Abstract: An algorithm is presented which, given the parameters of an Iterated Function System (IFS) which uses affine maps, constructs a closed ball which completely contains the attractor set of the EFS. These bounding balls are at least as small as those computed by existing methods, and are sometimes much smaller. The algorithm is numerical in form, involving the optimisation of centre-point and radius relationships between the overall bounding ball and a set of smaller, contained balls which are derived by analysis of the contractive maps of the IFS. The algorithm is well-behaved, in that although it converges toward an optimal ball which it only achieves in the limit, the process may still be stopped after any finite number of steps, with a guarantee that the sub-optimal ball which is returned will still bound the attractor.