Fractals are typically not self-similar

An explanation of fractal dimension. Home page: https://www.3blue1brown.com/ Brought to you by you: http://3b1b.co/fractals-thanks And by Affirm: https://www.affirm.com/

Music by Vince Rubinetti: https://soundcloud.com/vincerubinetti/riemann-zeta-function

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.
... https://www.youtube.com/watch?v=gB9n2gHsHN4