(Part Nineteen)

We now conlude our investigation concerning the generation of group elements of the Plane Groups.

In the previous eighteen documents we have generated the group elements of the 17 Plane Groups, i.e. of the seventeen fundamental two-dimensional periodic patterns. We have interpreted these patterns as two-dimensional *crystals*. Although our goal was to give the *conceptual* generation of *real* (3-dimensional) crystals, we have done this by means of considering imaginary *two-dimensional crystals*. They give us already the basic features that we can expect in real crystals. The latter are much more complicated in their internal structure, and moreover very hard to illustrate graphically than their two-dimensional analogues.

There are exactly 17 different symmetry groups describing two-dimensional periodic patterns (and thus two-dimensional crystals). All of them have been treated in the previous documents.

So because of this definite number, these Plane Groups form a self-contained whole, just like each symmetry group in itself also forms a self-contained whole (Each group represents a certain algebra). This causes them to be *objective* structures in the sense that they must subsist 'somewhere' in some immaterial domain. The latter was described as the Implicate Order or (generative) orders, or, metaphysically, as 'hypostases', which are metaphysical levels, representing stages of 'emanation' from the ultimate source, which is called **'The One'**. The latter is __one__ in the sense of being an *absolute unity* without any parts or distinguishable features. There is a certain dynamic of outflowing from, and return to, **The One**. The outflow results in implicate orders with an increasing degree of fragmentation, until, finally, Prime Matter -- which is 'all the way down', and which in itself is pure potentiality for receiving form -- is being informed, resulting in *material substances* that are each for themselves in the strongest way possible separately existing, i.e. separate from (but interacting with) other such material substances. This material level is the Explicate Order. As has been said, the outflow from **The One** results in *implicate orders*. This is, however not a process of enfolding, but a process of unfolding (because **The One** is 'hyper-implicate'). This unfolding continues until it has arrived at prime matter. The latter is then informed, resulting in maximally unfolded structures. When, on the other hand, we look, from material structures -- informed matter -- in the reverse direction of this emanation (the return to **The One**), then we see these material structures becoming enfolded again and so becoming part of the Implicate Order. This enfolding intensifies in approaching **The One**, until these structures are totally absorbed by the latter.

The crystallization process can be described within the Explicate Order with a high degree of approximation (to the real process), but in crystallization there are phenomena that seem to point to a certain degree of 'cooperation' between the relevant chemical participants, in a way that seems not to be explicable in terms of chemistry and physics as we see them in the Explicate Order. These phenomena could therefore be indications that a deeper level is involved in determining aspects of the crystallization process. This involvement is such that we experience it as 'cooperation'.

In this crystallization process we have concentrated on one aspect only, namely the *internal symmetry* of the emerging and growing crystal. Together with the nature of the chemical participants, symmetry determines much of the emerging structure of a crystal. This internal symmetry can be described by exactly 230 Space Groups for real crystals, while our two-dimensional analogues can be described by exactly 17 Plane Groups. In considering these analogues we have assumed that their symmetry is in an abstract and enfolded way already subsisting in some implicate order or hypostasis. We can imagine the way it is so subsisting, as being present as symmetry groups in the form of a thought process (noetic process), i.e. as continually self-generating symmetry groups. In the eighteen previous documents we have described a version of this noetic process for each Plane Group. And this immaterial process corresponds to the mentioned deeper level, which is responsible for the crystallization processes as we see them happening in the Explicate Order, and which *reveals* itself by the phenomena of 'cooperation'. And because the 17 Plane Groups form a self-contained logical whole, we have treated them all, i.e. we have shown the noetic generation process for all of them.

We will now summarize the 17 fundamental two-dimensional patterns according to the 17 Plane Groups, and the way those patterns can be partitioned in tesselating areas that represent group elements.

These are then the 17 periodic patterns partitioned according to their group elements.

In the

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