Fractals are a subset of Euclidean space, i.e. the two-dimensional Euclidean plane. They are infinitely complex patterns that are self-similar across different scales. To put it simply, they are the embodiment of a never-ending pattern.

The first to use the word “fractal” was Benoit Mandelbrot, the word itself being derived from the Latin “fractus”, meaning broken or fractured. He used this word because of the nature of fractals – their fractional dimensions. Unlike two dimensional polygons, or three-dimensional shapes, fractals cannot be represented by the same paradigm of dimensions. Think of a line. If you rep-tile a line (divide into smaller line segments) into three pieces which are each a third of the original length, in doing so you get three equal pieces. If you rep-tile a square in the same way (dividing into smaller squares), you get 9 pieces. The reason for this is because you are scaling down by a factor of three in every dimension. 9 can also be represented as $3^2$, and if we continue this pattern by rep-tiling a three-dimensional shape into three equal pieces that are similar to the original, the result will be $3^3$ pieces, or 27. Essentially, for any $n$ dimensional shape, scaling down by a factor of $x$ results in $x^n$ pieces. With fractals, however, although the same rule applies, the result is different. A well-known example of a fractal is the Koch Curve, which consists of four equal pieces, scaled down by a factor of 3. However, log34 does not yield an integer solution for $n$, the dimension. Therefore, the conclusion we draw from this is that a fractional dimension must exist, otherwise known as the ‘fractal dimension.’ Fractals are hence self-similar, which means they are approximately similar to certain parts of themselves, and therefore, zooming into one forever yields the same pattern, infinitely, or a very similar pattern.  This can have some very interesting consequences.

If you wanted to measure a fractal’s perimeter, you would definitely have a hard time. Imagine laying a long length of measuring tape on the perimeter of the fractal. However, since the tape is not making contact with all parts of the sides, the measuring tape must be made longer and longer to measure the smaller and smaller sections of the fractal. Given the fractal’s self-similar identity, the parts of the perimeter that you attempt to measure will repeat infinitely as its scale decreases, meaning that all fractals have an infinite perimeter. With a smooth curve, as the precision of measurement increases, the perimeter tends to a single value, yet with a fractal, as the precision of measurement increases, the perimeter simply becomes greater and greater. A real-world example of this would be the UK coastline, where smaller segments of measuring equipment only serve to increase the perimeter.

A common type of fractal is a complex number fractal. Complex numbers are represented as a + bi, with a being a real number, b being imaginary, and i being the square root of -1. Complex numbers can be represented on a “complex plane” where the horizontal axis represents and is real, and the vertical axis represents and is imaginary. An example of a fractal generated on the complex plane is the Mandelbrot set.

Let’s say that c = a +bi. By using the recursive formula, which uses the preceding term to define the next term, we can create an equation which separates points outside the Mandelbrot set and points inside the Mandelbrot set. The equation we find is:

To use this equation to find whether a complex number c is within the Mandelbrot set, we propose: For any complex number c, find the sequence which has the equation  zn+1 = zn2  + c, where z0 = 0. If the sequence, on the complex plane, has a difference from the origin which is less than or equal to 2 units, then it is a part of the Mandelbrot set. If not, then it is not a part of the Mandelbrot set, as shown in the figure above. Let’s take, for example, the complex number 0.5iso that c = 0.5i. We can determine whether it is a part of the Mandelbrot set by plugging in the complex number into our equation of zn+1 = Zn2  + c, where Z= 0. Our first (or technically ‘zeroth’) term is just 0 as the equation states. We can use this to find our next term, as this is a recursive sequenceWe square the term z0 and add 0.5which gives us 0.5i, as z0 squared is just zero. We do this for each term, using the previous one squared and adding 0.5i.

z0  = 0

z1  = z02 + 0.5= 0.5i

z2   = z12   + 0.5= -0.25 + 0.5i

z3  = z22  + 0.5= (-0.25 + 0.5i)2 = -0.1875 + 0.25i

z4  = z32  +0.5= (-0.1875 + 0.25i)2 + 0.5= -0.02734375 + 0.40625i

I have shown you the calculations for the first four terms not including the zeroth term, but if we went on, and we can still see now, that it is staying at a small value which does not go beyond 2 units from the origin on the complex graph, which shows that the complex number 0.5is part of the Mandelbrot set.

Many people who see the intricate patterns of fractals are unaware of the even more beautiful mathematics behind them, and as time goes on, perhaps they may play an important role in technology and engineering. Indeed, fractals are already being used in many fields of science, with ecologists making use of fractals to study animal behaviour! It goes without saying that fractals quite literally are capable of infinite possibilities.

Citations:

‘Generating Fractals With Complex Numbers | Mathematics for the Liberal Arts Corequisite’, https://courses.lumenlearning.com/mathforliberalartscorequisite/chapter/generating-fractals-with-complex-numbers/.

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