Introduction Definition of a fractal Special fractals:*The Mandelbrot set* The Koch snowflake* Sierpiński triangle

Fractals in nature Conclusion Bibliography

Presentation outline

The word "fractal" often has different connotations for laypeople than for mathematicians, where the layperson is more likely to be familiar with fractal art than a mathematical conception. The mathematical concept is difficult to define formally even for mathematicians, but key features can be understood with little mathematical background.

Introduction

 

Definition of a self-similarity

A fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similar across different scales. They are created by repeating a simple process over and over in an ongoing feedback loop. Nature is full of fractals, for instance: trees, rivers, coastlines, mountains, clouds, seashells, hurricanes, etc. Abstract fractals – such as the Mandelbrot Set – can be generated by a computer calculating a simple equation over and over.

Definition of a fractal

Benoit Mandelbrot was the first person to extensively study and fully appreciate the importance of this beautiful and complex mathematical object.

Definition of a fractal

I. The Mandelbrot set II. The Koch snowflakeIII. The Sierpiński triangle

Special fractals

 

The Mandelbrot set

The Mandelbrot set

The Koch snowflake is a remarkable geometric shape first studied by the Swedish mathematician Helge von Koch in the early 1900s.

The Koch snowflake

START. Start with a solid equilateral triangle of arbitrary size.(For simplicity we will assume that the sides of the triangle are of lenght 1)

The Koch snowflake

STEP 1. Procedure: Attach in the middle of each side an equilateral triangle, with sides of length one third of the previous side. When we are done, the result is a „star of David” with 12 sides, each of lenght 1/3

The Koch snowflake

STEP 2. For each of the 12 sides of the star of David in Step 1, repeat procedure: in the middle of each side attach an equilateral triangle (with dimesions one third of the dimensions of side). The resulting shape has 48 sides, each of lenght 1/9.

The Koch snowflake

STEP 3,4, etc. Continue repeating procedure ad infinitum

The Koch snowflake

The Koch snowflake

The constructionSTART: Start with an arbitrary solid

triangle ABC

The Sierpiński triangle

STEP 1. Procedure: Remove the triangle whose vertices are the midpoints of the sides of triangle. We’ll call this triangle the middle triangle. This leaves a white triangular hole in the orginal solid triangle, and three solid triangles, each of which is a half-scale version of the orginal.

The Sierpiński triangle

Wacław Sierpiński was a Polish mathematician. He was known for outstanding contributions to set theory number theory, theory of functions and topology

The Sierpiński triangle

STEP 2. For each of the solid triangles in the previous step, repeat procedure. This leaves us with 9 solid triangles (all similar to the original ABC) and 4 triangular white holes.

The Sierpiński triangle

STEP 3, 4, etc. Continue repeating procedure on every solid triangle, ad infinitum.

The Sierpiński triangle

The Sierpiński triangle

The Cantor set is a prototype of a fractal. It is self-similar, because it is equal to two copies of itself, if each copy is shrunk by a factor of 3 and translated.

Curiosity

Fractals are the kind of shapes we see in nature.

Fractals in nature

Natural objects also tend to be ’roughly’ self-similar appearing more or less the same at different scales of measurement.

Fractals in nature

I. P.Tannenbaum, R. Arnold „Excursions in modern mathematics”

II. M.Batty, P Longley „Fractal Cities”

III. http://ecademy.agnesscott.edu/~lriddle/ifs/ksnow/ksnow.htm

IV. http://pl.wikipedia.org/wiki/Trójkąt_Sierpińskiego

V. https://www.youtube.com/watch?v=uZD-KXOp9Xs

Bibliography

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