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), the fill color and line color, and the line thickness. You can change the fill color by clicking in the color box or button and choosing a new color. If you right click on the color box or button, the fill color will also be used for the line 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.

When the deterministic method is chosen in the Draw menu, you can choose New/Deterministic Fractal Movie to open a dialog box to construct a sequence of movie frames that step through a sequence of iterations. This is saved to a file as an animated gif and can be viewed in the Fractal Movie Viewer. See the individual help file for creating a deterministic fractal movie for more details.