Fractals are typically not self-similar

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 off one size fits all ad networks are Some content creators will surely protest 'im a creator stead a salesman' which is a fair complaint but I'd argue 3blue1brown really brings an artistry to the personal ad network the channel is creating.. self curated ads has the potential to be another medium of self expression (o) [LINK] .. an aside: the ad that YouTube ran in front of the video I linked when I first clicked on it was offensive and jarring, it depicted an animated Albert Einstein literally scribbling on a black board while a student looked on in dread and three words came in the screen, two of which I can remember 'math ___ sucks' the middle was either: teachers, lessons, or tutors; this ad actually upset me.. contrast that with this great 3blue1brown ad(i) and I am more confident in m

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: [VIDEO]

"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-obvious answer. But it's not that there are 0 possibilities, it is that there is very many. In a nutshell, the word "dimension" does not simply describe 3-dimensional space, and all other uses are somehow wrong. Dimension applies to any independent element of a vector, and using it to describe spacetime is a tiny, tiny subset of the valid uses of the term. Another very common example is that describing the movement of a free floating in space rigid body is a twelve dimensional value; you need to describe x, y, and z (or equivalant), you need to describe the rotation around x, y, and z, and then you need to describe the velocity of each of those components. That's 12 dimensions. In fact, it's 12 dreadfully mundane dimensions. Incidentally, not included in valid uses of the term is the sci-fi sen

"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: [VIDEO] That's a 3Blue1Brown video explaining why "fractal" and "self-similar" are not the same thing.

Actually, you make a good point. The social dynamics are themselves a dimension. Social network graphs can behave as fractals. In a gymnasium filled with people, there is a significant 'fractallyness' to the social network. In traffic, the fractallyness is very limited to just about all the people in the road near you. The social graph, though more fluid in a traffic jam than a pep-rally, 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] [VIDEO]

There is a common misconception, repeated here, that fractals are defined by their self-similarity. This is not true. Fractals are mathematical objects which have non-integral dimensionalities: that is, they exist in a space somewhere between (n)D space and (n+1)D space for some integer n .[1] The exact Hausdorff dimension, the number quantifying their dimensionality, is not in general the same for each construction. For the Koch curve, the 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] [VIDEO] [2] [LINK]

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

This is a more detailed explanation of what fractals actually are, and what mathematics actually mean by 'fractals are everywhere in nature': [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!)

Here is are two very clear explanations of fractal dimension by the man himself, Grant from 3b1b. [1] Fractals are typically not self-symmetric: [VIDEO] [2] Fractal charm: Space filling curves: [LINK]

3blue1brown has some good fractal videos; one in particular being relevant to this discussion: "Fractals are typically not self-similar" - [VIDEO]

"Fractals are not self-similar" [VIDEO] If you want to learn more about the fascinating world of fractals.