Sphere inversion fractals

79 UN CO RR EC Multiple methods exist to calculate and represent 3-D fractals. For example, spectacular, artistic images can be obtained using quaternion algebra [1]. In this article, I will briefly discuss three relatively simple methods to calculate fractal shapes consisting only of spheres. All these methods use iterative sphere inversions. A sphere inversion is the 3-D equivalent of a circle inversion. It is a transformation that maps the outside of a circle to the inside and vice versa. Circle inversions map circles to circles, and fractal shapes can be obtained by iterative inversions of a set of well-chosen initial circles in a set of inversion circles [2]. In Fig. 1, the blue circles are the inversion circles, and the green circles are the initial circles. Figs. 2 and 3 show the result after 1 and 5 iterations, respectively. Note that the initial circles do not overlap and, hence, that none of the calculated circles will overlap. A self-similar fractal pattern of Appolonian circles is generated. The same principle can be applied in three dimensions. A sphere inversion will map a sphere to a sphere. If the initial spheres do not overlap, then neither will any of the calculated spheres. A sphere that is orthogonal to an inversion sphere will not be affected by the inversion transformation.