Q in the title - thanks for your help
Fractals are pretty cool. They're geometric shapes with a circumference that is infinite.
How is that possible, you say?
Lets have some fun with Google Maps. Bring up the whole of Australia, and draw a quick line around the coast of the mainland. How long is that line?
Now zoom in some more. Can you see where you have skipped over some features and cut off some headlands etc? Do it again, and it will be longer.
Now Zoom so you can only see, say Victoria. You'll see your line isn't really all that accurate. If you correct it, it will get longer. Not by a lot, but still, it needs to be accurate
Now zoom into Port Phillip Bay, where Melbourne is. Argh, your line skips all those bays and totally cut off the Bellarine Peninsula, better go correct it. Its not going to add a huge amount to the length, but it will still count.
Zoom in further, til you can see St Kilda beach. Argh, the line STILL doesn't fit the coast line. Make it fit better, and the line will be longer.
If you zoom in as far as you can to St Kilda beach, you can put your line as close as you can to the actual water line, and that will add some length to your line as well.
But we don't need to stop with Google maps. Imagine if you were on the beach in real life, you'd have to travel over lumps and bumps in the sand, there'd be rocks, etc, and that would add to the length of the line.
And if you were to take an ant's perspective, then climbing over each and every grain of sand would add up. Its still technically the circumference of Australia, but by now it is much longer than your original circumferce.
But we don't have to stop there. What about the view of bacteria, on the surface of those sand grains. Or even the atoms that make up those sand grains.
In theory you could keep going forever, and the circumference would be getting longer and longer and closer and closer to being accurate, and yet, it would never be absolute.
Fractals are the same. You can zoom in forever.
Some fractals look the same no matter how far you zoom in. Some look similar, but not the same, no matter how far you zoom in, and for our Australia case, are infinitely different at different magnifications.
They appear in nature all the time, and they help model growth patterns, and this can be useful for all sorts of things from 3D rendering of trees to cancer research to how cities spread out.
They can be a form chaos: they're not random, but and not precisely predicable, but predictable enough to make useful models of real life stuff, like how the Universe is moving, or drug take up or the frost patterns on a window, for example.
And above all, they look REALLY cool.
This is a really good explanation, but it is worth pointing out that the real world does have a lower limit, since at a certain point, there stops being smaller particles that stuff is made of. Not so in ideal math land, where a fractal edge actually can go on to infinite detail.
Just mentioning before someone else nitpicks it. It’s still a really great example, just sadly runs into the limits of reality.
Fractals can have bounds too.
The only true “infinite fractal structure” we can reference is the universe as a whole. We believe it to be infinitely large in its boundaries, and (to our knowledge) as small as quarks in subatomic particles.
It's always a pleasant surprise to see the neighborhood I grew up referenced in such a big sub
That line on the coastline? It’s squiggly. If you zoom in, it’s still squiggly. It is actually infinitely squiggly- no matter how much you zoom in, it’s still squiggly.
Fractals are patterns that repeat at different scales. When you zoom in on part of a fractal, it still looks similar to the whole shape. You can see this in things like snowflakes, trees, and coastlines.
They matter because they help us understand and model complex systems in nature and science that don’t follow simple geometric shapes. Fractals are also used in computer graphics, physics, and even medicine.
Like fern leaves!
Fractals are things (of any sort) that have a dimensional rating that's not a whole number. A flat fractal in a plane might have a dimensional rating of 1.4 or 1.5, a '3-D' fractal in real life might have a dimensional rating of 2.6 or whatever. It's a proof of complexity in the universe that physical things are fractals. Fractals are easily recognized because a fractional part of the fractal looks like the whole fractal. In the physical world we live in, coastlines and trees are good examples of that sort of fractal: the branch of a tree looks like a whole tree, and a short piece of coastline, magnified, looks like a larger coastline. In mathematics, you have the Mandelbrot set, the Julia sets (fun fact: the Mandelbrot set can be defined as the collection of points where the Julia sets of those points are one piece and not split into parts), and Seirpinski Triangle.
Fractals show up everywhere. Biology (trees and circulatory systems), physics, geography, math.. They have interesting and often useful properties, so people devote a lot of time to studying fractals. the more we know about a given fractal, the more we can understand its structure, and thus, perhaps, how to make or mimic it. Believe me, I haven't even scratched the surface here, fractals are weird and wonderful.
A fractal is something that, when you drill down into it really precisely, appears to be similar to the overall picture before you drilled down (it doesn't have to be a picture at all, but that's a convenient example).
Imagine a coastline. If you zoom into it, down to a microscopic level, it looks a bit like a coastline still. Ridges and edges and cliffs and roughness, etc.
Why are they important? Because finding something that looks similar to something else, but which can be expressed in a smaller amount of mathematical data is extremely useful.
How useful?
The real name for a JPEG image is a JFIF - the JPEG Fractal Image Format.
JPEG works by finding smaller "equations" that end up looking similar to the picture you wanted to store... but which take up less space.
An average JPEG might be, say, 2-3 MBytes. If it wasn't in JFIF format, that same image might well consume 100's of MBytes.
So why are they important? Well, if you want to download this page, look at images from other Reddit users, or store photos on your camera or computer? They're pretty important.
Similarly, MP3 and MPEG are basically the same kind of algorithms, in 1 dimension (audio) and 3 dimensions (moving images). They're pretty important in modern life too. It's why we can download the entire Beatles back catalogue in a few minutes, or why we are able to cram hundreds of digital TV channels into the space that only used to be able to hold a small handful.
This is not technically correct but I love the examples of self similarity in compression cause recursive problems like that are fascinating!
Anyways here’s a video explaining it. Turns out self-similarity is not the defining feature of fractals, but instead being “rough” no matter how far you zoom in.
Well, yes, because a line would be "fractal" otherwise.
The real name for a JPEG image is a JFIF - the JPEG Fractal Image Format
This is factually incorrect.
JFIF stands for JPEG File Interchange Format. (Your comment is pretty much the only search results that calls it 'fractal interchange format'
JPEG/JFIF files don't use fractals in any way. JPEG uses discrete cosine transform.
JPEG works by finding smaller "equations" that end up looking similar to the picture you wanted to store... but which take up less space.
This is correct-ish. When jpeg image is saved, encoder splits the image in 8x8 chunks and compares each chunk to a pre-determined list of patterns.
But this isn't fractal.
While fractal encoding exists, it relies on self-similarity.
Most basically, a fractal is any geometric object that has a fractional dimension larger than its topological dimension. Without getting too mathy, what this basically means is that as you change the scale (think "zoom in"), the level of complexity changes.
A classic example is the "coastline paradox". If you haven't heard of this before, basically what it states is that the more you zoom in on a given coastline, the more complex it gets. In other words, if you were to try and measure the coastline with the resolution of a kilometer, you get one answer, if you go down to a meter stick you get a bigger answer, if you go down to centimeters you actually get a much bigger answer, if you go down to milimeters you get an even bigger answer, which suggests the more you zoom in to a coastline, the more complex it gets, and the "longer" the coastline is when trying to measure it using smaller scales. Zoom in enough and you're measuring the outline of each grain of sand along the beach and the "true" length of the coastline appears to grow infinitely.
What fractional dimensions represent, more generally, is how this complexity grows as you zoom in to something. The Koch snowflake is a simple fractal object that's created by starting with an equilateral triangle. Then, you add an equilateral triangle to each flat edge with a base that's 1/3 of the length of that side. You keep doing this forever. This is created using lines, so technically its topological dimension is one. But for things to be one dimensional, you have to be able to measure the distance between two points. Turns out, the distance between any two points on a Koch snowflake is actually infinite. So it can't be a 1-dimensional object by the topological definition of what a one-dimensional object is. Even though it's made with lines, there's no small portion of it you can look at that behaves line-like. To account for this, fractional dimensions were created, and the fractional dimension of the Koch snowflake is something like 1.26. Things that behave this way are generally referred to as fractals.
Many have mentioned what they are and how they're used - but looking up some famous ones visually can help a bit.
Iirc the Mandelbrot set hides the fibbonacci numbers within it. The sipierski triangle slowly filters out binomial numbers/pascal's triangle when constructed in a particular way etc. Koch's snowflake is a good one to see just in it showing you how from a basic premise an increasing complex shape forms(as some has said about the coastline paradox the rock snowflake sort of illustrates that one we'll over time).
Take a triangle with sides that have length 1. In the middle of each side put another triangle with sides that have length 1/3. In the middle of the two open sides of each of those 3 triangles put another 2 triangles with sides that have length 1/9. Continue- at each stage you will place more triangles, but also smaller ones. You will end up with a very, very wrinkly shape- something called a Koch snowflake. If you examine any single triangle except the first one, it looks the same- this is called "self-similarity".
One of my favorite examples of why this matters is the so-called Kleber-West scaling. As organisms get bigger and bigger, you might think that they would move slower and slower- because the energy you can push into an organism is limited by the surface area of its gut. An organism that is twice as large has eight times the mass to support, but only four times the gut area...unless the gut area and cell layout gets more "wrinkly" as you get bigger. This ends up meaning that the organism ends up with effective "extra dimensions" and instead of the total need for food going as the mass falls off more slowly with mass.
Fractals are the visual representation of recursion.
Romanesco broccoli is a cool example of a fractal in nature.
Great explanations above! I would add
"If you think about a line, its a two dimension thing - it has a start point and an end point. Just a pair of X/Y points. We get one edge between them and that line has no meaningful 'width' needed to represent it. Lets say the length of this edge is "1.00".
What about a curve? It's just two X/Y points right? Two dimensional - but you need meaningful "width" in one of the dimensions to represent the edge/bulge. Stil very much 2d, but, now that edge needs area to have meaningful representation, and that edge is now LONGER than the simple distance between the two X/Y points by maybe 5%? (Depends on the radius of the curve of course)
What about an S-curve? While still very much 2d, it's edge needs more area to meaningfully represent, and it's edge is even longer by a few more percentages of fractions.
What about a path that "turns left, goes 1/2 the remaining distance, turn right, go 1/2 the remaining distance a few times" and then unfolds in the same anti-clockwise path to arrive at the 2nd point? (think of Tron LightCycles, or the old classic Snake Game) You've extended the length of this edge a fractional amount more, but since you've started to need to fold away and back again enough times to wind up at the target-point, you've started to create a limit on the "area" that your path is going to take.
You're still very much 2d, but, you have a complex function that draws just a line between them, and the more complex the function the higher the fractional growth of that edge's length becomes. You can fold/unfold that line in REALLY COMPLEX but geometrically SIMPLE ways, and the patterns that start to arise are the ones we see in nature (the outside edges of a fern leaf are the classic example). You can easily have a line-length thats 10x, 100x, 1000x longer than the simple distance between the two X/Y points, and despite that, the overall "area" of that complex path is paradoxically bounded by the type of path folding you select. Overall, you kinda need a partial "third dimension" to represent the 2d definition. You're fractionally inching into 3d from a 2d geometry.
You can then extend this idea into 3d. Take a sheet of paper. It has an easy area to express. Its volume is zero. Crinkle it up. It's still a nominal 2d object - right? Same area. But is its volume really zero? No, because the crinkles sorta need a Z dimension to represent them - but they're still very constrained to the 2d geometry of the sheet of paper. What if you infinitely fold that sheet of paper to oragami-infinitum? Notice how the area shrinks but the volume fractionally increases.
So what value is this geometric-acid trip? In pure abstract terms it has applications in telecommunication where you predict the error rates in a data transmission (which is a "line" of data through Time), and can make assumptions about "in theory how small of a data element can i send and recover on the other side, by changing the speed of transmission" and in practical engineering terms: Metal surfaces evaporate heat based on surface area. The more crinkles you have on a metal-part the more surface area that can evaporate heat. The question becomes, "what cutting path can i program into a CNC machine to maximize the surface area without having to manually configure each cut"
Okay so you know how the world is like, super duper complex? With like stars and planets and brains and birds and weather and trees and grass?
Yet the maths that runs physics is very simple. Like you probably know E=mc² from Einstein, but how does something so simple explain so much about light, gravity and nuclear fusion in suns.
This was a big problem in the philosophy of science. How do we get a very complex universe when the rules that ‘control’ it are very simple?
Enter Fractals.
You’ve probably seen the Mandelbrot set, the famous fractal pattern that looks like a trippy butterfly. It’s created by the simple rules that are a bit more complex than this : f_{c}(z)=z{2}+c. The resulting pattern isn’t just VERY complex, it’s INFINITELY complex. The more you zoom it the more patterns that are expressed.
This proved once and for all that a very simple rule could make something infinitely complex. Suddenly the complexity of the universe makes sense.
When we look all around nature we see lots of fractals. It’s why a tiny finite amount of dna in a tree can explain the super complex branches and leaves of a tree. It’s all around us.
They're also great for computer applications because they can generate arbitrary complexity from a simple (meaning: compact and easy to store and compute) start.
Minecraft, the entire world, is generated from a few basically fractal equations overlapping each other. That creates a function of the form "Give me a chunk's x and y coordinate and I will give you a chunk that 'looks right' and meshes smoothly to the other neighboring chunks." That gives you functionally infinite world to explore no matter what direction you go, because the function always works (*).
(*) handwave-handwave floating point accuracy.
There's a bbc doc The Secret Life of Chaos which talks about life/nature, morphogenesis, fractals and the history of all.
https://youtu.be/RWb70e2AhJA?si=RG0UkzYApvdOCSCV
This is a must watch. And it is presented by the amazing Jim Al-Khalili!
I love the way you write!!
They didn't write it.
?
Chat gpt
It clearly isn’t
Not everything you're jealous of is written using ChatGPT.
It was several people?
Probably they're just using "they" singular to refer to a person whose gender is not known (as I just did here).
I’m five years old. I have not seen the Mandelbrot set.
Then you are missing out! It's pretty trippy: https://mandel.gart.nz/
It’s like those Zoom In drawings that were so popular a while back!! So cool
I'm 5 an confused, what am I looking at?
It's basically a graph of a formula that generates a very complicated shape, and as you zoom in the computer uses that same formula to refine the shape further and further and keeps adding more complexity.
The idea with fractals is that no matter how much you zoom in, you could always add more detail according to the formula. Although this particular example website does have a limit to how far you can zoom in. But that's inherent to the software, not to the math of the fractal.
A shape. If you zoom in, you see that shape again. No matter how far you zoom in it's self-similar. A shape with this property is called a fractal. There are even higher dimension fractals too e.g. a Menger Sponge.
Yeah, that really got me pissed
It clicked for me one day when I was hiking. I stopped at a stream to fill my camelback. While bent over I looked down at a rock that had moss growing on it and some water trickling through the moss.
It looked literally identical to an aerial view of a river running through a forest on some nature documentary I had just seen. Like literally the exact same thing. Indistinguishable.
I realized everything is relative to the vantage point.
Great summary! If you’re not an educator, you should be