MC Escher's "Circle Limit" pictures

[Max Sinister]

That's also true. Well, even the experts haven't agreed on how to define fractals.

[Max Sinister]

But I'm wondering about what kind of dimensions are we talking about? if it's the fractal dimension as in Hausdorff Dimension, lots of fractals have an integer dimension, like the mandelbrot set with 2.

Just checked the German WP article again and found something we missed:

Mandelbrot himself defined fractals like this: A fractal is a set which Hausdorff dimension is bigger than its Lebesgue covering dimension.
(German: "Ein Fraktal ist eine Menge, deren Hausdorff-Dimension größer ist als ihre Lebesgue’sche Überdeckungsdimension.")

Cantor dust has L(ebesque) 0, but H(ausdorff) 0.6309...

A Hilbert curve has L=1, but H=2.


This allows fractals with a dimension that's exactly 2.