It Almost Got Away

Grassmann was not pointing out a fact about area of parallelograms, but a fact about 3 Strecken connectd by being in the same figure. This figure involves a "triangle of Strecken, even if one side has to be produced to form it. In such a configuration the sum of any 2 "dynamically adjacent " strecken multiplied by a third Strecken always gives the same result no matter what connected strecken is chosen. This is in fact the property od areas of figures between prallel lines, and thus the importance and relevance of the parallelogram over the triangle. But the configuration must always include a triangle. Thus the law od 2 streaks and the law of 3 streaks in a parallelogram are combined.

I ALWAYS HAVE TO REMIND MYSELF THAT THE BOARD OF GEOMETERS, long since subsumed it seems into the general mathematical Board, seems to have imparted the idea that Elementary geometry should be taught as Static, when it is quite clear that it is dynamic. I also have to remind myself that mathematics tends ,due to this to be overbearingly visual, and magnitude is thought of spatially, visually and as objective. However when one removes the visual emphasis the whole theory crashes down to its real foundations: a combination of dynamic space and spatial intensity which is entirely subjective.

Thus, to bear this in mind at all times connects and conducts one not to the rulings of the Board of mathematicians, but to the Combinatorial and divisional outworkings of the natural context.

At this early stage in his thinking Grassmann had not appreciated the combinatorial decisions he was making, and how radically different they were. I have written before about the confusion in multiplication, and the basic combinatorial form called the multiole form in Euclid, and how prime factors of multip;e forms were distinguished, and how some exact forms were distinguished, by symmetry, from approximate forms. In this light Grassmann is following Euclid without realising, because he is following the natural combinatorics of our spatial context, not the board of mathematical governors.

The 2 streaks law and the 3 streaks law mean you have to think differently about geometry processing. One has to always notice the dynamism and use it.

The grand Pythagorean scheme is division of The monad into The Henads, and the return of the henads to the monad by Combination. So the 2 fundamental actions are division and combination. From these one derives instances called subtraction and addittion, and patterned rhythms called "division by a factor" and "multiplication by 2 factors".
Within and beyond that is the rotation and translation of the henads in these division and combinatorial actions, and proportional relationships.

Today i can define {-, +, i, mod(n)} as a set of combinatorial actions which can define all other combinatorial actions and notations. These are actions on the Henads, subjectively processed, and the process reults are both subjective and obective, symbolic and real. However not all subjective processes produce objective results directly, or act on the henads exclusively. Some act directly and specifically on the subjective process itself.

The subjective processing actions i have not researched fully, but would have to use computer programmng language terminology to fully describe, and as a rough guide Html, or Xml will serve as a rough model. I do not propose to go into it any deeper than i need to .

Logarithms and comparison are essential to the scheme and so i may nodify the set of actions accordingly when i establish a clear motion sequence that warrants it for these additional actions. In the meantime they sit on the edge between combinatorial actions objevtively and subjective processing actions.

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