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Fractal Geometry

“Fractal Geometry is not just a chapter of mathematics, but one that
helps everyman to see the same old world differently”. – Benoit Mandelbrot
The world of mathematics usually tends to be thought of as abstract. Complex and imaginary numbers, real
numbers, logarithms, functions, some tangible and others imperceivable. But these abstract numbers, simply
symbols that conjure an image, a quantity, in our mind, and complex equations, take on a new meaning with
fractals – a concrete one. Fractals go from being very simple equations on a piece of paper to colorful,
extraordinary images, and most of all, offer an explanation to things. The importance of fractal geometry is that it
provides an answer, a comprehension, to nature, the world, and the universe. Fractals occur in swirls of scum on
the surface of moving water, the jagged edges of mountains, ferns, tree trunks, and canyons. They can be used to
model the growth of cities, detail medical procedures and parts of the human body, create amazing computer
graphics, and compress digital images. Fractals are about us, and our existence, and they are present in every
mathematical law that governs the universe. Thus,
fractal geometry can be applied to a diverse palette of subjects in life, and science – the physical, the abstract, and
the natural.

We were all astounded by the sudden revelation that the output of a very simple, two-line generating formula
does not have to be a dry and cold abstraction. When the output was what is now called a fractal, no one called it
artificial… Fractals suddenly broadened the realm in which understanding can be based on a plain physical basis.

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A fractal is a geometric shape that is complex and detailed at every level of magnification, as well as self-
similar. Self-similarity is something looking the same over all ranges of scale, meaning a small portion of a fractal
can be viewed as a microcosm of the larger fractal. One of the simplest examples of a fractal is the snowflake. It
is constructed by taking an equilateral triangle, and after many iterations of adding smaller triangles to increasingly
smaller sizes, resulting in
a “snowflake” pattern, sometimes called the von Koch snowflake. The theoretical result of multiple iterations is the
creation of a finite area with an infinite perimeter, meaning the dimension is incomprehensible. Fractals, before that
word was coined, were simply considered above mathematical understanding, until experiments were done in the
1970’s by Benoit Mandelbrot, the “father of fractal geometry”. Mandelbrot developed a method that treated
fractals as a part of
standard Euclidean geometry, with the dimension of a fractal being an exponent. Fractals pack an infinity into “a
grain of sand”. This infinity appears when one tries to measure them. The resolution lies in regarding them as
falling between dimensions. The dimension of a fractal in general is not a whole number, not an integer. So a
fractal curve, a one-dimensional object in a plane which has two-dimensions, has a fractal dimension that lies
between 1 and 2. Likewise, a fractal surface has a dimension between 2 and 3. The value depends on how the
fractal is constructed.

The closer the dimension of a fractal is to its possible upper limit which is the dimension of the space in
which it is embedded, the rougher, the more filling of that space it is. Fractal Dimensions are an attempt to
measure, or define the pattern, in fractals. A zero-dimensional universe is one point. A one-dimensional universe is
a single line, extending infinitely. A two-dimensional universe is a plane, a flat surface extending in all directions,
and a
three-dimensional universe, such as ours, extends in all directions. All of these dimensions are defined by a whole
number. What, then, would a 2.5 or 3.2 dimensional universe look like? This is answered by fractal geometry, the
word fractal coming from the concept of fractional
dimensions. A fractal lying in a plane has a dimension between 1 and 2. The closer the number is to 2, say 1.9,
the more space it would fill. Three-dimensional fractal mountains can be generated using a random number
sequence, and those with a dimension of 2.9 (very close to the
upper limit of

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