![]() P_vartype=z p_const=0/0 p_colorby=magnitude f_fofvar=ident Maxiter=1 percheck=off filename=”jp.ufm” entry=”pixeljp” Title=”Basic Pixel” width=500 height=500 layers=1Ĭaption=”Background” opacity=100 method=multipass Here are some parameters to copy and paste into Ultra Fractal so you can follow along, and the resulting image: It can be very useful for exploring coloring algorithms since it omits the complexity of fractal iterations. Set Maximum Iterations to 1 to make all the points inside or to 2 to make them outside. The one I use the most often is “Pixel (jp)” from jp.ufm, which generates an orbit with a single point for each pixel. There are several Pixel formulas available, with different properties. To understand the details of how Ultra Fractal works, let’s start with a much simpler formula called a Pixel formula, which just passes the point associated with a pixel to the coloring algorithm. This coloring won’t work for the inside since the orbit never escapes the inside coloring is Lyapunov, which computes a mathematical construct called the Lyapunov exponent to color the inside points. Outside points are colored using Smooth (Mandelbrot), which uses the escape time modified by the final z value to smooth the result. ![]() For example, the following fractal uses the same formula used to produce the stark black and white image above. These algorithms are independent of the fractal formula, so can be used in different combinations to produce various effects. Since the orbits for inside and outside points behave very differently, separate inside and outside coloring algorithms are used. A coloring algorithm is then used to compute what color the point should have. Ultra Fractal uses the fractal formula to compute the orbit and determine if a point is inside or outside. Note that the lines connecting the points are to show the order, and are not part of the orbit. The 14th point of the outside orbit is at the bottom right of the drawing, the 15th and all subsequent points are outside the drawing. Notice that the inside orbit points are always inside the figure and the outside orbit points are always outside the figure. The second points are the farthest points to the right, and are beginning to separate. The first points in the orbit are near the left edge, still very close to each other. The green arrow shows the initial points, which are very close to each other so overlap in the drawing. The inside points are black.The first few iterations of the orbits of two points, one inside (blue squares) and one outside (red diamonds) are shown. The above figure shows a representation of the formula (using the formula Julia). ![]() (Don’t confuse this with the z coordinates present in three dimensional systems Ultra Fractal only uses two dimensions.) An Ultra Fractal formula can either use z as a complex value (the language Ultra Fractal provides for writing formulas supports complex arithmetic and functions), or it can split the real and imaginary parts into x and y coordinates to compute the new value separately. ![]() This allows the point that is iterated to be stored in a single variable, named z. Ultra Fractal represents points as complex numbers: the real part is the x coordinate and the imaginary part is the y coordinate. Points that “escape” (usually by exceeding some threshold) are “outside” and points that don’t are “inside” the fractal. Be sure to load it into your copy of Ultra Fractal by running Options->Update Public Formulas.Īs we saw in Basic Fractal Concepts, escape time fractals are defined by iterating a formula on a set of points. There is an extensive public formula database that contains most of the formulas that will be covered in this blog. We’ll be focusing on the concepts and formulas here, not the mechanics of the user interface (Ultra Fractal comes with a set of tutorials to learn that). It is commercial software (a free trial version is available) available at. For the Fractal Formulas blog, we will be using the program Ultra Fractal for two dimensional escape time fractals. ![]()
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