Fractals were coined by French mathematician Benoit Mandelbrot in 1975, coming from the Latin word fractus, meaning broken or fractured. Of course, fractals existed long before humans ever designed or defined them - in nature. The examples previously shared: stars, coastlines, and lightning… are all good examples to look out for.

If these are the only places you think they exist, think again. They show up everywhere from plants and trees to the lungs you use to breathe. They exist in the rocks, in the water, in the skies, the mountains... everywhere, and at all levels. If you stare at fractals enough, you may start to notice their levels of self-similarity (how smaller pieces of the whole look like the whole). This is actually one of the easiest ways to invent and create fractals and is a very common theme for all the fractals that show up in nature.

Notice how each creek in a river looks like the whole, how every time you peer deeper into the stars there only seems to shine at you more, how the branches of a tree only keep branching into leaves that do too... How the canyons only seem to connect one unto another until you can't tell from where they came, snowflakes seem to branch into endless directions and the clouds only have more and more texture the closer you look… (Gunther, 2024).

This pattern of growing the same way at each level generates fractals and happens all over the place in nature for a variety of reasons. Whether it's about efficiently covering the most area possible, the way erosion happens, the way things flow into one another due to the slope of the land, expanding to have the most surface area for absorption, or even coalescing to have the minimal surface area. There are a lot of reasons why; challenge yourself - go outside and look around for things that are fractal-like, and try to understand why.

        - Often in plants and animals, fractal growth is efficient in that it takes little genetic information to code or it leads to a pattern of beneficial growth (funneling water, evening sunlight…), (Weibel, 1991).

Generally speaking, there are many types of fractals. The first and most ubiquitous types that inspired Mandelbrot to start using computers for fractals are the ones that nature provides , such as coastlines or lightning (Gleick, 1987). The coastline paradox was explored by British Mathematician Lewis Richardson when he wanted to find a definitive answer on the length of a country's coastline (to measure if the length of a coastline affected the likelihood of war in a country), (Vulpiani, 2014). The problem that arose was that when Richardson used more precise measurements or lengths to remeasure the coastline, it only kept on growing, seemingly without bounds. Richardson further noticed that for more wiggly coastlines, such as Norway with its enormous count of fjords, this rate of increase in coastline was faster, (Hey, 2019).

Mandelbrot got curious about these natural shapes that seemed to be incredibly complex, so much so that simple geometry couldn’t define them, and so he formalized the idea of fractals: shapes with some degree of self-similarity and roughness. Whether it was the count of stars, the perimeter of a river system, the detail of a cloud, or the shape of Romanesco broccoli, Mandelbrot wanted to model these shapes with something that could capture their detail.

Other types of fractals are those that are made from idealized forms of geometry, such as branching segments to form a tree or fern, repeating triangles or squares to form a mesh or increasing the number of side lengths to form the endless detail on the edge of a snowflake. These fractals work by scaling and shifting copies of a fractal back onto itself, and can very quickly explode in detail, (Patrzalek, n.d.).

The last main type of fractals are those made by iterations of complex numbers and are by and large sketched by computers to allow our minds to enjoy and explore them (Patrzalek, n.d.). The most famous ever made is the Mandelbrot set which follows a relatively simple recursion scheme:

$$Z_{new} = Z_{old}^2 + Z_0$$ where, $Z_{old}$ is the complex number's current value, and $Z_0$ is the orignal value of the complex number

Variations on this scheme, such as Julia Set, or higher-dimensional Mandelbrot set variants (Barrallo, 2010), are regularly made on computers because of their relative simplicity and beauty (but extreme computational load). The detail in these fractals can become far more intricate than self-similarity, and to many, these fractals are described as divinity.

Below are some examples of these fractals (along with the lightning at the top of this page)

Generates Julia Set Fractals and the Mandelbrot Set

Recall how fractals have nearly infinite complexity. The rate at which this complexity grows when a fractal is scaled (or zoomed into) is what determines its fractal dimension . As Sanderson (2017) points out, the dimension of a shape has to do with the power on its scalar as you scale a side length. For example, if you double a length, it merely doubles, but as soon as you double the side length of a square, there are now 2^2 or 4 times as many squares (or four times the area). Similarly, if you double the side length of a cube there are now 2^3 or 8 cubes (hence 8 times the volume). Notice that the exponent of the scalar (2), defines the dimension (1,2 or 3).

We can extend this into the realm of a fractal by asking: if we scale the fractal by some constant (that makes sense for the fractal), how much does the fractal perimeter or area scale by? Notice how if we double the size of the Serpenski Triangle, it becomes 3x as large in perimeter. Thus, its defining feature perimeter, grew by 3 instead of 2. What power of 2 is 3? Well, we can use logarithms to figure that out, giving us that $log_2(3)$ = 1.585 (Sanderson, 2017).

Let’s also look into the Serpenski Carpet to clarify this idea some more. Since the Serpenski Carpet has three equal parts on its base, imagine tripling its base length so that one 1/9 of the figure becomes the whole thing. When we triple a shape, we typically expect its perimeter to triple, but instead, the perimeter becomes eight times larger as that one Carpet now becomes 8 of them. Thus, the fractal dimension of the Serpenski Carpet is $log_3(8)$ ≈ 1.892789.

This notion of dimension can be approximated by asking how many boxes a curve intercepts as it gets larger or the boxes get smaller (since this process approximates length). Thus, when you double the relative size of the UK (to the boxes)., how many times longer is the coastline? Sanderson (2017) found this power on the scalar for the U.K. to be 1.21. Mathematician Lewis Fry Richardson was looking into these measures only a century ago (Vulpiani, 2014). Stay curious, and try out this method of analyzing fractal dimensions of other fractals and see what you find!

This world has all manners of complexity, big and small, and allowing ourselves to play with the ideas of infinity to describe it is a wonder that extends us beyond the limitations of those who came before us. Isaac Newton is often accredited to saying“If I have seen further, it is by standing on the shoulders of giants.” Imagine what great wonders we will see if we allow ourselves to climb those heights and look upon the great wonders this universe provides.

Lastly, here are some places where fractals have been applied or used in the world around us. Of course, Mandelbrot (1982) thought fractals are a good way to describe many natural phenomena such as clouds, coasts, and lightning. In my experience, these are the most wonderful and curious fractals around, even among the limitless and perfect fractals of mathematical creation.

In many places of mathematics, paradoxes linger, questioning if mathematics is a reasonable model of our world or if it accepts too much to be true. Fractals extend what we accept in mathematics and turn it into an art form. Many call fractals a form of poetry, that reflects at us all that is beautiful in our creation. Fractals stand as edge cases with weird dimensions, infinite perimeters, and all levels of self-similarity. They exist at the boundary of what Mathematics can wrap its mind around and entreat us with the curious wonders of unfathomable complexity. Fractals are intrinsically broken - it’s in their name- and so they give us new ways to make art out of mathematics.

Furthermore, fractals are commonplace in many applications such as modeling biological systems where incredible amounts of surface area, pathways, or complexity from little information are needed. This goes as far as asking if fractal geometry is perhaps a basic design principle of biology, (Weibel, 1991). We see these patterns play out all over the place in ferns, the branching of plants and trees, the structures of fungus, lichen, moss, and much much more. Fractals may even be within and the structure of our DNA, (Patrzalek, n.d.).

Fractals have similarly been used to create compact radio receivers, to model mountains and planets in movies such as Star Trek and Star Wars, and to model the heavens above us, (Patrzalek, n.d.).

Photos credit to Edyta Patrzalek (link in the references)

From the humble beginnings of fringe mathematics to the forefront of magazines and the wonder of mathematics in the public eye, fractals have been a curiosity that the machinery of mathematics has only recently been able to uncover. Today, we have so many resources to document and build fractals, which has enabled and boosted the creation, development, and application of fractals. They serve many important roles in understanding the world around us and are wonderfully complex things to get caught up in.

Hopefully, you’ve enjoyed a look through these fractals and find them hiding all around you. Perhaps nothing is intrinsic in mathematics, not counting, not real numbers, nor complex ones. If you open your mind and allow a little bit of the bizarre to model this world, you’ll find you have another excellent tool to capture this peculiar world with your imagination.

Barrallo, J. (2010). Expanding the Mandelbrot Set into Higher Dimensions. The
      Bridges Archive. Retrieved November 25, 2024, from
      https://archive.bridgesmathart.org/2010/bridges2010-247.pdf

C A O. (n.d.). Fractals in Nature. Retrieved November 25, 2025, from
      https://fractalsaco.weebly.com/fractals-in-nature.html

Fox, K., Johnson, A., Gran, R., & Garner, R. (2023, September 24). NASA's First
      Asteroid Sample Has Landed, Now Secure in Clean Room. NASA. Retrieved November 25,
      2024, from https://www.nasa.gov/news-release/nasas-first-asteroid-sample-has-landed-now-secure-in-clean-room/

Fractal Foundation. (n.d.). What are Fractals? Fractal Foundation. Retrieved
      November 25, 2024, from https://fractalfoundation.org/resources/what-are-fractals/

Gleick, J. (2011). Chaos: Making a New Science. United States: Open Road Media.

Gunther, S. (2024, May 30). 9 Amazing Fractals Found in Nature. Treehugger. Retrieved November 25, 2024, from
      https://www.treehugger.com/amazing-fractals-found-in-nature-4868776#:~:text=There%20are%20many%20examples%20of,evolutionary%20adaptations%20for%20efficient%20growth

Hey, J. (2019, March 2). The Coastline Paradox. sketchplanations. Retrieved
      November 25, 2024, from https://sketchplanations.com/the-coastline-paradox

Livio, M. (2011, August 1). Why Math Works. Scientific American. Retrieved
     November 25, 2024, from https://www.scientificamerican.com/article/why-math-works/

Mandelbrot, B. B. (1983). The fractal geometry of nature. Henry Holt and Company.

Patrzalek, E. (n.d.). Fractals: Useful Beauty. Introduction to Fractal Geometry.
      Retrieved November 25, 2024, from https://www.fractal.org/Bewustzijns-Besturings-Model/Fractals-Useful-Beauty.htm

Sanderson, G. (2017, January 27). Fractals are typically not self-similar.
      YouTube. Retrieved November 25, 2024, from https://www.youtube.com/watch?v=gB9n2gHsHN4&ab_channel=3Blue1Brown

Vulpiani, A. (2014). Lewis Fry Richardson: Scientist, visionary and Pacifist.
      Lettera Matematica, 2(3), 121–128. https://doi.org/10.1007/s40329-014-0063-z

Weibel, E. R. (1991, December). Fractal geometry: a design principle for living
      organisms. American Journal of Physiology: Lung Cellular and Molecular Physiology, 261(6), L361-L369.
      https://doi.org/10.1152/ajplung.1991.261.6.L361