How To Really Understand Fractals? - An image showing a Romanesco Broccoli on the left, and a unique triangular pattern on the right. There is an arrow point left-to-right at the centre. The triangular pattern involves a big triangle and three differently sized smaller triangles inside. Some of these triangles are shaded pink, while others are shaded white.

Fractals are generally seen as geometric shapes that self-similarly contain themselves at arbitrary scales. Imagine a vector space where you keep zooming into a geometric pattern, and the geometry contains smaller versions of itself recursively.

The title image features a Romanesco broccoli, which features a self-similar form. It is an approximation of a natural fractal. You can see clearly that its smaller features resemble the entire thing (and vice versa)!

So, how do fractals work?

The trouble with fractals is that things can get really complex really fast. So, in this essay, I aim to keep things as intuitive as possible, yet enable a reasonable level of understanding of the phenomenon/concept.

For this purpose, we will be leveraging the mathematical concepts of fractions and geometry. If either of these scares you, worry not. The trick is that when you combine fractions and geometry, things get much simpler and easier to grasp (even if math is not your thing). Without any further ado, let us begin.

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How are Fractions and Geometry Related?

Let us start by considering the shaded equilateral triangle below. Say that it occupies an area of 1 units².

How To Really Understand Fractals? — An illustration showing an equilateral triangle shaded in pink.
Illustrative art created by the author

Given this premise, let us cut out a chunk of this triangle in the following manner:

How To Really Understand Fractals? — An illustration showing a smaller inverted trinagle with black borders inside the former pink triangle. The region inside the black bordered triangle seems to be white, while the pink shade remains inside the rest of the bigger triangle. The smaller triangle consequently splits the bigger triangle into four equal parts, out of which three are shaded pink and one is shaded white.
Illustrative art created by the author

As you can see, the geometry of the inverted triangle we introduced into the bigger triangle splits it into four equal areas. Since we removed the chunk at the centre, we are left with ¾ (three-fourths) of the original area of the bigger triangle. And this new area (shaded region) amounts to ¾ units².

Understanding Fractional Decay

Now, what if we multiply ¾ by ¾? Will the resulting number be smaller than ¾ or greater than ¾? This is where fractions can get tricky for a non-math person. However, there is a simpler way of thinking about this.

When you multiply the original area (1 units²) by ¾, you are asking the following question:

What is three-fourths of the original area (1 units²)?

Similarly, when you multiply ¾ by ¾, you are asking the following question:

What is three-fourths of the three-fourths of the original area (¾ units²)?

We can picture this as follows in our triangle:

How To Really Understand Fractals? — The three smaller triangles that were shaded pink from the previous image each now have a smaller inverted black triangle splitting them into 4 equal parts. Similar to the previous process, the area contained by the smaller black triangles is now shaded white.
Illustrative art created by the author

You can see that we have now taken a one-fourth chunk out of each of the smaller shaded triangles. If you count the number of shaded smaller triangles, you will get 9 of them. If you stack the smaller triangles visually, you will notice that the original triangle can be constructed using 16 of the smallest triangles.

In other words, our new area is 9/16 small triangles (or nine sixteenths of the original area). This is how you can make sense of the following calculation:

(¾)*(¾) = (3²)/(4²) = 9/16

What’s more, each time we multiply by ¾, the resulting number will get smaller and smaller. This is what is known as fractional decay. Now that we’ve seen how fractional decay works, let us move onto a slightly more advanced concept.

How to Understand Fractals using the Sierpiński triangle

We just saw what happens to our triangle if we multiply ¾ with itself twice (that is: (¾)²). What if we choose to construct (¾)³? Our triangle would look like this:

How To Really Understand Fractals? — The process of splitting the smalelr pink shaded triangles into 4 equal parts, out of which the one central part is shaded in white is repeated once more. As a result, we now have 27 shaded small triangles, while the rest of the area is white.
Illustrative art created by the author

What in the blue moon is THAT? You see, we are starting to approach a construct that is known as the Sierpiński triangle. The Sierpiński triangle is an example of a fractal. It was described first by Polish mathematician, Waclaw Sierpiński in 1915. Here is an animation of its construction in action:

How To Really Understand Fractals? — An animated image shows the same process we have been doing so far in one flow step after step. The animation includes 9 steps/iterations.
Construction of the Sierpiński triangle (image from WikiCC)

As the order of the exponent of (¾) increases (in other words, as the number of times we multiply by ¾ increases), the triangles get smaller and smaller. As a result, when the exponent approaches infinity, we approach just the outlines of the triangles.

To be precise, all the images we have seen so far are just depictions of fractals. A true fractal would enable you to keep zooming in whilst not losing ANY level of detail.

This is truly one of the hallmarks of fractals: they preserve ALL of their details at any arbitrary scale!

Wait — There’s More to Fractals

My aim with this essay was just to give you an intuitive understanding of fractals. We first covered the notion of fractional decay as a geometric construct. Then, we applied fractional decay recursively to construct the Sierpiński triangle. This is a great start.

But fractals need not be restricted to geometries or fractions. They can be applied to any structure. Stay with me here; structures may form in any medium that we might be able to sense. What this means is fractals can also emerge as sounds or taste (for example)!

When I first wrapped my head around this concept, it truly blew my mind! We as human beings, seem to have a liking to fractals; research indicates that we are most suited to fractal patterns with dimensions between 1.3 and 1.5 (reference linked at the end of the essay).

I know that I haven’t covered fractal dimensions yet, but for now, think of it as the fractal’s behaviour change at different scales.

I could keep going, but it would be too much for an entry level essay on fractals. So, I’ll wind-up by letting you in on a little secret. I often try and apply fractal patterns to my own essays.

As an example, check out my essay on how to perfectly predict impossible events. Pay attention to the sub-headings of the essay in addition to the context of the topics covered. Did you spot any pattern? If so, what did you spot?

Reference: Taylor, R.P.

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Further reading that might interest you: How To Really Understand Recursion and Why LaMDA Is Not Really Sentient?

If you would like to support me as an author, consider contributing on Patreon.

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