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#fractals#fractaldimension#hausdorffdimension

Fractals are typically not self-similar

3Blue1Brown

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  • Recommended by 7 people: 7 on Hacker News.
  • A classic. People kept recommending it from 2017 to 2024.

Intermediate. Assumes working knowledge.

4.5M views9 years ago7 recommendations

Computer scienceGraphics and audioMath for programmersRendering

This video explains the concept of fractal dimension and demonstrates why many fractals are not strictly self-similar.

An explanation of fractal dimension. Help fund future projects: patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: 3b1b.co/fractals-thanks And by Affirm: affirm.com/careers Home page: 3blue1brown.com/ One technical note: It's possible to have fractals with an integer dimension. The example to have in mind is some *very* rough curve, which just so happens to achieve roughness level exactly 2. Slightly rough might be around 1.1-dimension; quite rough could be 1.5; but a very rough curve could get up to 2.0 (or more). A classic example of this is the boundary of the Mandelbrot set. The Sierpinski pyramid also has dimension 2 (try computing it!). The proper definition of a fractal, at least as Mandelbrot wrote it, is a shape whose "Hausdorff dimension" is greater than its "topological dimension." Hausdorff dimension is similar to the box-counting one I showed in this video, in some sense counting using balls instead of boxes, and it coincides with box-counting dimension in many cases. But it's more general, at the cost of being a bit harder to describe. Topological dimension is something that's always an integer, wherein (loosely speaking) curve-ish things are 1-dimensional, surface-ish things are two-dimensional, etc. For example, a Koch Curve has topological dimension 1, and Hausdorff dimension 1.262. A rough surface might have topological dimension 2, but fractal dimension 2.3. And if a curve with topological dimension 1 has a Hausdorff dimension that *happens* to be exactly 2, or 3, or 4, etc., it would be considered a fractal, even though it's fractal dimension is an integer. See Mandelbrot's book "The Fractal Geometry of Nature" for the full details and more examples. Music by Vince Rubinetti: soundcloud.com/vincerubinetti/riemann-zeta-function Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that). If you are new to this channel and want to see more, a good place to start is this playlist: youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC- Various social media stuffs: Twitter: twitter.com/3Blue1Brown Facebook: facebook.com/3blue1brown/ Reddit: reddit.com/r/3Blue1Brown

Why it's here

7 people recommend this

What people say. Recommended as an exceptionally clear explanation of fractal dimension and the distinction between fractals and self-similarity.

SavantIdioton Hacker News • Recommends

On “A Closer Look at Fractals”

Here is are two very clear explanations of fractal dimension by the man himself, Grant from 3b1b. 1 Fractals are typically not self-symmetric: link to this video 2 Fractal charm: Space filling curves: link
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FullMtlAlcoholcon Hacker News • Recommends

On “The Hindu Temple as a Model of Fractal Cosmology (2015)”

When something is referred to as fractal, does it always imply a degree of self-similarity? As I understand it, Mandelbrot developed fractal geometry as a tool to examine the "roughness" of reality as opposed to the smoothness of calculus and fractal dimension is a measure of complexity at every scale, with a self-similar pattern being an easy example to follow. If that is so, that the universe has a high degree of complexity at every scale, is that really controversial? Here is an excellent 20 min introduction to fractals: link to this video
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Balgairon Hacker News • Recommends

On “Ford CEO says the company 'overestimated' self-driving cars”

…is still fairly small. 3Blue1Brown has a great video on the dimensionality of fractals 0 that is useful here. I'm just going to spitball, but I'd bet that the dimensionality of the social graph of a traffic jam is really close to 1. So in the end, the DoFs of a traffic jam are probably very close to 3, but just a bit over. Say, 3.14-ish. ;) 0 link to this video…
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jerfon Hacker News • Recommends

On “Recursion and Fractals”

Trying to explain fractional dimensions is probably not conducive to an HN post. but fortunately, 3blue1brown converted this: link to this video Which also nicely and conclusively shows why self-similarity is just a special case of things that have fractional dimension.
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jamesharton Hacker News • Recommends

On “Cellular Automata in the Browser”

This is a more detailed explanation of what fractals actually are, and what mathematics actually mean by 'fractals are everywhere in nature': link to this video It might help connect some dots. Short version: simple rules -> can create self-similar shapes -> which can have fractal dimension -> cellular automata can create fractals but nature -> creates shapes which have fractal dimensions does not mean that the processes that create fractal shapes in nature are based on simple rules or cellular automata (but that's not to say they aren't!)
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coremoffon Hacker News • Recommends

On “Fern leaves and cauliflower curds are not fractals”

3blue1brown has some good fractal videos; one in particular being relevant to this discussion: "Fractals are typically not self-similar" - link to this video
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jrvieiraon Hacker News • Recommends

On “The Matrix: Infinite-Horizon World Generation with Real-Time Interaction”

"Fractals are not self-similar" link to this video If you want to learn more about the fascinating world of fractals.
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u/jerfin r/haskell • Mentions

On “Reducing usage of `fromIntegral`”

In addition to Ed's points, it isn't just Haskell that objects to this. Even a language as different from Haskell as Go won't do this conversion for you automatically. There's also a unit error of a sort going on here, in the physics sense of units perhaps moreso than Haskell types. If `length 1, 2, 3` is returning "the number of elements in a list", what does `length 1, 2, 3 * 1.5` really _mean_? What is 4.5 "elements in a list"? You could have some fun defining some mathematical meaning of that (mathematicians _love_ taking things that seem like they could only be integers and making them reals), but back in the real world with your real RAM and your real CPU it doesn't have meanin…
16View on Reddit

u/Endercraft_Oin r/compsci • Mentions

On “What is "efficiency" in terms of represent things in a certain no of dimensions?”

I'm not certain if it is correct but I'll provide my interpretation of this. First is your question about efficiency. Efficiency in this context most likely means how much space is used compared to how much space is available to be used. Like if you pack a square box with cubes it would be 100% efficient since it uses all the space provided. But packing it with spheres can only be 74% efficient at most since there is always unused space. So in this article the efficiency is how much information can be stored in the space compared to how much space there is. (I'll go into what information space is later) **Fractional Dimensions** Next is your doubt about non-integer dimensions. For that I…
3View on Reddit

u/GeorgeneKeckin r/compsci • Mentions

On “Snowflake Simulation with Cellular Automata (CA)”

Speaking of fractals, several months ago, after watching this video: [video I decided to implement the methodology in Rust and implement all the algos myself: https://github.com/Chubek/SummerSavory In fact if you go there, I have implemented several of the vision algos. But then I realized I need a png parser, so I decided to write a PNG parser, then I realized I need zlib, etc etc.
3View on Reddit

justifieron Hacker News • Mentions

On “Ask HN: Anyone monetizing their sideproject with direct ads / sponsorships?”

I am unable to answer your questions for you, but I wanted to talk on the topic because it is something I've been thinking a lot about lately There recently was a lot of talk and reporting surrounding YouTube 'changing their algorithm' and hurting a lot of YouTubers I watch frequently One of my thoughts was that unfortunately the best solution would probably mean content creators would need to seek their own advertising'sponsorships I can tell you the only ads I have clicked on in any recent memory are all, every single ad, of the ones that 3blue1brown places at the end of thaer videos(o) Considerate effort like this really made me think how inconsiderate and detrimental the current hands of…
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jerfon Hacker News • Mentions

On “The special data device SpaceX’s Falcon Heavy sent to orbit is just the start”

"Would people consider a colour photograph as encoding four dimensional information when its pixels are represented as (x, y, brightness, colour) for example?" Actually, yes, they do. Computer vision people will envision a photograph as containing the number of dimensions equal to the number of independent numbers used to describe it, so a 1000x1000 RGB image would be considered 3,000,000-dimensional. This is perfectly routine in that world. Writing would be considered n-dimensional. link In non-math terms you'd probably be justified in saying the question is ill-defined, as what constitutes an orthogonal vector component in a piece of writing is not something that has an universally-obvio…
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jerfon Hacker News • Mentions

On “Could 'Oumuamua be an icy fractal aggregate ejected from a protoplanetary disk?”

"Perhaps calling it a fractal is an unnecessary spin," It is, or at least calling it self-similar is, as well as what the image implies: link to this video That's a 3Blue1Brown video explaining why "fractal" and "self-similar" are not the same thing.
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uoaeion Hacker News • Mentions

On “Recursion and Fractals”

…Hausdorff dimension is ln(4)/ln(3) = log_3(4) approximately 1.2619.2 This is because for each iteration of the curve's construction, the perimeter of the curve grows by a factor of 4 while the area enclosed by the curve only grows by a factor of 3. That makes it fill space faster than a growing 1D line, but slower than a growing 2D surface. 1 link to this video 2 link…
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