Golden section

design theory.
5

mercredi, 27 avril 2011

Golden section


Golden Section

The Golden Mean or Golden Section is a ratio that is present in the
growth patterns of many things; the spiral formed by a shell or the curve of
a fern, for example.
We will call the Golden Ratio after a Greek letter, Phi (Φ) .
The Golden Mean or Golden Section was derived by the ancient Greeks.
Like "pi", the number 1.618... is an irrational number. Both the ancient
Greeks and the ancient Egyptians used the Golden Mean when designing
their buildings and monuments.


A bit of history...

Euclid, the Greek mathematician of about 300BC,
wrote the Elements which is a collection of 13 books on Geometry.
It was the most important mathematical work
until this century, when Geometry began
to take a lower place on school syllabuses,
but it has had a major influence on mathematics.
It starts from basic definitions called axioms
or "postulates" (self-evident starting points).

An example is: the fifth axiom
that there is only one line parallel
to another line through a given point.


A bit of history...

Euclid develops more results (called propositions)
about geometry based purely
on the axioms and previously
proved propositions using only logic.
The propositions involve constructing
geometric figures using a straight edge
and compasses only, so that
we can only draw straight lines and circles.


A B

Book 1, Proposition 10 to find the exact centre of any line AB


A bit of history... The golden ratio

In Book 6, Proposition 30, Euclid shows how to divide a line in mean and
extreme ratio.

Euclid used this phrase to mean the ratio of the smaller part of this line,
GB to the larger part AG (the ratio GB/AG) is the SAME as the ratio of the
larger part, AG, to the whole line AB (the ratio AG/AB). If we let the line AB
have unit length and AG have length g (so that GB is then just 1–g) then
the definition means that……


The golden ratio :
By geometry : method 1

1/2

1


The golden ratio :
By geometry : method 2 (golden rectangle)


The golden ratio :
By mathematics :

GB/ AG = AG / AB or 1–g / g = g / 1 so that 1–g = g2

here we have g² =1–g or g²+g =1.
We can solve this in this way and we find that:
g = (–1 + √5) /2 or g = ( –1 –√5) /2 = 0.6180339... = - Ø
1/0.6180339… = 1.6180339…= Ø
This is our friend phi


The golden ratio : Why is it unique
Let’s compare a line divide in any proportion with line divide with golden
ratio.

The smaller AB / The larger BC The smaller AB / The larger BC
2 / 5 = 0.4 0.618 / 1 = 0.618
The larger BC / The whole AC The larger BC / The whole AC
5 / 7 = 0.71 1 / 1.618 = 0.618

The two equations give different The two equations give identical
answers. answers.


A bit of geometry...

How to build a Pentagon from a square


The golden section..?


The golden section..?

AC/AB or AC/BC = 1.618
BD/DE -1,618


BD/DE -1,618


The Pentagram

One thinks that it was the rallying symbol of Pythagorean. The pentagram
was considered by the ancient as an universal symbol of perfection and
beauty. It is found in artistic creations, on some currencies, in the rosettes of
cathedrals, on flags and the badges of some sects.
BD/DE -1,618



Take three golden rectangles and assemble them at 90 degree angles to
get a 3D shape with 12 corners:

This is the basis for two geometric solids



Dodecahedron Icosahedron
The 12 corners become the 12 The 12 corners can also become
centers of each of the 12 the 12 points of each of the 20
pentagons that form the faces of a triangles that form the faces of a
dodecahedron. icosahedron.


A polyhedron, considered as a solid is convex if and only if the line
segment between any two points of the polyhedron belongs entirely to the
solid. However, if we admit a polyhedron to be non-convex, there exist four
more types of regular polyhedra !

The two Kepler polyhedra

The Small Stellated Dodecahedron The Great Stellated Dodecahedron


A polyhedron, considered as a solid is convex if and only if the line
segment between any two points of the polyhedron belongs entirely to the
solid. However, if we admit a polyhedron to be non-convex, there exist four
more types of regular polyhedra !

The two Poinsot polyhedra

The Great Dodecahedron The Great Icosahedron


The Small Stellated Dodecahedron


The Great Stellated Dodecahedron


The Great Dodecahedron


The Great Icosahedron


Penrose Tiling
Tiling in 5-fold symmetry was thought impossible!
Areas can be filled completely and symmetrically with tiles of 3, 4 and 6
sides, but it was long believed that it was impossible to fill an area with 5-
fold symmetry, as shown below:

3 sides 4 sides 5 sides leaves gaps 6 sides


The solution was found in Phi
In the early 1970's, however, Roger Penrose discovered that a surface
can be completely tiled in an asymmetrical, non-repeating manner in five-
fold symmetry with just two shapes based on phi, now known as "Penrose
tiles.“
Phi plays a pivotal role in these constructions. The relationship of the
sides of the pentagon, and also the tiles, is Ø, 1 and 1/Ø.

The triangle shapes found within a One creates a set of tiles like this.
pentagon are combined in pairs.


Penrose Tiling

The arrows give the rule of assembly


Golden Section & Fibonacci series

Leonardo Fibonacci discovered the
series which converges on phi.

In the 12th century, Leonardo Fibonacci discovered a simple numerical
series that is the foundation for an incredible mathematical relationship
behind phi.
Starting with 0 and 1, each new number in the series is simply the sum of
the two before it.
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, . . .
The ratio of each successive pair of numbers in the series approximates
phi (1.618. . .) , as 5 divided by 3 is 1.666..., and 8 divided by 5 is 1.60.
The table below shows how the ratios of the successive numbers in the
Fibonacci series quickly converge on Phi or Ø. After the 40th number in
the series, the ratio is accurate to 15 decimal places.


Golden Section & Fibonacci serie
0 0
1 1
2 1 1.000000000000000 +0.618033988749895
3 2 2.000000000000000 -0.381966011250105
4 3 1.500000000000000 +0.118033988749895
5 5 1.666666666666667 -0.048632677916772
6 8 1.600000000000000 +0.018033988749895
7 13 1.625000000000000 -0.006966011250105
8 21 1.615384615384615 +0.002649373365279
9 34 1.619047619047619 -0.001013630297724
10 55 1.617647058823529 +0.000386929926365
11 89 1.618181818181818 -0.000147829431923
12 144 1.617977528089888 +0.000056460660007
13 233 1.618055555555556 -0.000021566805661


Golden Section & Fibonacci serie
14 377 1.618025751072961 +0.000008237676933
15 610 1.618037135278515 -0.000003146528620
16 987 1.618032786885246 +0.000001201864649
17 1,597 1.618034447821682 -0.000000459071787
18 2,584 1.618033813400125 +0.000000175349770
19 4,181 1.618034055727554 -0.000000066977659
20 6,765 1.618033963166707 +0.000000025583188
21 10,946 1.618033998521803 -0.000000009771909
22 17,711 1.618033985017358 +0.000000003732537
23 28,657 1.618033990175597 -0.000000001425702
24 46,368 1.618033988205325 +0.000000000544570
25 75,025 1.618033988957902 -0.000000000208007
26 121,393 1.618033988670443 +0.000000000079452
27 196,418 1.618033988780243 -0.000000000030348


Fibonacci numbers Fibonacci numbers
in plant spirals in plant branching

Here a flower seed illustrates Here a plant illustrates that
this principal as the number of each successive level of
clockwise spirals is 55 (marked branches is often based on a
in red, with every tenth one in progression through the
white) and the number of Fibonacci series.
counterclockwise spirals is 89
(marked in green, with every
tenth one in white.)


Fibonacci's Rabbits

The original problem that Fibonacci investigated (in the year 1202) was
about how fast rabbits could breed in ideal circumstances.
Suppose a newly-born pair of rabbits, one male, one female, are put in a
field. Rabbits are able to mate at the age of one month so that at the end
of its second month a female can produce another pair of rabbits. Suppose
that our rabbits never die and that the female always produces one new
pair (one male, one female) every month from the second month on. The
puzzle that Fibonacci posed was...

How many pairs will there be in one year?


Fibonacci's Rabbits
1 - At the end of the first month, they mate, but there is still one only 1 pair.
2 - At the end of the second month the female produces a new pair, so now there are 2 pairs of
rabbits in the field.
3 - At the end of the third month, the original female produces a second pair, making 3 pairs in
all in the field.
4 - At the end of the fourth month, the original female has produced yet another new pair, the
female born two months ago produces her first pair also, making 5 pairs.


Golden Section

Why do some "beautiful" objects quickly lose their appeal, while others
seem to have a more lasting allure? Do lasting truths guide our perception
of what is beautiful?
Mankind has been fascinated by the notion of beauty since before
recorded history. When the Egyptians, Greeks and other ancient cultures
erected structures conforming to the proportions of the golden section,' it
wasn't because they applied rigorous mathematical formulas; few of them
had any awareness of the "power of numbers." Rather, collective opinion
told them the designs looked correct.

From article of : Ronald B. Kemnitzer, IDSA, and Augusto Grillo


Golden Section

We now recognize that mathematical proportions and systems drive
nature at all levels, from particles and atoms to micro-organisms to the
universe and beyond. The logarithmic spiral of the golden mean perfectly
describes the outwardly expanding nebulae of universes. The same spiral
can be found in the smaller world of shells.

From article of : Ronald B. Kemnitzer, IDSA, and Augusto Grillo


Golden Section

It was used in the design of
Notre Dame in Paris


Golden Section
It was used in the design of the Parthenon


Golden Section
It was used in the design of the Pyramid


Golden Section
In one object used by billion of people all over the world ?

The Credit Card is made with golden ration, it is a perfect golden rectangle


Golden Section
More examples :
Braun


Golden Section
More examples :
Your hand shows Phi and the Fibonacci Series
Each section of your index finger, from the tip to the base of the wrist, is larger than the
preceding one by about the Fibonacci ratio of 1.618, also fitting the Fibonacci numbers
2, 3, 5 and 8.

The ratio of your forearm to hand is Phi
Your hand creates a golden section in relation to your arm, as the ratio of your forearm
to your hand is also 1.618, the Divine Proportion.


Golden Section
More examples : Modulor by Le Corbusier


Golden Section
More examples :
Violin


Golden Section
More examples :
Art….

Advertising..


Golden Section
More examples :
Cars…


Golden Section
More examples :
……


Golden Section: A Controversial Issue

There are many books and articles that say that the golden rectangle is
the most pleasing shape and point out how it was used in the shapes of
famous buildings, in the structure of some music and in the design of
some famous works of art. Indeed, people such as Corbusier and Bartók
have deliberately and consciously used the golden section in their
designs.
However, the "most pleasing shape" idea is open to criticism. The golden
section as a concept was studied by the Greek geometers several
hundred years before Christ, as mentioned on earlier pages, But the
concept of it as a pleasing or beautiful shape only originated in the late
1800's and does not seem to have any written texts (ancient Greek,
Egyptian or Babylonian) as supporting hard evidence.


Golden Section: A Controversial Issue

At best, the golden section used in design is just one of several possible
"theory of design" methods which help people structure what they are
creating. At worst, some people have tried to elevate the golden section
beyond what we can verify scientifically. Did the ancient Egyptians really
use it as the main "number" for the shapes of the Pyramids? We do not
know.
So this course has lots of speculative material on it and would make a
good Project for a Science Fair perhaps, investigating if the golden section
does account for some major design features in important works of art,
whether architecture, paintings, sculpture, music or poetry.

It's over to you on this one!


Golden Section:
Some link
http://goldennumber.net/
http://www.ee.surrey.ac.uk/Personal/R.Knott/Fibonacci/fib.html
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibmaths.html
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibBio.html
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/fibnat.html
http://galaxy.cau.edu/tsmith/KW/golden.html
http://www.fortunecity.com/westwood/smith/338/mobaing.html
http://galaxy.cau.edu/tsmith/KW/goldenpenrose.html
http://www.akasha.de/~aton/PENROSEtile.html
http://staff.lib.muohio.edu/~jgoode/penrose/
http://www.beautyanalysis.com/index2_mba.htm


Golden section

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Golden section