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”.
On category theoretic treatments of the self-similarity found in fractals in terms of terminal coalgebras:
Tom Leinster: A general theory of self-similarity [arXiv:1010.4474]
Prasit Bhattacharya, Lawrence S. Moss, Jayampathy Ratnayake, Robert Rose: Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra, in: Horizons of the Mind — A Tribute to Prakash Panangaden, Lecture Notes in Computer Science 8464, Springer (2014) [doi:10.1007/978-3-319-06880-0_7]
Victoria Noquez, Lawrence S. Moss: The Sierpinski carpet as a final coalgebra, Theory and Applications of Categories, 45 2 (2026) 33-129 [tac:45-02, arXiv:2110.06404]
See also:
Last revised on January 6, 2026 at 18:27:57. See the history of this page for a list of all contributions to it.