Fractals — A Visual Treat

A fractal can be called as a never-ending pattern that repeats itself at different scales (also called “Self-Similarity”). Simply put, they are created by repeating the same pattern again and again

In the above picture, if you keep on zooming in, you will keep getting back to the initial image. That is what fractals are

If you thought Fractals exist only in the world of mathematics, you are wrong. Fractals exist in the real world also — through nature. Some examples are given below

Koch Snowflake

The Koch snowflake was described by Holge van Koch in 1904. It is very simple to create the Koch Snowflake. Start with an equilateral triangle, divide each side into 3. Remove the middle portion and construct another equilateral triangle in that place. Keep repeating this process and you will end up with what is known as a Koch Snowflake

The Sierpinski Triangle

The Sierpinski Triangle is another famous example of a fractal. In the case of Koch snowflake, we took an equilateral triangle and increased its area. In the case of the Sierpinski Triangle, we take an equilateral triangle and cut out smaller triangles from it

If you notice, the bigger triangle is made up of 3 identical copies of itself. Each of these copies again consists of even smaller copies. In this way, you can continously keep zooming into the Sierpinski Triangle and the samea pattern would keep repeating

The Mandelbrot Set

One of the most famous examples of fractals is the Mandelbrot Set, named rightly after mathematician Benoit Mandelbrot, one of the pioneers in fractal geometry.

The Mandelbrot set is obtained from the equation

where z0 = C, C is the set of points in the complex plane which do not diverge to infinity. The above equation is a recurrence relation (the next value depends on the previous value). Substituting for k=2 (i.e squaring) yields the above picture

Though the math behind the Mandelbrot set is very complex(pun intended), it is very famous outside the realm of mathematics for its sheer beauty and visual appeal

I mean like, who cannot fall in love with this

Other variants of the Mandelbrot set also produce beautiful patterns

Instead of raising to power of 2, trying to raise to the power of 3 & 4 also produces stunning patterns

(Top) Patterns obtained for k=2,3,4. (Bottom) Patterns obtained by substituting for z with its complex conjugate

Fractal Dimensions

Dimension can be defined as a measure of any object(like length, height). A line segment has a dimension of 1. A square has a dimension of 2. Another way to look at dimension is, if we increase the length of one side, by what factor does the area change?

If you double a line segment, its length also doubles since it has a dimension of 1 (2¹ = 2)

If you double the sides of a square, its area increases by a factor of 4 since it has a dimension of 2 (2² = 4)

Doubling the length of a square increases its area by 4

Similarly, doubling the length of a cube increases its area by 8 times (2³ = 8)

But if you take the Sierpinski triangle, if you double the length, the area increases by a factor of 3. This implies that 2^d = 3 => d = 1.585.

Doubling the length of a square increases its area by 4

Is it possible that something could have a dimension that is not an integer? Yes. This is one of the weird and interesting properties of fractals. Along with ‘self-similarity’, fractals have another property: fractional dimension

There are more examples of things that are not exactly fractals, but similar to them. They include lightning bolts in the sky, blood vessels in the retina etc. Those interested can read up more about fractals in this link: Fractals — Mathigon

Those interested in knowing more about the Mandelbrot set and how it is graphed can refer to this video from Numberphile: What’s so special about the Mandelbrot Set?

Thanks for reading!!

References

1. https://mathigon.org/course/fractals/introduction
2. https://mathworld.wolfram.com/MandelbrotSet.html

Originally published at http://infinitesimallysmallcom.wordpress.com on March 21, 2021.