Deterministic Algorithm

Note: This is also called the Multiple Reduction Copy Machine (MRCM) algorithm in some books.

Make sure that Deterministic is chosen in the Draw menu. The deterministic algorithm will take an initial compact set A displayed in the Fractal Window and apply the function

F(A) = f1(A) U f2(A) U ... U fn(A)

where f1, f2,...,fn are the function in the IFS. It will then plot the new set F(A). The next iteration will plot F2(A) = F(F(A)). Continued iteration will produce a sequence of sets A, F(A), F2(A), F3(A),... which will converge to a fixed point in the Hausdorff topology on the set of compact sets in the plane if each function is a contractive mapping. This fixed point is the attractor of the IFS.

You must first choose an initial set for the iteration. This is done from the Draw/Add Picture submenu. You can use the current polygon from the Design Window (this is the default choice the first time you choose the deterministic option). You can display the current polygon with a fill color. You may also start with a rectangle, a triangle (equilateral or right), a regular polygon of up to nine sides, or a circle. Each of these choices will bring up a dialog box in which you can choose the size of the rectangle, triangle, polygon, or circle, the style (outline or filled), and the color. You can change the color by clicking in the color box and choosing a new color. Another option is to load a bitmap, GIF, of JPEG file. It is also possible to paste an image from the clipboard.

If Colors/Use IFS Colors is checked, then fk(A) will be plotted using the corresponding color for function k. If Colors/Use Image Colors is checked, then fk(A) will be plotted using the colors of the set A. If you add a picture to the Fractal window from a file, Use Image Colors will be the default choice for the color scheme.


 Use IFS Colors
 Use Image Colors

If Overlay Images are checked, each new set will be drawn on top of the previous sets. The color used for each iteration is determined by the current IFS color palette. You can edit the palette and save it, then load that palette at a later time.