Strecken is…

A Strecken is a symbolic notation. The mark is usually a gramme a drawn line which is subjectively motivated to represent a line, or a direction or a length traveled, or a sequence or a duration or a flow or a motion, or an action. Such a symbol is manageable by language referents, letter referents and subjective feedback.

Grassmann in the law of 2 Streaks deliberately leaves the referent AC as a line, not a Streak. It was derived from a measurement situation where the result sought was a length and or a direction. The law of 2 streaks is not a vector law. It is a law for 3 points. It can be so easily analogised that the vector sum is derived from it by analogy. Any analogy has to be verified.

WE can use the law of 2 Streaks to represent the circular arc , particularly in the definition of the radian. AC is the arc, the 2 strecken AB and BC are symbols of the journey to and from the centre of the circle B . The additional restrictions on the Strecken comes on what lengths can be attached .

The sum of 2 Streaks is a line.
The product of 2 Streaks is a Form
The sum of a Streak and a Twist is a line of fixed magnitude in any orientation
The product of a Streak and a Twist is a sector form

This combination can also be used to define a Twist, starting at the centre a Streak followed by a Twist, gives a resultant line BC of a fixed length but varied relative orientation. The law of 2 streaks with a circular arc does not allow the 3 points to ever be on a line with C in between A and B. There is also a conflation in the notation when A, B, C are in a straight line. AC represents a semi circular arc, but it could double for the 2 Streaks as a line. Clearly this is a switch of referents, and this switch has to be explored and verified. At this Stage the referent is to arcs and Twists.

We can now look at the product. Remember the product is to find the area. The law of 3 strecken can be applied, but this only gives the triangular area not the sector area. We can however devise a modified law, The law of 2 Strecken and a twist.

The area is a twist AC*Strecken BC*1/2 = (AB + BC)*BC*1/2 = AB*BC*1/2 + BC*BC*1/2

This equates the area of a sector to the area of parallelograms.

As you can see, the formulations show the similarity between the circular area and the parallelogram area. Also the Action of the twist produces area. like the sliding of the streaks produces area. Again these are not vector formulations, but analogous vector forms can be derived.

The point here is that Grassmann is not doing anything other than applying the law of 3 streckens to formalise the method of "producting" twists and Strecken. This is not new, as The radian is used in precisely this way, but it is consistent with the Ausdehnungslehre approach.

In 1877 Grassmann wrote about the Ausdehnungslehre 1844 in a second impression foreword: "I kept it as it was , removing any typographical errors of course, as an example of how an initial idea is carried through to a State of expression as a Theory, how this process is handled, the insights and modifications it entails etc. Consequntly i have heavily annotated it with reference to the newer ideas and applications and expression in the 1862 version and developments even beyond that".

Grassmann's final words in that Auflage show him to be harried by the richness of developmental work that still remained before him . He died later that year.

These foundational laws are therefore in some sense supersceded, but i think never actually changed. To grasp these 2 simple laws is to be able to see the richness of their application in development.

The essential idea of Grassmann was development by iterating simple rules. This of course is the heart of Fractal Geometry.


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