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Golden treasure Golden treasure we sharewe share
In the world of mathematics
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Definition: What is Phi?Definition: What is Phi?
If the ratio of the sum of the quantities to the larger quantity is equal to the
ratio of the larger quantity to the smaller one.
φ = (a+b)/a = a/b
You can see it included in everything around you, but it
isn’t obvious.
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Calculating PhiCalculating Phi
11
11
11
11
11
11
11
11
11
1 ....
12
2
151
Φ=1,61803…
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Golden section in Golden section in the history of the history of mathematicsmathematics
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Ancient times
It was discovered by Egyptians and Greeks and was used in architecture
and art-like pyramidsThe ancient Greeks used it too, they
based the design of the Parthenon on Phi
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Middle Ages
Leonardo Fibonacci:In his book "Liber abaci", he
introduced the so called rabbits problem and the
Fibonacci-series
Leonardo Da Vinci presented the Golden Ratio through his
famous painting Marcus Vitruvius.
He used the golden ratio to define all the fundamental
proportions.
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CuriositiesCuriosities
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How can be calculated the stocksHow does colors pass to each other
How appears the Phi in population growth, and how it can be used, to predict urban
populations. How does Phi’s mysteriousness appear in
solar system…
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Construction of Construction of geometrical formsgeometrical forms
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Special geometrical forms can be created by using the golden ratio:
GOLDEN TRIANGLE
GOLDEN PENTAGON
AND DECAGON
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CoveringsCoverings
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A tessellation or tiling of the plane is a
collection of plane figures that fills the
plane with no overlaps and no gaps.
A tessellation or tiling of the plane is a
collection of plane figures that fills the
plane with no overlaps and no gaps.
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A non-periodic tiling can only fill the plane by A non-periodic tiling can only fill the plane by turning and mirroring the pieces, not just pushing turning and mirroring the pieces, not just pushing
them.them.
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We made it:We made it:
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Golden section in Golden section in the computer the computer
workingworking
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Text editionText editionThe golden s. is often
used in book printing
This book was designed after the plan above.
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Picture editionPicture editionThe golden section can be found in
photography, advertising and generally in mediaIsn’t it fantastic?
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FractalsFractalsA fractal is "a rough or fragmented geometric shape that can be
split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. Some
fractals follow the Fibonacci-series (Mandelbrot) or Fibonacci-spiral
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THANKS FOR YOUR THANKS FOR YOUR PARTICIPATIONPARTICIPATION!
SEE YOU IN APRIL!SEE YOU IN APRIL!