How to Design a Symmetric Fractal
Symmetric fractals are created using
either the cyclic group Z_{n} of order n or the dihedral group D_{n} of order 2n. These are both symmetry groups. The group Z_{n} consists of counterclockwise rotations through angles that are multiples of 360°/n. The group D_{n} consists of symmetries of a regular polygon with n sides, including both the same rotations as Z_{n} and also reflections. The dihedral group D_{2} is an exceptional case, however, known as the Klein fourgroup. The figures below show the symmetry reflections for a triangle (D_{3}) and a square (D_{4}).
To design a symmetric fractal you start by either defining a contractive affine transformation to use as the base or selecting an IFS from the list in the Fractals menu.
In the first case, suppose f is the selected affine transformation. Let G be either a cyclic group or a dihedral group. Let g_{k} be the elements of the group G for k from 1 to order(G), the order of the group (either n or 2n). Then we take as the iterated function system the set of functions {g_{k}f : k = 1 to order(G)}, where g_{k}f is the composition of the symmetry g_{k} with the transformation f. The attractor for this IFS will have the symmetry corresponding to the group G. If instead you select an IFS in the Fractals menu to use in the construction, then the new IFS is obtained by applying the symmetry group multiplication just described to each of the functions in the selected IFS.
 Select Design\Examples\Symmetric Fractals
 Select either to use a base affine transformation or to use an IFS from the Fractals list.
 For the first case, enter the matrix values and translation vector for the base affine transformation.
 For the second case, select an IFS from the drop down menu. You can refresh the list if any changes have been made while the dialog box has been open.
 Select either the cyclic group Z_{n} or the dihedral group D_{n}, then select the value for n from the drop down list.
 Click on "Create IFS" when done. If the preview window is open you can see a rough approximation of the fractal.
 The best coloring for symmetric fractals is often obtained by using the Gradient Color Options (Edit/IFS Color Scheme menu) with pixel counting. This is because there are often many overlapping regions in a symmetric fractal constructed this way.
 You can copy the image in the picture box to the clipboard with the standard Copy menu command (or type ctrlC), or you can click in the picture box with the right mouse button to get a contextual menu with options to copy the image or save it to a file in gif, png, jpeg, or bitmap format.
Z_{5} symmetry

D_{4} symmetry (using Koch curve IFS)

For more information on symmetric fractals, see the book Symmetry in Chaos: A Search for Pattern in Mathematics, Art, and Nature (2nd Edition) by Michael Field and Martin Golubitsky, SIAM, 2009. [See Preview at Google Books.]