Understanding Fractals

Podcast Transcript

Some of the most beautiful things in the world of mathematics are fractals.

Fractals are unique geometric objects that are both easy to comprehend and have complicated mathematics.

Moreover, fractals are not just a theoretical mathematical construction. Fractals can be found everywhere in nature, including trees, beaches, and even your own body.

Learn more about fractals, what they are, and how they work on this episode of Everything Everywhere Daily.

Even if you don’t know it, you are familiar with fractals. You don’t need an advanced degree in mathematics to know and appreciate what fractals are.

Fractals are beautiful, important, and can help explain how the natural world works.

So, what exactly is a fractal?

The textbook definition of a fractal is a geometric object or set that can be subdivided into parts, each of which is a reduced-scale copy of the whole. This property is known as self-similarity. Fractals often have non-integer dimensions, meaning they exist somewhere between traditional one-dimensional, two-dimensional, or three-dimensional spaces.

The first part of that definition is relatively easy to understand; the second part requires more explanation.

The idea of fractals goes back well before the word was ever coined.

The 17th century provided the earliest roots of fractal geometry, although the concept of fractals as we know them today had not yet been developed. During this time, mathematicians were deeply engaged in exploring the foundations of calculus and geometry, which would later serve as a basis for understanding irregular and infinitely detailed structures.

One of the key figures of this era was Gottfried Wilhelm Leibniz, the co-creator of calculus. Leibniz was particularly intrigued by recursive patterns and the idea of infinite division. While studying curves and mathematical structures, he noticed that some shapes appeared to repeat their forms on smaller and smaller scales.

Though he didn’t formalize these observations into a theory of fractals, his work on infinitesimal calculus and recursive processes hinted at self-similarity principles fundamental to fractal geometry.

The 19th century saw a more explicit shift toward understanding irregular and non-differentiable shapes. In 1872, Karl Weierstrass created a function that was continuous everywhere but differentiable nowhere. This function appeared to oscillate wildly at every point, presenting a ” roughness ” level that traditional calculus could not capture.

Let me explain this briefly for those who aren’t versed in calculus. If a function is continuous, if you look at a graph of the function, it simply means that there are no gaps.

If you remember back to my episode on calculus, the derivative of a function can be thought of as the slope of a tangent line at any particular point on a function. However, some points don’t have a single tangent line. Imagine a curve that looks like an upside-down V. At the tip of the upside-down V, an infinite number of tangent lines can be drawn, so at that point, it is non-differentiable.

Weierstrass’s function was both continuous and non-differentiable at every point, which was thought to be impossible.

Shortly thereafter, in 1874, Georg Cantor introduced a set that exemplified self-similarity, a hallmark of fractals. The Cantor set was constructed by repeatedly removing the middle third of a line segment, leaving behind an infinite number of points arranged in a highly irregular pattern.

The late 19th century also saw contributions from mathematicians like Felix Klein and Henri Poincaré. Their work hinted at the chaotic and self-referential behaviors that would later be associated with fractal geometry. Poincaré, in particular, laid the foundations for chaos theory, which shares deep connections with fractals. Chaos theory may be the subject of a future episode.

One of the biggest breakthroughs in the understanding of fractals came in 1904. Helge von Koch introduced the Koch snowflake, one of the first explicit geometric constructions of a fractal.

A Koch Snowflake is a geometric shape that is mind-bending, yet not that difficult to understand or even make.

Here’s a step-by-step guide to creating a Koch Snowflake:

Draw an equilateral triangle where all sides are equal and all angles are 60°. This is called the initiator.

On each side of the triangle, mark two points that divide the side into three equal segments.

On each side, replace the middle segment with an outward equilateral triangle such that the new peak forms a “bump.”

Take each side of the new shape (now with “bumps”) and divide it into three equal parts.

Replace the middle third with another outward equilateral triangle.

Repeat the process as many times as desired or mathematically an infinite number of times. With each iteration, the shape becomes more detailed, resembling a snowflake.

Koch demonstrated that an infinitely long boundary could enclose a finite area. This paradoxical result challenged traditional notions of dimension and perimeter, pushing mathematicians to rethink the boundaries of geometry.

Koch snowflake opened the doors to a host of other extremely exotic shapes.

The Sierpi?ski Triangle, also known as the Sierpi?ski Sieve, was created in 1915 by the Polish mathematician Wac?aw Sierpi?ski. It is a fractal and geometric figure that exhibits self-similarity, meaning it looks the same at different scales.

It is constructed by starting with an equilateral triangle and recursively removing smaller triangles from its structure. To create it, the middle triangle formed by connecting the midpoints of each side is removed from the initial triangle, leaving three smaller equilateral triangles. This process is repeated infinitely for each of the remaining triangles, producing an increasingly intricate pattern.

What you wind up with is a shape that has zero area.

A three-dimensional version of this would be the Menger sponge. The Menger sponge was developed by mathematician Karl Menger in 1926,

It starts as a cube, and in each iteration, the cube’s center and each face’s centers are removed, leaving 20 smaller cubes. This process is repeated infinitely for all remaining cubes, creating a structure with infinite surface area but zero volume.

The most famous fractal shape is probably the Mandelbrot Set.

It was created by Benoît Mandelbrot in 1980, using computers to visualize patterns in a type of math called complex numbers, on which I’ve done a previous episode.

To create the Mandelbrot Set, you start with a simple formula and repeatedly apply the formula to see what happens.

Mandelbrot used computers to perform these calculations for millions of points and plotted the results on a graph, coloring the points inside the set black and using vibrant colors to represent points outside the set, depending on how quickly they escaped to infinity.

This process revealed the Mandelbrot Set’s intricate, infinitely detailed boundary, full of spirals, swirls, and self-similar shapes. Its beauty and complexity brought fractals to the attention of the public.

What all of these odd shapes have in common is that they involve simple rules that are iterated over and over and over. Do it an infinite number of times and you can end up with bizarre and counterintuitive results.

No matter how much you zoom in, everything looks the same, or as it is described, it is self-similar.

The word fractal comes from the fact that these shapes technically have a fractional dimension. The Koch Snowflake, for example, lies between a one-dimensional and a two-dimensional shape.

The technical dimension is approximately 1.26186…

You might now be wondering, is this all just mental games for mathematicians? Is there any point to any of this in the real world?

The answer is…..yes.

Fractals are abundant in nature, appearing in many structures and patterns that exhibit self-similarity and complexity across different scales.

A classic example is the branching pattern of trees, where the trunk splits into branches, and each branch further divides into smaller branches, resembling the original structure. Similarly, the veins in leaves show a fractal pattern, with larger veins branching into smaller ones to distribute nutrients efficiently.

The human body contains fractal patterns, such as in blood vessels and lungs. The branching of blood vessels ensures efficient circulation, and the fractal structure of the lungs maximizes surface area for oxygen exchange. Additionally, fractals can be seen in natural growth processes, like the spiral patterns of shells and the arrangement of seeds in a sunflower, which follow fractal-based mathematical principles.

Clouds, rivers, lightning bolts, and snowflakes also display fractal properties, showing that nature uses fractal geometry to create efficient, intricate, and beautiful structures. These patterns arise naturally from processes like growth, erosion, and flow dynamics, making fractals an essential tool for understanding the natural world’s complexity.

Why are these mathematical oddities found in nature? The reason is very straightforward. It requires taking a simple rule and replicating it to create complex objects. It is much simpler than trying to make complex rules to make something complex.

There is one other very real-world example of fractals. It is known as the coastline paradox.

The coastline paradox is a phenomenon that demonstrates how the length of a coastline is not a well-defined measurement.

This paradox arises because the measured length of a coastline depends on the scale of the measuring tool or unit used. When measuring with a large unit, such as 100 kilometers, smaller features like bays and inlets are ignored, resulting in a shorter measurement.

However, as the measuring unit becomes smaller, finer details of the coastline, such as smaller curves and irregularities, are included, causing the measured length to increase. If the measuring unit continues to decrease in size, capturing even tinier features like rocks and crevices, the total length appears to grow infinitely.

You could continue this almost infinitely, or at least to a point where it becomes impossible to measure using physical tools. You could measure around grains of sand, and beyond that, the surface of sand grains, and then molecules and atoms.

This happens because coastlines are not smooth, straight lines but irregular, fractal-like shapes that exhibit self-similarity. Each segment of a coastline contains smaller and smaller features that resemble the whole, creating an infinitely complex boundary.

In mathematical terms, the fractal dimension of a coastline lies between one and two, meaning it occupies a space more complex than a straight line but less than a two-dimensional area.

The coastline paradox originated from the work of mathematician Lewis Fry Richardson in the 1950s during his studies on the measurement of geographical boundaries and borders. Richardson’s research initially focused on political geography, particularly on whether the length of national borders or coastlines influenced the likelihood of conflict between neighboring countries.

His research into the paradox stemmed from Portugal reporting their border with Spain as being 987 kilometers or 613 miles. Yet, Spain reported their border with Portugal as being 1,214 kilometers or 754 miles.

They had very different measurements for the exact same thing, which by definition had to be exactly the same.

This isn’t just a theoretical problem. For example, the Congressional Research Institute, the CIA, and the National Oceanic Atmospheric Administration came up with wildly different answers for the length of the coastline of the United States. The Congressional Research Institute put it at 29,093 miles. The CIA put it at 19,924 miles, and NOAA put it at 95,471 miles.

You can find fractal-like structures all over if you just look for them. Certain types of sea shells, Romanesco broccoli, snowflakes, pine cones, rivers, foam, and many other things all exhibit fractal-like behavior.

Fractals are much more than abstract mathematical constructs; they are vital tools for understanding and modeling the complexity of the real world. From their fascinating self-similarity to their diverse applications, fractals serve as a bridge between the realms of order and chaos, simplicity and complexity.