golden treasure we share in the world of mathematics

Golden treasure Golden treasure we sharewe share

In the world of mathematics

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.

Calculating PhiCalculating Phi

11

11

11

11

11

11

11

11

11

1 ....

12

2

151

Φ=1,61803…

Golden section in Golden section in the history of the history of mathematicsmathematics

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

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.

CuriositiesCuriosities

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…

Construction of Construction of geometrical formsgeometrical forms

Special geometrical forms can be created by using the golden ratio:

GOLDEN TRIANGLE

GOLDEN PENTAGON

AND DECAGON

CoveringsCoverings

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.

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.

We made it:We made it:

Golden section in Golden section in the computer the computer

workingworking

Text editionText editionThe golden s. is often

used in book printing

This book was designed after the plan above.

Picture editionPicture editionThe golden section can be found in

photography, advertising and generally in mediaIsn’t it fantastic?

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

THANKS FOR YOUR THANKS FOR YOUR PARTICIPATIONPARTICIPATION!

SEE YOU IN APRIL!SEE YOU IN APRIL!