### Fractal Geometry

Fractal Geometry Fractal Geometry Fractal geometry is a branch of mathematics having to do with fractals. Fractals are geometric figures, just like rectangles, circles and squares, but fractals have special properties that those figures do not have. In geometry two figures are similar if their corresponding angles are congruent in measure. Fractals are self-similar meaning that at every level the fractal image repeats itself. An example of self-similarity would be a triangle made up of triangles that are the same shape or are similar to the whole. Another important property of fractals is fractional dimensions.

While in Euclidean geometry figures are either zero dimensional points, one dimensional lines, two dimensional planes, or three dimensional solids, in fractal geometry figures can have dimensions falling between these whole numbers, that is being made up of fractions. For example a fractal curve would have a dimension between one and two depending on how much space it takes up as it twists and curves. The more a flat fractal fills a plane the closer it is to being two-dimensional. As few things have basic shapes, fractal geometry provides for the complexities of these shapes and allows the study of them better then Euclidean geometry which is only successful in accommodating the needs of regular shapes. Fractals are formed by iterative formation, meaning one would take a simple figure and operate on it in order to make it more complex, then take the resulting figure and repeat the same operation on it, making it even further complex. Algebraically fractals are the result of repetitions of nonlinear-equations. Using the dependent variable for the next independent variable a set of points is produced. When these points are graphed a complex image appears.

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One does not have to try very hard in order to experience fractals first hand in the real world as they are ever present in nature. For example in the instance of a river and it’s tributaries, each tributary has it’s own tributaries so that it’s structure is similar to that of the entire river. Many of these things would seem irregular, but in fractal geometry they each have a simple organizing principle. This idea of trying to find underlying theories in what seem to be random variations is called the chaos theory. This theory is applied in order to study weather patterns, the stock market, and population dynamics.

Fractals can also be used in order to create computer graphics. It was found that the information in a natural scene can be concentrated by identifying it’s basic set of fractals and their rules of construction. When the fractals are reconstructed on a computer screen a close resemblance of the original scene can be produced. The first person to study fractals was Gaston Maurice Julia, who wrote a paper about the iteration of a rational function. This work was essentially forgotten until Benoit Mandelbrot brought it back into the light in the 1970’s. Mandelbrot, who now works at IBM’s Watson Research Center, wrote The Fractal Geometry of Nature that demonstrated the potential application of fractals to nature and mathematics. Through his computer experiments Mandelbrot also developed the idea of reconstructing natural scenes on computer screens using fractals.

In conclusion fractals are irregular geometric objects made of parts that are in some way similar to the whole. These figures and the study of them, Fractal geometry, allow the connection between math and nature. Bibliography Bibliography M. Barnsley, Fractals Everywhere, 2d ed, 1992 T. Vicsek, Fractal Growth Phenomena, 1992 http://www.ncsa.uiuc.edu/edu/fractal/fgeom.html Mathematics.

### 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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