derived from the Latin word ‘fractus’
Fractal
Fractus means
Fragmented or Broken
“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.”
Fractals
is a never-ending, self-repeating pattern — kind of like a pattern that keeps echoing itself at smaller and smaller scales.
Fractal
What are the characteristics of a Fractal?
- Self-Similarity
- Fractal Dimension
- As a Mathematical Equation, a fractal is nowhere differentiable.
Its parts have the same form or structure as the whole, at least approximately,if not exactly.
Self-Similarity
The associated degree of complexity of shape, structure, and texture of fractals are quantified in terms of
Fractal Dimension
The bigger the fractal dimension,
the more rough the structure is.
can be made by repeating a simple process involving plane transformations.
Geometric Fractals
When a process is repeated over and over, each repetition is called
Iteration
What are the types of Geometric Fractals?
- Cantor Set
- Sierpinski Triangle
- Sierpinski Carpet
- Menger Sponge
- Koch Snowflake
Who discovered Cantor set?
Henry John Stephen Smith in 1874
Who introduced Cantor Set?
German mathematician Georg Cantor in 1883.
Each segment is replaced by two segments.
Cantor Set
The Sierpinski triangle was named after the Polish mathematician
Waclaw Sierpinski
When was Sierpinski Triangle discovered and invented?
1915
This is a famous fractal first described in 1916 by Waclaw Sierpinski (1882-1969).
Sierpinski Carpet
This was used in designing antennas in cellphones as the number of scales allows for a wide range of receptions.
Sierpinski Carpet
A famous fractal solid that is the three-dimensional equivalent of the one dimensional Cantor set and the two-dimensional Sierpinski carpet.
Menger Sponge
Who first described Menger Sponge?
Karl Menger in 1926
first appeared in a paper published by the Swedish mathematician Niels Fabian Helge von Koch in 1906.
Koch Snowflakes
Branching fractals are also observed in algebraic fractals. It is a never-ending process called
bifurcation
What are the types of fractals in algebra?
- The Mandelbrot Set
- Julia Sets
One of the most intricate and beautiful images in mathematics is the
Mandelbrot set
