Early Simple Fractals

Koch Snowflake

David Hilbert's famous space filling curve.

It was perplexing for the mathematicians of the day to find there was a one dimensional entity, (a curve),  that seemed to fill a two dimensional space.

Waclaw Sierpinski

Source: http://spanky.triumf.ca/www/fractal-info/f-hist.htm

Weierstrass Function: Continuous everywhere, but differentiable nowhere

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http://www.shu.edu/projects/reals/cont/fp_weier.html