Fractal Science Kit ##TOP## Download
Niccoletta Boyer
Unconventional 20th-century mathematician Benoit Mandelbrot created the term "fractal" from the Latin word "fractus" (meaning irregular or fragmented) in 1975. You can find this type of irregular and fragmented geometric shape or pattern all around.
You can create fractals with mathematical equations and algorithms, but there are also fractals in nature. At their most basic, fractals are a visual expression of a repeating pattern or formula that starts out simple and gets progressively more complex.
One of the earliest applications of fractals came about well before the use of the term. Lewis Fry Richardson was an English mathematician in the early 20th century studying the length of the English coastline.
Carry this to its logical conclusion and you end up with an infinitely long coastline containing a finite space, the same paradox put forward by Helge von Koch in the Koch Snowflake. This fractal involves taking a triangle and turning the central third of each segment into a triangular bump in a way that makes the fractal symmetric. Each bump is, of course, longer than the original segment, yet still contains the finite space within.
Weird, but rather than converging on a particular number, the perimeter moves toward infinity. Mandelbrot saw this and used this example to explore the concept of fractal dimension, along the way proving that measuring a coastline is an exercise in approximation [source: NOVA].
These self-similar patterns are the result of a simple equation or mathematical statement. You create fractals by repeating this equation through a feedback loop in a process called iteration, where the results of one iteration form the input value for the next.
Now take all of that, and we can plainly see that a pure fractal is a geometric shape that is self-similar through infinite iterations in a recursive pattern and through infinite detail. Simple, right? Don't worry, we'll go over all the pieces soon enough.
When most people think about fractals, they often think about the most famous one of them all: the Mandelbrot set. Named after the mathematician Benoit Mandelbrot, it's become practically synonymous with the concept of fractals. But it's far from being the only fractal in town.
The spiral of a seashell and the crystals of a snowflake are two other classic examples of this type of fractal found in the natural world. While not mathematically exact, they still have a fractal nature.
A relatively simple way for measuring this is called the box-counting (or Minkowski-Bouligand Dimension) method. To try it, place a fractal on a piece of grid paper. The larger the fractal and more detailed the grid paper, the more accurate the dimension calculation will be.
In this formula, D is the dimension, N is the number of grid boxes that contain some part of the fractal inside, and h is the number of grid blocks the fractal spans on the graph paper. However, while this method is simple and approachable, it's not always the most accurate.
One of the more standard methods to measure fractals is to use the Hausdorff Dimension, which is D = log N / log s, where N is the number of parts a fractal produces from each segment, and s is the size of each new part compared to the original segment.
On the surface, chaos theory sounds like something completely unpredictable, but fractal geometry is about finding the order in what initially appears to be chaotic. Start counting the multitude of ways you can change those initial equation conditions and you'll quickly understand why there are an infinite number of fractals.
Some fractals start with a basic line segment or structure and add to it. That's how you make a dragon curve. Others are reductive, beginning as a solid shape and repeatedly subtracting from it. The Sierpinski Triangle and Menger Sponge are both in that group.
More chaotic fractals form a third group, created using relatively simple formulas and graphing them millions of times on a Cartesian Grid or complex plane. The Mandelbrot set is the rock star in this group, but Strange Attractors are pretty cool, too. These images are all expressions of mathematical formulas.
After Mandelbrot published his seminal work in 1975 on fractals, one of the first practical uses came about in 1978 when Loren Carpenter wanted to make computer-generated mountains. Using fractals that began with triangles, he created an amazingly realistic mountain range [source: NOVA].
In the 1990s, Nathan Cohen became inspired by the Koch Snowflake to create a more compact radio antenna using nothing more than wire and a pair of pliers. Today, antennae in cell phones use such fractals as the Menger Sponge, the box fractal and space-filling fractals as a way to maximize receptive power in a minimum amount of space [source: Cohen].
While we don't have time to go into all the uses fractals have for us today, a few other examples include biology, medicine, modeling watersheds, geophysics and meteorology with cloud formation and air flows [source: NOVA].
The Fractal Science Kit fractal generator is no longer available for download or purchase. This web site is provided for informational purposes only, and to provide online documentation for existing Fractal Science Kit users. All downloadable products have been removed and the purchase page disabled.
I continue to license fractal images for commercial use and I would be happy to work with you to license any of my fractal images. I can provide you with large, high quality image files to your specifications. Contact me atrj.h...@verizon.net for details.
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Ross Hilbert
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The Fractal Science Kit fractal generator is a Windows program to generate a mathematical object called a fractal. The term fractal was coined byBenoit Mandelbrot in 1975 in his book Fractals: Form, Chance, and Dimension. In 1979, while studying the Julia set, Mandelbrot discovered what is now called theMandelbrot set and inspired a generation of mathematicians and computer programmers in the study of fractals and fractal geometry.
Like other mathematical ideas, fractals involve numbers and equations. Unlike most other mathematical ideas, fractals can be used to generate complex, beautiful images that appeal to mathematicians and children alike. Swirling spirals, endless self-similar repetitions receding into the distance, geometric objects arranged in infinitely complex patterns, plant-like creations, geologic designs, clouds, and more, comprise the fractal landscape. These wondrous patterns defy logic yet owe their very existence to mathematics and computers. See theFractal Image Gallery for some examples of the myriad of fractal designs possible.
A fractal image is created by evaluating a complex equation or by performing a sequence of instructions, and feeding the results back into the equation over and over again. During the iteration, you accumulate statistics and map the resulting data to colors, creating the fractal image. By varying the equation or the instructions, you can create Mandelbrot Fractals,Orbital Fractals, L-System Fractals, Orbit Traps, and more.
The Fractal Science Kit fractal generator provides a rich framework for exploring the world of fractals. It handles the common processing steps required to generate a fractal image so that you can concentrate on the fun part; developing the fractal formulas/equations, complex transformations, and coloring schemes that define the fractal.
This is not to say that you must write code to use the Fractal Science Kit. On the contrary, hundreds of Built-in Programs are available and most of these provide options that yield countless variations. A fractal image is the result of combining an equation with data collection programs, complex transformations, and color controllers (the instructions that map the data to colors). By choosing different combinations of these programs/options, you can generate more fractal images than you could ever hope to view in your lifetime without ever writing a single line of code. See theFractal Image Gallery for examples of what you can produce using only the Built-in Programs.
The Fractal Science Kit fractal generator supports many different Fractal Types including:Mandelbrot, Julia,Convergent,Newton, Orbit Traps,Sierpinski Triangle, IFS,Strange Attractors, Rep-N Tiles, Symmetric Icons, Symmetric Attractors, Frieze Group Attractors,Wallpaper Group Attractors, Hyperbolic Attractors,Apollonian Gasket, Circle Inversion, Mobius Dragon IFS,Mobius Patterns, Grand Julian IFS, Elliptic Splits IFS, Schottky Group, Kleinian Group, L-System and many more. Hundreds of built-in equations, transformations, orbit traps, and color controllers, allow the casual user to produce stunning fractal images while providing the experienced fractal developer a rich set of illustrative examples on which to build his/her own fractal programs.
The Fractal Science Kit fractal generator provides an interactive programming environment withApplication Windows for viewing the fractal image, modifying the properties that define the fractal, examining the data behind the fractal, and viewing/editing the fractal programs, macros (inline functions/methods), and color gradients, used by the Fractal Science Kit to produce the final image.
The Properties Pages allow you to view/edit all the properties associated with a fractal. Properties control every aspect of the resulting fractal image and the Fractal Science Kit fractal generator supports a rich set of properties for choosing colors, controlling image processing tasks (e.g., smoothing, sharpening, embossing, anti-aliasing), controlling Data Normalization (e.g., contrast stretching, histogram equalization, data scaling via a transfer function), selecting/editing the Fractal Programs (equations, data collection programs, transformations, and color controllers), and much more.
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