"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."
- *Benoit Mandelbrot, introduction to The Fractal Geometry of Nature*

Fractals are infinitely complex patterns that are self-similar across different scales and sizes. Fractals are patterns formed from chaotic equations containing self-similar patterns of complexity increasing with magnification. If we divide a fractal pattern into parts you get a nearly identical reduced–size copy of the whole. The mathematical beauty of fractals is that infinite complexity is formed with relatively simple equations. By iterating or repeating fractal generating equations many times, random outputs create beautiful patterns that are unique, yet recognizable. In a mathematical fractal, such as the famous Mandelbrot set, each pattern is made up of smaller copies of itself, and those smaller copies are made up of smaller copies again. They can be generated by a computer calculating a simple equation over and over.

“If I wanted to construct a fractal, I would take a simple rule, like adding three small triangles on the edge of a big triangle. Then, the rule is repeated again and again.”--- Frederi Viens

Self–similarity may be manifested as exact self-similarity which is identical at all scales; e.g. Koch snowflake or Quasi self–similarity which approximates the same pattern at different scales but may contain small copies of the entire fractal in distorted and degenerate forms; Statistical self–similarity, repeating a pattern stochastically; qualitative self–similarity as in a time series and multi fractal scaling. Fractals have a fractional or fractal dimension greater than its topological dimension and Fractals as mathematical equations are no where differentiable.

Nature is full of fractals. From sea shells and spiral galaxies to the structure of human lungs, the patterns of chaos are all around us. These fractals come in the form of branching and spiral patterns. Trees, ferns, the neurons in the brain, the blood vessels in our lungs, lightning bolts, rivers branching, as well as the coastline and rock formations are examples of branching types of fractals. Spiral fractals can be seen in the nautilus shell, weather patterns such as a hurricane, spiral galaxies, the spiral of pine cones and sunflowers.

## Examples of fractals in nature:

Flowers, Ferns, Leaves, River channels, Lightning, Snowflakes are some of the examples of fractals in nature. Romanesco Broccoli a variant of cauliflower is the ultimate fractal vegetable. Its pattern is a natural representation of the Fibonacci or golden spiral, a logarithmic spiral where every quarter turn is farther from the origin by a factor of phi, the golden ratio. The Fibonacci sequence, a common and beautiful numeric pattern in nature creates the Golden Ratio.

Ferns are a common example of a self-similar set, meaning that their pattern can be mathematically generated and reproduced at any magnification or reduction. The mathematical formula that describes ferns, named after Michael Barnsley, was one of the first to show that chaos is inherently unpredictable yet generally follows deterministic rules based on nonlinear iterative equations. In other words, random numbers generated over and over using Barnsley's Fern formula ultimately produce a unique fern–shaped object. Many plants follow simple recursive formulas in generating their branching shapes and leaf patterns.

"All the branches of a watercourse at every stage of its course, if they are of equal rapidity, are equal to the body of the main stream." - Leonardo da Vinci

The planet Earth has fractal river networks that transport rainfall from the land to the oceans. Like all fractals, these complex, self–similar patterns are formed by the repetition of a simple process of channel formation due to erosion. River network collects a huge amount of rainfall from a very large land area and condenses it into a small area.

Lightning does not travel in straight lines but follows a chaotic and jagged path. Lightning can be very large, spanning several miles, but it is formed in microseconds. Thunder is a fractal sound. It is caused by the superheating of air. Because the pathway of the lightning bolt is a jagged fractal in 3D space, the time it takes to reach our ears varies and hence the sound we hear is a fractal pattern.

Galaxies are the largest examples of spiral fractals known. A single spiral galaxy may contain a trillion stars. The spiral arms do not contain a greater number of stars but still the spiral arms are brighter because they contain many short-lived extremely bright stars, formed by a rotating spiral wave of star formation. The waves of star formation are made visible because they contain many young and very bright stars that only live a short time, perhaps 10 million years, as compared to the more common stars, such as our sun which live for several billion years. Hurricanes or cyclones are the largest spirals here on Earth. The plant kingdom is also full of spirals, as evidenced in many cacti, flowers, fruit, pinecones, etc. A nautilus shell is another example and in this, a simple combination of rotation and expansion creates the spiral. (The figures in this page depicts the patterns in nature. In the regularity of eclipses, the arrangement of seeds in the head of a sunflower, or the spiral of a Nautilus shell, nature exhibits **periodic patterns**. This principle exactly matches with the **arrangement** of electrons in atoms which recurs periodically, which causes many properties of the elements to recur periodically and, thus, allows us to predict physical and chemical behavior.)

Snowflakes are a very good example of fractals in nature. A snowflake begins to form when an extremely cold water droplet freezes onto a pollen or dust particle in the sky. This creates an ice crystal. As the ice crystal falls to the ground, water vapour freezes onto the primary crystal, building new crystals – the six arms of the snowflake. Snowflakes are symmetrical as water molecules in ice crystals join to one another in a hexagonal structure, an arrangement which allows water molecules to form together in the most efficient way.

Crystallizing water forms repeating patterns in snowflakes. The repetitive pattern is like a miracle which prompted claims about the power of consciousness to affect matter. The Koch snowflake represents one of the earliest fractal curves to have been described. However some people may not agree that snowflakes are fractal in nature as snowflakes are self–similar only through a few dimensions. Depending on the temperature and humidity of the air where the snowflakes form, the resulting ice crystals will grow into a myriad of different shapes. It is unlikely that any two snowflakes are alike due to the estimated 1019 (10 quintillion) water molecules which make up a typical snowflake, which grow at different rates and in different patterns depending on the changing temperature and humidity within the atmosphere.

The lungs are an excellent example of a natural fractal organ. The volume of a pair of human lungs is only about 4-6 litres but the surface area of the same pair of lungs is between 50-100 square metres. Surface area to volume ratio in lungs is very high and is very useful. This is possible only because the structure of lungs is fractal. There are 11 orders of branching, from the trachea to alveoli at the tips of the branches. Fractal branching geometry provides an incredibly useful way to make a very large surface area extremely compact.

Every cell in the body must be close a blood vessel, say within about hundred microns to be able to receive oxygen and nutrients. The fractal branching network of blood vessels down to the width of a capillary, which is about eight microns in diameter makes it possible. The length of blood vessels in human body could be about 150, 000 kilo metres of blood vessels as there are about 250 capillaries /mm^{3} of body tissue and the average length of a capillary is about six hundred microns.

The human brain comprises approximately 100 billion neurons with about 100 trillion synapses or connections among these neurons and on an average each neuron may have to communicate with about thousand cells at a time. The axons reach out to make synaptic connections with the dendrites of other neurons. It is the fractal branching pattern of the neuron’s axons and dendrites that allows them to communicate with so many other cells.

Many scientists are trying to find applications for fractal geometry, from predicting stock market prices to making new discoveries in theoretical physics. Fractals have more and more applications in science such as in astronomy, computer science (fractal compression), fluid mechanics, telecommunications (fractal antenna) and medical science. Sometimes fractals describe the real world better than traditional mathematics and physics.