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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.mercredi, 27 avril 2011
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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.mercredi, 27 avril 2011
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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.mercredi, 27 avril 2011
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A B Book 1, Proposition 10 to find the exact centre of any line ABmercredi, 27 avril 2011
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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……mercredi, 27 avril 2011
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The golden ratio : By geometry : method 1 1/2 1mercredi, 27 avril 2011
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The golden ratio : By geometry : method 1 1/2 1mercredi, 27 avril 2011
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The golden ratio : By geometry : method 1 1/2 1mercredi, 27 avril 2011
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The golden ratio : By geometry : method 1 1/2 1mercredi, 27 avril 2011
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The golden ratio : By geometry : method 1 1/2 1mercredi, 27 avril 2011
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The golden ratio : By geometry : method 1 1/2 1mercredi, 27 avril 2011
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The golden ratio : By geometry : method 2 (golden rectangle)mercredi, 27 avril 2011
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The golden ratio : By geometry : method 2 (golden rectangle)mercredi, 27 avril 2011
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The golden ratio : By geometry : method 2 (golden rectangle)mercredi, 27 avril 2011
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The golden ratio : By geometry : method 2 (golden rectangle)mercredi, 27 avril 2011
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The golden ratio : By geometry : method 2 (golden rectangle)mercredi, 27 avril 2011
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The golden ratio : By geometry : method 2 (golden rectangle)mercredi, 27 avril 2011
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The golden ratio : By geometry : method 2 (golden rectangle)mercredi, 27 avril 2011
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The golden ratio : By geometry : method 2 (golden rectangle)mercredi, 27 avril 2011
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The golden ratio : By geometry : method 2 (golden rectangle)mercredi, 27 avril 2011
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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 phimercredi, 27 avril 2011
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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.mercredi, 27 avril 2011
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A bit of geometry... How to build a Pentagon from a squaremercredi, 27 avril 2011
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A bit of geometry... The golden section..?mercredi, 27 avril 2011
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A bit of geometry... The golden section..?mercredi, 27 avril 2011
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A bit of geometry... The golden section..?mercredi, 27 avril 2011
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A bit of geometry... The golden section..?mercredi, 27 avril 2011
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A bit of geometry... The golden section..? AC/AB or AC/BC = 1.618 BD/DE -1,618mercredi, 27 avril 2011
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AC/AB or AC/BC = 1.618 BD/DE -1,618mercredi, 27 avril 2011
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AC/AB or AC/BC = 1.618 BD/DE -1,618mercredi, 27 avril 2011
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AC/AB or AC/BC = 1.618 BD/DE -1,618mercredi, 27 avril 2011
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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 AC/AB or AC/BC = 1.618 cathedrals, on flags and the badges of some sects. BD/DE -1,618mercredi, 27 avril 2011
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A bit of geometry... 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 solidsmercredi, 27 avril 2011
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A bit of geometry... 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.mercredi, 27 avril 2011
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A bit of geometry... 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 Dodecahedronmercredi, 27 avril 2011
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A bit of geometry... 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 Icosahedronmercredi, 27 avril 2011
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A bit of geometry... The Small Stellated Dodecahedronmercredi, 27 avril 2011
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A bit of geometry... The Great Stellated Dodecahedronmercredi, 27 avril 2011
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A bit of geometry... The Great Dodecahedronmercredi, 27 avril 2011
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A bit of geometry... The Great Icosahedronmercredi, 27 avril 2011
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A bit of geometry... 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 sidesmercredi, 27 avril 2011
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A bit of geometry... The solution was found in Phi In the early 1970s, 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.mercredi, 27 avril 2011
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A bit of geometry... Penrose Tiling The arrows give the rule of assemblymercredi, 27 avril 2011
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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.mercredi, 27 avril 2011
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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.)mercredi, 27 avril 2011
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Fibonaccis 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?mercredi, 27 avril 2011
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Fibonaccis 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.mercredi, 27 avril 2011
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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 wasnt 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 Grillomercredi, 27 avril 2011
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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 Grillomercredi, 27 avril 2011
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Golden Section It was used in the design of Notre Dame in Parismercredi, 27 avril 2011
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Golden Section It was used in the design of Notre Dame in Parismercredi, 27 avril 2011
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Golden Section It was used in the design of Notre Dame in Parismercredi, 27 avril 2011
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Golden Section It was used in the design of Notre Dame in Parismercredi, 27 avril 2011
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Golden Section It was used in the design of Notre Dame in Parismercredi, 27 avril 2011
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Golden Section It was used in the design of Notre Dame in Parismercredi, 27 avril 2011
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Golden Section It was used in the design of Notre Dame in Parismercredi, 27 avril 2011
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Golden Section It was used in the design of the Parthenonmercredi, 27 avril 2011
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Golden Section It was used in the design of the Parthenonmercredi, 27 avril 2011
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Golden Section It was used in the design of the Parthenonmercredi, 27 avril 2011
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Golden Section It was used in the design of the Parthenonmercredi, 27 avril 2011
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Golden Section It was used in the design of the Pyramidmercredi, 27 avril 2011
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Golden Section It was used in the design of the Pyramidmercredi, 27 avril 2011
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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 rectanglemercredi, 27 avril 2011
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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 rectanglemercredi, 27 avril 2011
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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 rectanglemercredi, 27 avril 2011
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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 rectanglemercredi, 27 avril 2011
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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 rectanglemercredi, 27 avril 2011
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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 rectanglemercredi, 27 avril 2011
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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 rectanglemercredi, 27 avril 2011
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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.mercredi, 27 avril 2011
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Golden Section More examples : ……mercredi, 27 avril 2011
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Golden Section More examples : ……mercredi, 27 avril 2011
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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 1800s and does not seem to have any written texts (ancient Greek, Egyptian or Babylonian) as supporting hard evidence.mercredi, 27 avril 2011
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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. Its over to you on this one!mercredi, 27 avril 2011
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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.htmmercredi, 27 avril 2011
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