Fractal Geometry
Sreesha
“……Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line,…..” Benoit Mandelbrot
Then how can we define all these? We can’t explain these using Euclidean geometry.
Fractal geometry is a new branch of mathematics which makes math more interesting than the usual scenario of math as a body of complicated formulas and concepts. It mixes mathematics with arts. Fractal geometry is the best existing mathematical description of many natural forms.
A fractal is a mathematical set that exhibits a repeating pattern displayed at every scale. It is also known by the names “expanding symmetry” or “evolving symmetry”. An abstract definition for fractals is that it is a set for which Hausdorff – Besicovitch dimension strictly exceeds the topological dimensions. Generally a fractal is defined as a rough or fragmented geometric shape that can be subdivided in parts, each of which is atleast approximately a reduced size copy of the whole.
Important properties of fractals are self similarity and non-integer dimension.
- Self similarity
On magnifying them many times and after every step we can see the same shape, which is the characteristic of that particular fractal. Even if we zoom in, no new details appear, nothing changes and the same pattern repeats over and over.
For example,
If we look carefully at the branches of a tree, we can notice that every little branch-part of the bigger one, has the same shape as the whole.
- Non-integer dimension
In Classical or Euclidean geometry we have points in zero dimension, lines in one dimension, squares in two dimensions, cubes in three dimensions. Likewise we have fractals in fractional dimension. Many natural phenomena are better described using a dimension between two whole numbers.
For example,
A straight line has a dimension of one, while a fractal curve will have a dimension between one and two.
- Fractals are formed by the method of iterations.
- Fractals cannot be measured in traditional ways. Fractals as mathematical equations are ‘nowhere differentiable’.
Examples of Fractals
The well known cantor set is an example for geometric fractals. Another example is the Von koch curve. Mandelbrot set and Julia set are examples for mathematical fractals.
Natural Fractals
The world is full of chaotic patterns that are both surprising and complex. Fractal patterns are found throughout the natural world. The world seems to be in accordance with the laws that govern the creation of fractals. Natural fractals such as trees and ferns don’t keep repeating their patterns forever whereas the mathematical fractals such as Mandelbrot set repeats infinitely.
Application of Fractals
Fractals have applications in telecommunications, astronomy, art, and reproducing realistic images.
Fractals play an important role in crime investigation in the detection of shoe prints. If we get evidence of a part of shoe print from any crime scene, the entire shoe print pattern can be generated by fractals.
Fractals improved our precision in describing and classifying random or organic objects. Fractals have so much more depth than being just a piece of art. The discovery of fractals has allowed us to decrease the size of our cell phones every year. Without the discovery of fractals our technology, health etc would not be where it is today.
Read more: The Steinmetz solid
Check your knowledge
Evolving symmetry and expanding symmetry
Geometric Fractals – Cantor set, Von Koch Curve
Mathematical Fractals – Mandelbrot set, Julia set
Benoit Mandelbrot
- Self similarity
- Non-integer dimension
- Fractals are formed by the method of iterations.
- Fractals as mathematical equations are ‘nowhere differentiable’.
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