Fractal geometry is used to make the infinitely complex, rough shapes that fractals are. Shapes are made differently in fractal geometry than in classical geometry. In classical geometry, equations are used to create and define shapes! However, in fractal geometry, iteration is used to make fractal shapes! That means simple equations are run many times to make fractals. Also, fractal geometry uses different rules than classical geometry which helps create fractals. When a fractal is made, a rule about changing the shape is applied to the shape an infinite number of times as the equation is run. These rules are simple, but produces complex figures.
Dimensions
Fractals don’t exist in a whole dimension so they exist in a fractional dimension. Each fractal exists in their own unique dimension. They do not exist in whole number dimensions like 1D, 2D, etc. Fractals exist in non-integer dimensions. Felix Hausdorff extended the definition of dimension in a 1918 paper, and his definition allowed for there to be dimensions that were not whole numbers for sets of numbers. The dimension of a fractal measures how quickly the perimeter of the fractal grows every iteration which makes the fractal more complicated. The dimension of a fractal is very important to know when studying fractals.
Finding the Dimension of a Fractal
When a fractal is scaled, its area, length, and its contents change by a factor of the factor to the power of the dimension, or x to the power of d. The equation is:
d = log x * x^d
where x is the factor of scaling. The Hausdorff Dimension is another way to calculate the dimension of a fractal. It is
d = log N / log s
where N is the number of parts that a segment produces in each iteration and s is the size of each new part compared to the size of the original part.
Finding the Dimension of a Fractal
When a fractal is scaled, its area, length, and its contents change by a factor of the factor to the power of the dimension, or x to the power of d. The equation is:
d = log x * x^d
where x is the factor of scaling. The Hausdorff Dimension is another way to calculate the dimension of a fractal. It is
d = log N / log s
where N is the number of parts that a segment produces in each iteration and s is the size of each new part compared to the size of the original part.
Area and Perimeter
Lewis Fry Richardson was a mathematician that lived during the 20th century. He tried to measure the length of a coastline and he found that the length changed based on what kind of tool was used to measure the coast. This helped discover the fact that the perimeter of a fractal is infinite. All fractals have infinitely long perimeters because every time they are iterated, segments are added and the added length makes the perimeter grow. Since fractals are iterated infinitely, the perimeter grows infinitely many times so it is infinitely long. On the other hand, as a fractal is iterated, its area comes closer to 0.
Randomness
Fractal Geometry is about finding order in chaos. Randomness can be a part of fractal shapes whereas it cannot be incorporated into classical shapes. Barnsley Fern and Sierpinski's Gasket can both be generates using randomness. Since many random events happen in nature, one reason why fractals can be found all throughout nature is because they can use randomness in the equations. This is also why things from nature can be generated easily to perfection using fractals.