Infinitely complex patterns that are self-similar across different scales, bridging the gap between mathematics, nature, and art.
Fractals are geometric patterns that repeat at every scale. Zoom into a fractal and you find smaller copies of the whole, a property called self-similarity. The term was coined by Benoit Mandelbrot in 1975, from the Latin fractus, meaning broken or fragmented.
Unlike smooth Euclidean shapes, fractals capture the roughness and complexity of the real world. Coastlines, mountain ranges, blood vessels, and even stock market charts exhibit fractal geometry.
Fractals have a fractional dimension. A coastline, for example, is more than a one-dimensional line but less than a two-dimensional plane, existing in a strange in-between space that fractals elegantly describe.
From pure mathematics to the natural world, fractals appear in astonishing variety. Here are some of the most iconic forms.
Each branch splits into smaller copies of itself, mirroring how real trees grow their canopies.
Remove the centre triangle, then repeat. An infinite regression of triangles within triangles.
A curve with infinite length enclosing finite area, created by adding triangles to each edge.
The most famous fractal, a set of complex numbers whose boundary reveals infinite detail at every zoom level.
Electrical discharge and river networks branch in fractal patterns dictated by energy dissipation.
Nature's most stunning fractal vegetable, with spiralling buds that repeat at smaller and smaller scales.
Adjust the branch angle and depth to see how simple rules create complex organic shapes.
Discover more pattern families that share mathematical beauty with fractals.