Romanesco Broccoli/Cauliflower may be my favorite example of fractal design.
By strict definition (per Wikipedia) a fractal is:
"...a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between the integers. Fractals are typically self-similar patterns, where self-similar means they are "the same from near as from far". Fractals may be exactly the same at every scale, or they may be nearly the same at different scales. The definition of fractal goes beyond self-similarity per se to exclude trivial self-similarity and include the idea of a detailed pattern repeating itself."
Yeah, that's a mouthful!
To me, I know a fractal when I see it represented visually. Here is a classic image of a fractal called a Mandlebrot Set:
I am very interested in the forms of patterns in nature. Permaculture is about modeling nature, and fractals are one pattern found in many forms all over the natural world. Maybe these would be design motifs for a garden, an orchard, a wall, hedgerow, irrigation system, paddock system, or... who knows. But they are beautiful.
Lightning is a common fractal pattern.
River mouths and networks are fractals (Ganges River, India)
Yes, even clouds are patterned... fractals.
Mountains in Tibet.
Fractal leaf design.
Grand Prismatic hydrothermal spring in Yellowstone National Park, USA.
Cut red cabbage showing a fractal design.
Fractal pattern found in Agate, a type of stone.
Yes, even Slime Mold follows fractal pattern design!