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Before they were called "fractals", infinitely intricate shapes like the Koch snowflake and the Sierpiński gasket were known as "monsters". At least to mathematicians who kept seeing their buddies invent ever more complicated versions of the things. But until the 1980s, fractals were bound to the realm of so-called "pure" mathematics. No one thought they existed in nature, and there were no real-world applications. This is, of course, not true. Fractals are everywhere, from the edges of clouds, to snowflakes, coastlines, even broccoli. And what can you do with fractals? It turns out, quite a lot.
Over the last few decades, engineers have been busy using fractals to solve all kinds of hard-hitting problems… from boosting your 5G signal, to generating worlds in Minecraft. [♪INTRO] Before we get into any actual applications, let's take a quick trip back to the 80s, when Benoit Mandelbrot was working at IBM. By the way, Benoit used to joke that his middle initial was B, and that the B in Benoit B. Mandelbrot stood for Benoit B. Mandelbrot.
He made a fractal out of his own name. Using some of the most powerful computers available at the time, he started playing around with some mathematical functions that would, under the right circumstances, act really weird. Basically, all he had to do was take the output of a function and then plug it back in to get a new output, then plug that back in, over and over. When he used a computer to print what one set of functions actually looked like on a graph, out popped one of the most iconic shapes in all of math: the Mandelbrot set. By studying shapes like these, Mandelbrot discovered what made them special: you can pick one spot on the edge and no matter how long you zoomed in, that bit of edge
would never smooth out into a boring straight line. You'd just see intricate shape after intricate shape. Mandelbrot then noticed the same characteristic appears in real-world shapes that don't perfectly repeat themselves, like coastlines or eddies in swirling water. It was he who named these kinds of shapes fractals, and kicked off a revolution in finding, understanding, and even creating new ones. There are, in fact, infinitely many fractals to keep mathematicians busy. But we don't have an infinite amount of time, so let's jump into our first application.
Many electronic devices use antennas to send and receive signals. Whether they stick out like rabbit ears on top of a vintage television, or get buried inside your smartphone, they're usually a piece of metal whose electrons wiggle in response to being hit by a radio wave. Those moving electrons create an electric current that your device can convert into text, images, and sounds. And your device can, in turn, drive a current through the antenna to send radio waves somewhere else. It turns out, if you shape an antenna like a fractal, it can act like multiple different antennas in one awesome looking design.
Those classic antennas that were basically just a straight line… or maybe a handful of straight lines sticking out of one another… worked just fine for picking up a limited set of specific radio wave frequencies. Think one metal stick for each frequency. But in the 21st century, we often want our devices to keep track of a bunch of different frequencies. Like if you want to pair your phone with your wireless earbuds, call your mom, and book her tickets for a BTS show in Vegas all at the same time.
Without fractals, you'd need a whole bunch of different antennas on the same device. And on devices we want to keep light… say, microchip ID tags or drones… all that extra weight would be a problem. But some fractals, like the Hilbert curve, snake around with a whole bunch of length in a fairly tiny space. This makes them really efficient at packing an antenna into a small space, a property mathematicians call a space filling curve. In fact, even without creating a perfect fractal, engineers are currently using patterns like those in the Hilbert curve to squeeze more antenna length into smaller electronic spaces. But the real magic comes from a fractal's repeating patterns.
Thanks to a physics phenomenon called resonance, different parts of the antenna with the same pattern, just larger or smaller, can wiggle in sync to create a clearer signal. You couldn't get that by just scrunching up an antenna randomly into a small space! Plus, when you've got the same patterns repeating on different scales, your antenna can resonate with different radio wave frequencies. For example, in one paper from 2022, a research team describes an antenna they made that's just a few centimeters wide. One side features a kind of Sierpinski gasket, and the other a Hilbert Curve.
The team found their antenna did a good job receiving signals over three different bands of frequencies without any unwanted interference, which they say makes it great for handling some kinds of 5G signal. In the future, this might be used to improve reception on our phones even more. Which, honestly, there could use here in the mountains of Montana. Fractals don't just help us communicate, of course. Engineers have also applied their weird patterns to help keep tiny devices cool. Stuff like fuel cells, or microchips that are used to mix tiny amounts of chemicals in laboratory settings.
One traditional way of keeping technology from overheating, be it a nuclear reactor or a refrigerator, is to add plumbing that pushes a cooler-than-everything-else liquid into a place it can absorb unwanted heat, and then carries that heat away. But this plumbing trick gets, well, tricky when we shrink things down. The viscosity of liquids, or how easily they flow, starts to matter a whole lot more. The further the liquid flows down those tiny tubes, the more and more pressure you need to make it move, and the harder it is to carry away heat properly.
Unless, of course, you make the whole thing a fractal! Engineers have created heat exchangers with a whole bunch of branches that splinter off into tinier parts, sort of like the branches on a tree, or even a spider's web. At each branching-off point, the liquid that's getting pushed through the tubes experiences what scientists call pressure recovery. It encounters a greater surface area and a smooth taper that decreases the amount of pressure needed to push it through each branch.
Plus at every one of those branches, all the stuff flowing in the tubes gets mixed up. This helps bring the hotter liquid traveling along the center of the tubes out towards the walls, where it can lose its heat more easily. It's a lot like how stirring some piping hot tea helps it cool down. Research suggests that with a fractal layout, these kinds of heat exchangers can be as much as 25% more efficient. Which might not sound like a lot, but in tiny devices that tend to malfunction when they get too hot, every bit helps. Up next, a quick ad, brought to you by the antenna in your device that may or may not be a fractal.
This SciShow video is supported by JMP: the statistical analysis software trusted by medical writers. JMP is all about making it easier to tell the story of your findings. And if you're a medical writer, your job is basically putting together patient narratives, AKA telling the story of a patient's data. That can be a painstaking, time-consuming process with significant consequences for inaccuracies or delays. Which is why medical writers use JMP Clinical. With JMP Clinical, they can automate the whole enchilada. JMP specializes in taking a big data set and bringing the data together in ways that are more digestible.
Just look at all of their interactive dashboards and web visualizations! To reap the benefits of visual statistics for yourself and get a 30-day free trial, visit jmp.com/scishow. How did those researchers get the idea to cool stuff down using fractal plumbing? By noticing the fractal plumbing inside of you. Many biological circulatory systems are fractal-like in nature, from blood vessels, to sweat ducts, to the veins in a leaf. And biomedical researchers have used this to develop diagnostic tech
that can tell you if you have a greater risk of developing a disease. But to explain how this technology works, we have to talk about a weird concept called fractal dimension. Remember, the key feature of fractals are their intricate patterns on lots of different scales. Mathematicians want to find a way to capture that intricacy and represent it with a single value, a single "dimension", which they can then use to compare fractals against one another. One such way is called the box-counting dimension. It starts with something straightforward: drawing boxes.
Imagine taking a sheet of paper and drawing a square grid over the entire thing… unless you've still got some graph paper from math class lying around. Draw a 2D shape. A circle, a hexagon, whatever. Then, count up how many of the grid's boxes overlap with your shape's boundary. For any overachievers out there, you can also imagine doing the 3D version of this, with a 3D grid and drawing a sphere or the ever nerdy d20. Now, you may have realized that for the exact same shape, you'll get a different number of overlapping boxes based on how small the boxes are. But whatever the size of grid you're using,
for simple shapes like lines, hexagons, and d20s, the total number of overlaps will always scale with the dimension each shape is in: 1, 2 or 3. That doesn't happen with fractals. A Sierpinski gasket, for example, comes out as 1.585. Like it's halfway between being a one-dimensional shape and a two-dimensional shape. The coastlines of various countries, for another example, have a fractal dimension of about 1.1 to 1.3. Researchers in the real world don't have to draw their own grids and shapes, of course.
They can program computers to do it for them, count up the digital boxes and calculate a fractal's dimension. Which brings us back to blood vessels. In a 2009 study, published in the journal Diabetes Care, over 700 young people with Type 1 diabetes had photographs taken of their retinas, and the network of arteries and veins in the back of their eyes. After cross referencing each participant's unique fractal dimension with their own health data, the team found that every 0.01 increase in dimension carried a 40% increase in risk of developing retinopathy, a condition that causes blurry vision, or even blindness if left untreated. Then, in a 2022 study from Frontiers of Digital Health, a different group of researchers used the same idea
with more than 6,000 patients with Type 2 diabetes. The results showed a higher fractal dimension in retinal arteries came with an increased risk of dementia, but a higher dimension for veins decreased risk of Alzheimer's. Time will tell how this research may turn into yet another test you take at the optometrist. But in the meantime, I'm gonna ask my doctor to print out my retina on some graph paper. I want to try something… For our final entry, let's cover something a bit more fun. Or agonizing, if the hiss of a Creeper gives you a rush of anxiety.
Fractals are how games like Minecraft can present players with endless systems of mountains and caves to explore, without also being like 50 billion terabytes in size. It all comes back to those funky math functions Mandelbrot was studying in the 80s. The ones that repeatedly act on their own output. With just a tiny set of rules programmed into a computer, you can create amazing looking fractals. Like with a mere four equations, you can generate a shape known as the Barnsley fern. Programmers can use tricks like this to easily generate all kinds of shapes with super fine details. For example, to create its cave systems,
Minecraft uses a technique called Perlin noise. By adding tiny, random changes to the numbers that are plugged back into the functions at each step, Perlin noise can make detailed structures on finer and finer scales. And what's extra nice is you don't need a bunch of complex structures stored in computer memory, waiting for a player to stumble across them. You can generate landscapes on the fly when a player gets close enough! If the player is far away, you use just a few iterations to get something vaguely mountain-looking, then apply more iterations to add further detail as the player approaches.
Which is way kinder to your computer processor. The same idea is used to give computer generated objects more realistic textures, too. That is, admittedly, less of a thing you have to worry about in Minecraft. But it's also not exclusive to infinitely sprawling video game worlds, or video game worlds at all. Which I mention because of the people who are currently shouting at the screen, "Why didn't you talk about CGI?!" Like I said, the number of fractals is infinite, but our time here today isn't. [♪OUTRO]
