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Doesn’t fractured glass look pretty to you? Fine lines extending outward with cracks upon cracks? These repeating patterns, called fractals, are not only beautiful and incredibly useful, but they are literally everywhere! Lightning, mountains, waves, and even your body. I’ve analyzed them for years, and I’ve learned that as you get to understand what fractals are and how they work, you start to gleefully point them out wherever you go: “oooh look - there’s another one!”

Audience

My audience is people who are generally unfamiliar with fractals and programming. They are between the ages of 12 and 30 and want something to satisfy their curiosity. These individuals are meandering the internet, looking for interesting ideas or intriguing images. Those with short attention spans will have access to various media, and those who want to explore in depth will have access to detailed explanations. We provide both. Users will be on their PC to maximize image quality, though mobile users will be accommodated to the best of our ability.

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Heading Font: Play

This site should seem high-tech. I want it to feel powerful and captivating. Sans-serif fonts help accomplish this.

Paragraph Font: Josefin Sans

I want both fonts to be sans-serif so they're easily readable. Josefin Sans is highly readable, and still reflects the high-tech, captivating, powerful feel I want for this site.

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The best Whitewater Rafting in Colorado, White Water Rafting Company offers rafting on the Colorado and Roaring Fork Rivers in Glenwood Springs. Since 1974, we have been family owned and operated, rafting the Shoshone section of Glenwood Canyon and beyond.

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Trips vary from mild and great for families, to trips exclusively for physically fit and experienced rafters. No matter what type of river adventures you are seeking, White Water Rafting Company can make it happen for you.

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Fractals in Nature

Fractals in Computing

Build Your Own Dragon Curve

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Home page

Tl;dr: Fractals are things that contain near-duplicates of themselves. They’re breathtaking, carry some intriguing properties, and are literally everywhere.

Think of a cloud. What’s the general shape of it? Whether you imagine it as lumpy or wispy, it probably has a general shape, with layers of added details. Start with a big lump, add lumps to that lump, add lumps to those lumps, and smaller lumps on those lumps until you get a cloud-like texture. Well, that’s a fractal. Each of those smaller lumps is pretty much the same as the bigger lump, and they add together to make a complete cloud!

Another quick example: a tree. Start with one branch (the trunk) and add two branches to it. Now add two branches to the ends of those, and two branches to each of those, and on and on. Suddenly you've got yourself a tree!

Self-similarity. The idea that a fractal is similar to its parts is partly why fractals are so mind-blowing. With a high-quality fractal you could zoom in on it for days and it would always look pretty much the same! And yet each piece is a unique and distinct part of the whole. But why does nature use them? And what possible use could they be to us?

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Fractals in Nature

Nature seems to have an obsession with fractals. Even more so than I do! Here’s a list of some common fractals in nature:

Trees (branches upon branches)
Waves (crests upon crests)
Mountains (peaks upon peaks)
Clouds (billows upon billows)
Ferns (leaves upon leaves)
Snail Shells (swirls upon swirls)

Pretty Cool, right? As you drink in the details of each of nature’s masterpieces, you’ll begin to notice them everywhere you go. Want to hear something trippy? You’re a fractal. You are a line with 5 pieces sticking out (legs, arms, and head) and your arms and legs each have 5 pieces sticking out. If you allow me to bend the rules a bit, your head even has two ears, two eyes, and a nose.

But why? Why do fractals occur so often in nature? One reason is because they’re designed to grow. As trees get bigger they can easily split their outermost branches to make more branches. As clouds continue to gather mass and condensate, the main billow gets bigger and bigger and produces mini-billows on its outer edge. When those billows get too big, they grow mini-billows on themselves as well!

Another reason They’re incredibly simple. By following a short list of instructions over and over, an organism can effortlessly create a tiered home (snail shell), or an object with superiorly high surface area (lungs), or a complex and highly efficient network for transporting resources (roots)!

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Fractals in Computing

Built For This Computers are incredibly well-equipped for fractals. Why? Because fractals repeat so much! If there’s one thing computers know how to do, and how to do well it’s repeat a set of instructions over and over and over. Exhibit A: The most famous fractal is by far the Mandelbrot set, named after the man who coined the term ‘fractal.’ To create this design, take a point in the real-imaginary plane and multiply it by itself.

Forever.

If it gets infinitely big, then it’s outside the set - let’s color it a shade of blue to yellow based on how fast it got there. But what about if it caves in and never gets there? Then it’s inside the Mandelbrot set and we’ll leave it black.

Take a Look at That What you end up with is a beautifully abstract rendering of blobs. But as you zoom in on parts of it, the cool factor ramps up to 11! It twists and swirls, splits and explodes into hair-like fractures. And if you zoom in far enough the thing implodes on itself and creates a Mandelbrot set within a Mandelbrot set.

The Dragon Curve Another popular fractal is called the Dragon Curve. It’s created from an infinite list of 1s and 0s the computer treats as instructions to ‘turn left’ or ‘turn right’ as it draws lines. It generates this list by starting with some 1s and 0s and reading them backwards. Starting from the end, if it reads a 1, it’ll add a 0. If it reads a 0, it’ll add a 1. When it reaches the beginning of the list it goes to the back of the whole list (including the digits it added) again and begins reading - 1 adds a 0, 0 adds a 1 - until you tell it to stop or just go insane.

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