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Table of contents
This simple formula is responsible for all of the infinite complexity, and infinite beauty, in the Mandelbrot set. FracTest provides controls to pick which area in the complex plane is mapped to the screen specified as its centre and size. In FracTest, the size of an area of the fractal is defined by what we call the "radius", which is actually half the height.
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Starting from the overall view of the Mandelbrot set, you can explore it by picking an interesting point to zoom in on, usually by using a selection. Then, repeat the process. Very zoomed-in views are referred to as "deep", and going deep needs a lot of maths to compute the image. The way a Mandelbrot set point is converted to a visible colour is called the representation function.
There are many possible representation functions, but FracTest only supports a few; these can be used in any combination, and the colours generated by them will be averaged.
You can select the function or functions to use in the user interface. This is the most widely-used and simple function; although it's easy to compute, it gives a useful idea of how "close" a given point is to the Mandelbrot Set, and hence gives a good idea of the true shape of the set. This function illustrates how regions of the Mandelbrot set are influenced by nearby satellites. It does not show the shape of the set itself, but it is of interest when exploring the relationships between structures within the set.
Note that when using this function, the bailout value should be set to a high value to obtain a detailed plot. Each of the infinitely many points in and around the Mandelbrot set has a corresponding Julia set Julia Sets Wikipedia article on Julia sets.
FracTest allows these Julia sets to be computed. The difference is that z starts at c , the screen point we're displaying; but the additive term is r , the Julia set reference point, which is the same for all points on the screen. The main effort by far in generating a fractal view is mathematical calculations, and many of them. A typical high-resolution view can easily involve many trillions of individual additions, subtractions, and multiplications, which might take hours or days.
Making these as fast as possible is obviously a key goal. However, the more deeply you zoom into the fractal, the more digits are required to correctly calculate the co-ordinates of the individual pixels so that they can be computed correctly.
Fractal Mode (Mode, book 2) by Piers Anthony
As you zoom deeper, you need more digits. The fastest way to do maths in a computer is by using its built-in arithmetic hardware. This is very fast, since it ultimately comes down to networks of transistors wired to produce the result electrically; however, the number of digits in the result is strictly limited. Specifically, the data type is called double , and it has a precision of 53 bits, which is around decimal digits. Any attempt to zoom farther will produce pixellated images. This might seem like a pretty remote limit, but in fact when exploring Mandelbrot in depth, it's pretty easy to get that far in and hit the wall.
Since FracTest is designed for in-depth exploration, it's important to have a means to perform arithmetic with more precision.
The answer is software arithmetic, where software is used to perform calculations instead of the machine's hardware. And it only gets slower as you zoom in, and the number of digits you need goes up. FracTest has two software maths packages, each offering the best available performance at particular resolutions.
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So in all, there are 3 modes for arithmetic:. Fixed mode is a little complicated. An individual Fixed value has a fixed precision, i. A Fixed value can be created at any size, but then stays at that size as we do arithmetic on it. As you zoom deeper, FracTest will automatically create Fixed values with larger precisions. If you leave the Math Mode at AUTO , this pretty much takes care of itself; but if you set the mode manually, remember that you will need to specify the appropriate precision as you zoom deeper.
The resolution indicator in the main window will show you how much precision you have left at the current mode. A practical consequence of all this is that if you leave the Math Mode at AUTO which is generally a good idea , then as you zoom in you will hit a point where the render suddenly becomes much slower, as FracTest switches to software arithmetic. It will then keep on getting slower as you zoom deeper.
An image computed at a depth of 10 -6, , using 21,bit maths. In practice, you will never reach this resolution, as it would take an unreasonably long time to compute fractals at that depth. The available precision will be enough for any reasonable purpose. Even at the very low resolution of x pixels, this took over 3. It's a widely-held belief that computers keep getting faster; and like many widely-held beliefs, it's wrong.
Pentium 4 Prescott CPUs reached 3. Synopsis: Four people with special qualities are brought across parallel universes to help their fifth member, Nona, the ninth child of a ninth child, overthrow the despots of her home universe and deliver power over to women.
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Title: Fractal Mode Mode, No. Book Description Ace. Mass Market Paperback. Condition: GOOD. Spine creases, wear to binding and pages from reading.
May contain limited notes, underlining or highlighting that does affect the text. Accessories such as CD, codes, toys, may not be included. Seller Inventory More information about this seller Contact this seller. Add to Basket. Light rubbing wear to cover, spine and page edges. Very minimal writing or notations in margins not affecting the text. Install using Pkg. This package provides: fatou , juliafill , mandelbrot , newton , basin , plot , and orbit ; along with various internal functionality using Reduce and Julia expressions to help compute Fatou.
FilledSet efficiently. Full documentation is included. The fatou function can be applied to a Fatou. Define object to produce a Fatou. FilledSet , which can then be passed as an argument to the plot function of PyPlot. Creation of Fatou. Define objects is done via passing a function expression in variables z , c and optional keyword arguments to juliafill , mandelbrot , and newton.