A flower like complex attractor.

Symmetric Icons

Symmetric icon is a name often given to symmetrical strange attractors, which Spirofractal creates using complex polynomials, formulas that involve calculating powers of complex numbers. They are called icons because a single, isolated, image is generated, in contrast to tiles (or symmetric quilts), where the pattern produced can be endlessly repeated without any seams showing. Spirofractal does not use this term: one reason is that symmetric C.A.T. fractal images could equally well be called symmetric icons. The other is that not all images of this type are symmetrical:

• The degree of symmetry is related to the degree of (the highest power in) the polynomial. If this is too low there is no rotational symmetry. You can find out more about in this on the Asymmetric Icons page.
• There are also cases where where a formula that would normally generate a symmetric icon breaks down, and generates an asymmetric one instead. See below.

Symmetric icons can vary almost as much as fractals in their appearance: perhaps most are approximately circular, but many approach the shape of a regular polygon. There are very often rays along the axes of symmetry, and these tend to be heavily visited, and sometimes almost stable: many iterations will go along a single ray, before the process branches off, seemingly at random.

Star shapes also occur, and various compound shapes.

The centre of an attractor is the other main area of variation. Sometimes there is a flower-like structure, sometimes a hole.These two yellow and green attractors are closely related. The parameters used to create them were slightly different so that the loops pointing towards the centre are big enough to meet in one and not the other.

The addition or loss of places where parts of an attractor intersect almost always makes a big difference to the appearance of the attractor. It can even cause the symmetry to disappear. (When this happens starting the iteration from another value would one of the missing parts of the attractor.) This loss of symmetry is another reason why Spirofractal uses the term Complex Attractor instead of Symmetric Icon.

You will notice that the asymmetrical attractor above is in more than one piece, or not connected in mathematical terms. This often happens when symmetry is lost, but this can happen anyway.

Although strange attractors usually look as though they fill a substantial part of the area they occupy, the reality is that at high magnification a network of almost parallel fine lines usually becomes apparent. This is most apparent in the attractors which resemble macrame.

 On to Asymmetric Icons