Fractals are not self-similar
3,202
by Super User, 7 years ago
What fractal dimension is, and how this is the core concept defining what fractals themselves are.
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(A few people have mentioned the affirm link doesn't work for them. You can also apply through a link like this: https://jobs.lever.co/affirm/8eb4a65b-df81-4a31-a0e2-da4aafedfa6d?lever-origin=agency&lever-source%5B%5D=3blue1brown . If you add those same query parameters to whatever particular application url you ultimately use, it has the same effect.)
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 surfaces 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. Pick a random fractal from a hat, though, and it will almost certainly have a non-integer dimension.
Special thanks to all of the following Patrons: Meshal Alshammari, Ali Yahya, CrypticSwarm, Yu Jun, Shelby Doolittle, Dave Nicponski, Damion Kistler, Juan Benet, Othman Alikhan, Markus Persson, Dan Buchoff, Derek Dai, Joseph John Cox, Luc Ritchie, Jerry Ling, Mark Govea, Guido Gambardella, Vecht, Jonathan Eppele, Shimin Kuang, Rish Kundalia, Achille Brighton, Kirk Werklund, Ripta Pasay, Felipe Diniz, dim85, Chris, David Wyrick, Hello friend, taking the time to actually look through these names, maybe you are as appreciative of these folks as I am, Rahul Suresh, Lee Burnette, David Kipper, John C. Vesey, Patrik Agné, Alvin Khaled, ScienceVR, Chris Willis, Michael Rabadi, Alexander Juda, Mads Elvheim, Joseph Cutler, Curtis Mitchell, Bright, Myles Buckley, Andy Petsch, Otavio Good, Karthik T, Steve Muench, Viesulas Sliupas, Steffen Persch, Brendan Shah, Andrew Mcnab, Matt Parlmer, Naoki Orai, Dan Davison, Jose Oscar Mur-Miranda, Aidan Boneham, Henry Reich, Sean Bibby, Paul Constantine, Justin Clark, Mohannad Elhamod, Ben Granger, Jeffrey Herman, and Jacob Young.
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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: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown
Check out Affirm careers here: https://affirmjobs.3b1b.co
Patreon page: https://www.patreon.com/3blue1brown
(A few people have mentioned the affirm link doesn't work for them. You can also apply through a link like this: https://jobs.lever.co/affirm/8eb4a65b-df81-4a31-a0e2-da4aafedfa6d?lever-origin=agency&lever-source%5B%5D=3blue1brown . If you add those same query parameters to whatever particular application url you ultimately use, it has the same effect.)
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 surfaces 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. Pick a random fractal from a hat, though, and it will almost certainly have a non-integer dimension.
Special thanks to all of the following Patrons: Meshal Alshammari, Ali Yahya, CrypticSwarm, Yu Jun, Shelby Doolittle, Dave Nicponski, Damion Kistler, Juan Benet, Othman Alikhan, Markus Persson, Dan Buchoff, Derek Dai, Joseph John Cox, Luc Ritchie, Jerry Ling, Mark Govea, Guido Gambardella, Vecht, Jonathan Eppele, Shimin Kuang, Rish Kundalia, Achille Brighton, Kirk Werklund, Ripta Pasay, Felipe Diniz, dim85, Chris, David Wyrick, Hello friend, taking the time to actually look through these names, maybe you are as appreciative of these folks as I am, Rahul Suresh, Lee Burnette, David Kipper, John C. Vesey, Patrik Agné, Alvin Khaled, ScienceVR, Chris Willis, Michael Rabadi, Alexander Juda, Mads Elvheim, Joseph Cutler, Curtis Mitchell, Bright, Myles Buckley, Andy Petsch, Otavio Good, Karthik T, Steve Muench, Viesulas Sliupas, Steffen Persch, Brendan Shah, Andrew Mcnab, Matt Parlmer, Naoki Orai, Dan Davison, Jose Oscar Mur-Miranda, Aidan Boneham, Henry Reich, Sean Bibby, Paul Constantine, Justin Clark, Mohannad Elhamod, Ben Granger, Jeffrey Herman, and Jacob Young.
------------------
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: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown
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Super User uploaded a new media, Fractals are not self-similar
7 years ago
