and therefore it must be the fiber of the wreath product in which the control group creates the subsequent generative process.

Let us take the control group to be the affine group on three-dimensional real space.²

The full structure, fiber plus control, is therefore the following wreath product:

Now, it is necessary to fix the group representation of this wreath product. First, by our theory of recoverability, the control group must have an asymmetrizing action. Thus proceed as follows: The particular fiber-group copy

corresponding to the identity element in the affine control group, must be the most symmetrical configuration possible. This exists only when the cube and the cylinder are coincident, with their symmetry structures maximally aligned. It will be called the alignment kernel.

Next, choose one of the two objects to be a reference object. This will remain fixed at the origin of the coordinate system. Let us choose the cube as the referent. Given this, now describe the action of the affine control group as providing an affine motion of the cylinder relative to the cube. Each fiber-group copy

for some member , of the control affine group, is therefore an arrangement of this system. In fact, any fiber copy will be called a configuration of the system. For example, figure 15 corresponds to a configuration. The crucial concept is this: The role of the affine control group is to transfer configurations onto configurations. The wreath product we have presented:

gives the complete symmetry group of the concatenated situation. It has all the internal symmetries of the objects individually, as well as their relationships.

Let us now understand how to add a further object, for example a sphere. First of all, the fiber becomes the following, with the added sphere group:

In such expressions, our rule will be that each object, encoded along this sequence, provides the reference for its left-subsequence of objects. Thus the cube is the referent for the cylinder-sphere pair, and the cylinder is the referent for the sphere.

Accordingly, there are now two levels of control, each of which is the affine group , and each of which is added via a wreath product. Thus we obtain the 3-level wreath product: