SYMMETRY AND AFFINE IFS GRAPHICS

These graphics combine symmetry, affine transforms and iterative function systems (IFS).

An affine transform (a plane transform, that is) is of the form:

xnew = ax + by +c
ynew = dx + ey +f

where c and f are the displacements. If

a=cost
b=-sint
d= sint
e=cost

the transform is a rotation of the plane through t radians counter- clockwise (around the origin), otherwise there are scaling, shearing and/ or reflection involved, too. It is normally convenient to define a contractive transform (one which maps shapes to shapes of a smaller perimeter- possibly excluding the first iteration.) Then:

a2 + b2 < 1
d2 + e2 < 1

That's nearly all the math necessary! The apparent texture effect is an artefact of the algorithm: It stems from the interaction between the calculation step increment and finite screen resolution.

Attractive though they look, these graphics are not fractals: Sure, there is self- similarity at all scales and several orientations. But, the length of the boundary is not infinite; nor is there any noteworthy sensitivity to initial conditions, another hallmark of chaos. If you really want stunning, captivating graphics, try superimposing the central part of any of these images on the corresponding part of another image.

The site will feature two animated symmetric affine IFS graphics eventually- but don't hold your breath!



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