                      5       2        2  
                                               
 "Let's suppose that in the fenced place there is the couple of the
rabbits (female male) in first day of January. This couple of the
rabbits reproduces the new rabbits couple in the first day of
February and then in the first day of each next month. Each
newborn rabbits couple becomes mature in one month and then
gives a life to the new rabbits couple each month after. There is a
question: how much rabbits couples will be in the fenced place
in one year, that is in 12 months from the beginning of
reproduction? "

                                                                Pascal’s triangle, Fibonacci sequence,
                                                                      binomial formula and phi

                                                                  1 2 3 5 8                   Fn  1
                                                                   , , , , ,              ,           
                                                                  1 1 2 3 5                    Fn

                                                                              1  5
                                                                   

                                                           Regular Pentagram

                                               a              b                    c  1

                                                              1                 1
                                                         d              1 
                                                                               
Leonardo da Vinci
    "The Mona Lisa," undisputably Leonardo's most famous
painting, is full of Golden Rectangles. If you draw a rectangle
whose base extends from the woman's right wrist to her left elbow
and extend the rectangle vertically until it reaches the very top of
her head, you will have a Golden Rectangle. Then, if you draw
squares inside this Golden Rectangle you will discover that the
edges of these new squares come to all the important focal points of
the woman: her chin, her eye, her nose, and the upturned corner of
her mysterious mouth. It is believed that Leonardo, as a
mathematician, purposefully made this painting line up with Golden
Rectangles in this fashion in order to further the incorporation of
mathematics into art. In the spirit of mathematics in art, it is also
worth mentioning that the overall shape of the woman is a triangle
with her arms as the base and her head as the tip. This is meant to
draw attention to the face of the woman in the portrait.

Leonardo's famous study of the proportions of man, "The Vetruvian
Man" (The Man in Action), is also full of Golden Rectangles.
Unlike the Mona Lisa, where all the lines of the Golden Rectangle
are assumed by the mathematician, in "The Vetruvian Man", many
of the lines of the rectangles are actually drawn into the image, at
least in part. There are three distinct sets of Golden Rectangles in
this painting: one set for the head area, one for the torso, and one
for the legs.
To find the first set of rectangles, the one for the head, draw a
rectangle whose base goes along the man's neck from shoulder to
shoulder (stop at the shoulder lines provided by Leonardo). The top
of the rectangle should meet the top of the man's head. This creates
the first Golden Rectangle. Once you have that, inscribe a square in
the left side of the rectangle, creating a smaller Golden Rectangle
on the right side of the man's head. Then do the same with the right
side of the original rectangle, creating a long, thin rectangle that
runs vertically through the center of the man's head. Note that the
smaller Golden Rectangles intersect with the focal points of the
head: the eyes.
The second set of rectangles is found in a similar way. This time, all the lines are provided by Leonardo
in the painting. Draw a rectangle which runs from elbow to elbow and from neck to waist. This creates
a Golden Rectangle. Then, in a similar fashion to the first set, inscribe a square in each side of the
rectangle, creating two more Golden Rectangles. Note this time that these new smaller Golden
Rectangles intersect with the innermost portion of the man's torso.
For the third set, draw a rectangle whose lower two vertices are at the places where the man's outermost
toes touch the outlying circle. The rectangle should extend vertically to the man's waist. This creates yet
another Golden Rectangle. Now inscribe squares in the sides of the rectangle as you did before. This
time it seems that the two smaller rectangles come to where the man's legs would be if they weren't
turned out in that fashion.
George Seurat

Salvador Dali
                                   Sacrament of the Last Supper

                           105.5” x 65.75” ≈ golden ratio &


                                           sunflower 34 petals

         Number of Petals                                            Flower
      3 petals (or 2 sets of 3)                   lily (usually in 2 sets of 3 for 6 total), iris
              5 petals                   buttercup, wild rose, larkspur, columbine (aquilegia), vinca
              8 petals                                       delphinium, coreopsis
             13 petals                                    ragwort, marigold, cineraria
             21 petals                                  aster, black-eyed susan, chicory
             34 petals                                     plantain, daisy, pyrethrum
             55 petals                                    daisy, the asteraceae family
             89 petals                                    daisy, the asteraceae family

There are exceptions to this list. Most fall into two categories; a doubling of the number of petals,
and/or a version of the Fibonacci Series called the Lucas Series (2, 1, 3, 4, 7, 11, 18, 29, 47, 76, etc.).
Mutations and variations from species to species also account for exceptions but when the number of
petals are averaged, the number will usually be a Fibonacci or Lucas Number.
By far the most fascinating appearances of the
Fibonacci Series in nature are the spirals that can
be seen in everything from sunflowers to pine
cones to pineapples. We are about to explain that
this phenomenon comes not from perfection
through evolution (which is, in itself, oxymoronic)
but from the dynamics of plant growth. To begin to
understand how these spirals come to be, one must
go back to the beginning; to where flowers and
fruits and seeds start: the apex. The apex is the tip
of the shoot of a growing plant. It is the bud on the
end of a stem on a tree and the bulb of a flower
before it blooms. Around the apex grow little
bumps called primordia. As more primordia
develop, they are pushed farther and farther from
the apex and they develop into the familiar features
of a plant, be it a leaf, a flower, or parts of a fruit.
Let us consider a sunflower with primordia
growing from the center. The first primordia to
develop end up being farther from the apex than
later primordia. Therefore, it can be deduced from Another appearance of the Fibonacci Series in
this in what order the primordia appeared. As it         seedheads like the one shown above, pincones,
happens, if one took the first and second primordia pineapples, etc., is that the number of spirals
and measured the angle between them with the             going in each direction is a Fibonacci Number. In
center of the seed head as the vertex, the angle         the diagram above, for example, there are 13
would be very close to 137.5 degrees.
                                                         spirals that turn clockwise and 21 curving
                                                         counterclockwise. On all other sunflowers, the
                                                         number of clockwise and counterclockwise
                                                         spirals will always be consecutive Fibonacci
                                                         Numbers like 21 and 34 or 55 and 34.

Diagram of a seed head with the golden angle,
about 137.5 degrees, inscribed.

                               Peter v. Sengbusch - Impressum

 Leaves are always organized around the shoot in a regular pattern that can be recognized best when
looked at from above. It is obvious in most cases that the leaves are arranged in the way of a screw and
that there is always the same angle measurable between subsequent leaves (angle of divergence). In its
simplest form, this angle is 180° which means that the leaves are grouped into two opposing rows along
the shoot. But other angles do occur, too: 120° (a third of a circle), 144° (five leaves in two circles), or
135° (eight leaves in three circles), etc.. The regularity of the positions of leaves has been discovered
in the last century by C. F. SCHIMPER and A. BRAUN. The occurring angles correspond to the
FIBONACCI -series, i.e. numerator and denominator of following fractions can be calculated by
adding the numerators and the denominators of the two preceding fractions:

                             2/5 = 1+1 / 2+3 ; 3/8 = 1+2 / 3+5 etc.

 The resulting series is:

                             1/2 1/3 2/5 3/8 5/13 8/21 13/34 21/55 34/89 ........

  Calculated down to the degree of the angles, a limit of approximately 137° 30' is achieved. This angle
is known to divide an arc of a circle with the golden section.
The advantage of a regular arrangement of the leaves is in the optimal yield of light gained.

                                  The Great Pyramid of Giza

                                                        Notre Dame in Paris

                                       United Nations
     Great Mosque of Kairouan

                             to 1050 places
1.6180339887   4989484820   4586834365 6381177203   0917980576     2862135448   6227052604
  6281890244   9707207204   1893911374 8475408807   5386891752     1266338622   2353693179
  3180060766   7263544333   8908659593 9582905638   3226613199     2829026788   0675208766
  8925017116   9620703222   1043216269 5486262963   1361443814     9758701220   3408058879
  5445474924   6185695364   8644492410 4432077134   4947049565     8467885098   7433944221
  2544877066   4780915884   6074998871 2400765217   0575179788     3416625624   9407589069
  7040002812   1042762177   1117778053 1531714101   1704666599     1466979873   1761356006
  7087480710   1317952368   9427521948 4353056783   0022878569     9782977834   7845878228
  9110976250   0302696156   1700250464 3382437764   8610283831     2683303724   2926752631
  1653392473   1671112115   8818638513 3162038400   5222165791     2866752946   5490681131
  7159934323   5973494985   0904094762 1322298101   7261070596     1164562990   9816290555
  2085247903   5240602017   2799747175 3427775927   7862561943     2082750513   1218156285
  5122248093   9471234145   1702237358 0577278616   0086883829     5230459264   7878017889
  9219902707   7690389532   1968198615 1437803149   9741106926     0886742962   2675756052
  3172777520   3536139362   1076738937 6455606060   5921658946     6759551900   4005559089

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