A fractal is generally "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole," a property called self-similarity. A mathematical fractal is based on an equation that undergoes iteration, a form of feedback based on recursion.Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex. Natural objects that approximate fractals to a degree include clouds, mountain ranges, lightning bolts, coastlines, snow flakes, even various vegetables (cauliflower and broccoli).
One of the most famous deterministic fractals was described by Von Koch and is called the Koch Curve. Consider starting with a single straight line that we will call the initiator.
The initiator
_______________
Next is to define a generator, or production rule, that states "Take the initiator, scale it down by a factor r=1/3, make N=4 copies and replace the initiator by these scaled down copies, oriented as shown".
The Generator
The generator is repeated...
and repeated...
and repeated.....
etc.....
After an infinite number of repetitions this curve will be everywhere non-differentiable and will also have infinite length.
Meet some of the most beautiful fractals found in nature.
Cauliflower
This is "Chou Romanesco" cauliflower.A special type of broccoli, this cruciferous and tasty cousin of the cabbage is a particularly symmetrical fractal.
Broccoli
Though not as famously geometric as its relative the Romanescu, broccoli is also a fractal.
Fern
The fern is one of many flora that are fractal; it’s an especially good example.
Fractal Tree
Pair of trees, grown from fractals.
Human Lung
Human lung, composed entirely of fractal algorithms.
Lightning
Lightning’s terrifying power is both awesome and beautiful. The fractals created by lightning are fascinatingly arbitrary and irregular.
Mountain Ranges
Both shorelines and mountain ranges are considered loosely fractal.
Peacock
Males of both the white peacock and standard peacock variety are resplendent examples of fractals in the animal kingdom.
Sea urchins and sea stars
Sea urchins are compact, almost artistic little creatures; sea stars are more commonly referred to as starfish.
Sea Shells
The nautilus is one of the most famous examples of a fractal in nature. The perfect pattern is called a Fibonacci spiral.
Shorelines
This stunningly complex fractal shoreline is none other than the pan handle of Florida.
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