Fractals are typically not self-similar

Fractals are typically not self-similar 3Blue1Brown An explanation of fractal dimension. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: https://3b1b.co/fractals-thanks And by Affirm: https://www.affirm.com/careers Home page: https://www.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. https://youtu.be/gB9n2gHsHN4