New video! Fractals are not self-similar

I understand - thanks! Allen’s rule is for animals of the same size who have different sized appendages like ears and limbs, so I thought that could correspond to a roughness type of measure which explains their different surface area/volume ratios

Meghna

The size won't affect the dimension, but what you're referencing is how the ratio between two quantities with different dimensions changes based on scale. For example, with respect to heat, it's better to be smaller so that surface area (2d) to volume (3d) is as low as possible. If something with a fractal dimension is involved, those same sorts of laws apply. If the oxygen receptors in your longs are 2.9 dimensional, then as you scale up the body the ratio of oxygen reception to mass will go down, but at a much smaller rate than it would if that lung-dimension had been, say, 2.1.

3blue1brown

So the dimensionality is related to how one measure changes with respect to other measures when scaled, right? I was wondering, with rougher two dimensional objects, the surface area is far higher with proportionate to its volume. I’m thinking about the biological rule Allen’s rule which talks about how the limbs and ears of animals in cold climates are smaller so the surface area isn’t too high compared to their volume. Would it be fair to say that the animals with smaller limbs are less ‘rough’ and hence a lower fractional dimensionality and hence the surface area to to volume ratio is lower?

Meghna

Yup! Which should make sense, because as a set it's completely equivalent to a square.

3blue1brown

What's the fractal dimension of space-filling curves? Is the hillbert curve 2 dimensional?

Matthew Hausmann

Loved this. I've always been fascinated by fractals, but had never been fascinated by them in this way.

Mark Mulvey

Probably at some point I'll do a behind the scenes video.

3blue1brown

fascinating animation! Will you make a video on how to make these animation?

As for what happens in limits, take a look at this clip: <a href="https://youtu.be/RU0wScIj36o?t=1m40s" rel="nofollow noopener" target="_blank">https://youtu.be/RU0wScIj36o?t=1m40s</a> It's definitely possible to have fractals with dimension n+1 that are very rough. The boundary of the mandelbrot set is an example of a 2-dimensional fractal curve (that doesn't fill space). When I said fractals are those shapes with non-integer dimensions, this is slightly inaccurate; the more technical definition is that fractals are shapes whose fractal dimension is bigger than their topological dimension. I just didn't see a way to work in a discussion of topological dimension without blowing up the video time. Basically, anything with non-integer dimension is a fractal, and "most" fractals fall in that category, though some fractals end up having an integer dimension. The mandelbrot set boundary, most Brownian motion, and the Sierpinski pyramid are all examples of fractals with integer dimension.

3blue1brown

Amazing, as always. The idea of non integer dimensions is certainly intriguing, mainly due to the contrast between how ridiculous the idea seems at first sight and how much sense your explanation makes. I'm now feeling really curious visualising what "happens" at the limit when increasing non integer dimensions approach the next integer. By which I mean, if you start at dimension 1 and then check all the "1.something dimensions". there is some sort of cohesiveness in that all their points can be defined by just one coordinate, no matter how much roughness increases, and then there's a frontier in the number 2, after which you need two coordinates to define points.in "2.something dimensions". How is the increasing roughness related to the appearance of a new coordinate, if at all? Could there be that excessive roughness "causes" new dimensions? And ( I feel embarrased, like I'm going full retard wiht this question) since 0.99999... equals 1, is it possible to represent a dimension n+1 as extremely rough n.something dimension? I don't even know if I'm making sense right now, or if you could even talk of the concept of "limits of dimension" (are dimensions even continuous? Or just non integers?). I don't know. I feel like a child who just learned about multiplication. I have so many questions, most of which are probably stupid...

Thank you very mwowch!

3blue1brown

Inorite?

john kraemer

Wownderful! (not a typo, I just realized where it comes from). Just a little bit embarassing at the end ;-). I used to work for a similar compary, but we were too ahead of time. I wish them all the luck they deserve.

Guido Gambardella

I was just about to ask for a triangle of power version...

Guido Gambardella

It's always cool to hear about this stuff used in practice. Hopefully, this helps future coders out when they're trying to measure roughness.

3blue1brown

I've spent a huge amount of my spare time developing a strange attractor rendering software, some of the core code I'd borrowed from a university professor, because never in my life would I have been able to come up with a way to measure the fractal dimension of a point cloud. His method was box counting, I think, and you just explained it to me, 15 years after I wrote that code without having a clue about what I was doing. So, Grant, really, thank you!

I wish you'd used the Triangle of Power!

john kraemer

Watching it now -- it's fantastic.

Luop90