A fractal is a space of sorts which is self-similar, to some extent, under rescalings.

There is a concept of fractal dimension which measures how the amount of detail of a given space changes with scale. For ordinary manifolds this fractal dimension coincides with the usual dimension, taking values in the natural numbers. For fractal spaces however the fractal dimension may be a non-negative rational number, in fact a real number, hence a fraction, whence the name “fractal”.

Prasit Bhattacharya, Lawrence S. Moss, Jayampathy Ratnayake, and Robert Rose, Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra, (pdf)