Fractals: Infinite Patterns from Simple Rules
Zoom forever. The pattern never ends.
Zoom forever. The pattern never ends.
Start with a triangle. Replace each line segment with a triangular bump. Repeat. Each iteration adds more detail, and the perimeter grows to infinity while the area stays finite. Drag the slider to add iterations.
Start with a filled triangle. Remove the middle. Remove the middle of each remaining triangle. Repeat forever. The amount of filled area approaches zero, but the shape never disappears.
A trunk splits into two branches. Each branch splits into two more. Adjust the angle and length ratio to create different tree shapes. Nature uses this same algorithm.
Benoit Mandelbrot coined the word "fractal" in 1975 from the Latin "fractus" (broken). He showed that coastlines, mountains, and galaxies all have fractal geometry. He asked: "How long is the coast of Britain?" The answer depends on your ruler size, and it keeps getting longer the smaller the ruler.
Nature, technology, and art all use fractal patterns.
Fractal antennas pack a long antenna into a small space by folding it in a fractal pattern. Your phone uses one to receive signals across multiple frequency bands.
Fractal algorithms generate realistic mountains, clouds, and coastlines in movies and games. A few parameters create infinite detail without hand-drawing.
Blood vessels, lungs, and neurons are fractal. Doctors measure fractal dimension to detect diseases: cancer cells have different branching patterns than healthy tissue.
You've seen how simple rules repeated infinitely create patterns of infinite complexity. Fractals are everywhere in nature: coastlines, trees, blood vessels, clouds, and snowflakes.
A fractal looks the same at every scale. Zoom into a piece and you see the whole shape repeated. A fern leaf is made of smaller fern leaves, which are made of even smaller fern leaves.
A fractal is defined by a short rule applied over and over. The Koch snowflake: "replace each line with a triangle bump." Four words generate infinite detail.
A line is 1D. A square is 2D. A fractal coastline is about 1.2D. Not a line, not a surface. Fractals exist between dimensions.
Trees branch into branches into twigs. Rivers split into tributaries into streams. Lungs branch into bronchi into alveoli. Nature uses fractal patterns for efficiency.
Put your new knowledge into practice!