Operands vs. Operations
When we choose to describe the symmetry of an object, We seek to define those transformations which appear to leave the object in a condition similar to the one we started with. We use translation, rotation and reflection almost without realising it. As we would imagining a checkerboard, circle or a simple figure of a face. The operations of symmetry do not depend on the orientation of the shape itself,.. merely those orientations that are transformed onto a similar orientation. Symmetry maps a shape onto itself, wheras mappings in general map a structure into another. Some mappings map onto another, and some mappings are not one to one.
The way to approach symmetry in the given model for the trinity is to consider throughout that If God needs to realise what it is like to have faith in a God such as himself, there must be at least three in the whole. The third must be the container within which the other two may have perfect understanding of the person of the two. So, any symmetry must be in cascade onto itself through the other two. This is not to say that structures like this need to be 1 to 1, but the Holy trinity is 1-1 in that we cease the cascade in similarity.
The difference between the operands and operators is this. Whilst we define non-commutativity as the difference between rotating and translating in one order or the other as within the dihedral groups, Each member of the trinity should have their own "group" of symmetry through which his own form may be commutatively found through either order of the other two. If the Holy Spirit is within the Father through the Son, and in the Son through the Father in a correspondence that leaves not only the Holy Spirit "fixed" in form without requiring a further operation, we can say that xy = yx. So, we do require a little thought here. There is no non- abelian group of order less than six and we require that the operation of one group of the trinity model be applied to the elements of the other two.
We could apply the operator of one such group member to operands made of the other two. If then the operation mapping upon those groups fixes the element initially applied as an operator in the first in a well defined manner, we are finished.
This seems very confusing, after all, the cascade is not an infinite one unless we require it to be viewed as such. If you have the time to consider that which a Lie group is, then do so. We dont require anything but the Trinity here, and were there a much larger way to view all such cascades, I would not have room for it here.
We have to start with our three "members" of the trinity. We have; the Klein four group, the Octal group and a cyclic group of order seven. Ideally we wuld wish to peceive the Klein four group as a set of subgroups of the Octal, which is easy enough for us to do. Finite fields allow us a little extra benefit in modelling the Octal combined with a cyclic group of order seven: but we do not wish to follow the standard description of finite fields in that we do not want to restrict ourselves to the concept of a fixed unity element, an element with the multiplicative properties of "one".
Also, within our model we have to examine what it means for a member to find himself transformed by the other two back onto an "image" of Himself. We will start in a rudimentary fashion and return to this as we progress.
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