Fractal Fern Growth

Overview

This fern emerged from repeated transformations applied to a single point. I used four different rules, each defined by equations. One transformation maps any x y to 0 and 0.16y, which flattens and pulls points downward into the fern’s stem. The others include scaling, rotation, and translation, such as 0.85x plus 0.04y and minus 0.04x plus 0.85y plus 1.6, that expand outward and create leaflets with different orientations. Each rule has a specific probability: 0.01, 0.85, 0.07, and 0.07. Over time, this layering of affine maps builds up the full shape of the fern.

I iterated over one million four hundred thousand points, starting from the origin. On each iteration, I randomly chose one transformation using the assigned probabilities and applied it to update x and y. The new position was mapped into pixel coordinates using px equals width over 2 plus x times width over 10, and py equals y times height over 11. These scalings help fit the structure into the frame. The vertical stretching and shifting are essential, because without the translation terms like plus 1.6 or plus 0.44, the fern would fold into itself instead of growing upward.

Each leaflet reflects the whole, built through repetition and precise transformation. The structure grows outward through a system of rules, not randomness. Every point is placed by logic, every curve a result of measured symmetry. This process embodies self-similarity, where complexity unfolds from the consistent layering of form.