In going from the gematria of the semeion to the gematria of the gramme we add one fundamentally important subjective processing tool: the motion trace.
This tool traces the path(Ω) of any selection of points in the sequential motion of the selection synchronously(usually) or otherwise. Such a trace is identifiable with the gramme, and thus a dynamic representation of every form in Euclid's Stoikeioon is revealed. Euclid is therefore never static!
We may identify this motion trace with Grassmann's idea of Strecken, and thus bring to the notion of Strecken a more general path than just a straight line or a cyclic dependency.
With motion trace we may see the independence of Length and Direction and indeed Orientation, and thus attain to the "true" or full freedom of subjective processing.
The notion of subjective vector herein advanced is defined on orientation with the transfer of information clearly being defined on direction: the path traveled in transference. There is no real implication that this path is a "straight line" although for convenience it may e thus represented, until that of course becomes or leads to substantive inaccuracy.
I may by differential means accommodate any path to a limit sum of such "straight lines", but this is done in order to apply calculation schemes based on straight lines to curved ones. If i may arrive at a calculation scheme based on curved lines i may apply it accordingly in a manner appropriate to accommodate it to the curve being investigated. Whichever is the simple in notation and process i may employ to th arriving at the relationships of form, whether they be in equilibrium, static or dynamic, or in ever changing flux of vorticity.
To this end, the attachment of the notion of monas to every aspect of form and to every aspect of motion trace will afford the user with a way of accounting or many relationships of form by a sequence of counting, and a construction of form with a sequence of ordering and arranging and translating, rotating relative to some semeion of reference or the subjective processing centre. To these actions magnification and deformation as well as the very special reflection through a centre of rotation, and a useful set of tools may e had to describe most dynamic forms and systems. The notion of length is but one monas that may be employed. Whereas the notion of motion race may be employed at all levels to give or remove extra"dimensionality" to a form to descrie some motion or position or situation or change.
Reflection occurs only in a centre of rotation. Thus a reflected image represents a summation over the centres of rotation of the reflected "ray". The processing of these "rays" to form the reflected image introduces its own processing order constraints.
It is to be added to the fundamental attributes of the subjective process in interacting with space, that the subjective vector, being equivalent to the semeia or analogous in strict terms, returns information by the contra path of the subjective vector and on the basis of the subjective vector orientation and focal distance etc. information is input into the subjective process. Thus the motion of the semeia is input via changes in the compass multivector network and on this asis path and direction which is not coequal or collinear with a subjective vector path is distinguished in the subjective process, and coincidence is distinguished.
Thus a set of attributes are inherent in the subjective process based upon orientation path and direction, information transference, and processing of sequential comparisons of the information. Special paths have been idenihfied and these go with special motions in the ever dynamic situation: translation, rotation and reflection throufg a centre of rotation. but what i draw attention to here, before continuing is that for any trace of motion collinearity must not obscure the contra direction of motion nor for any "point" as a semeion must coincidence obscure the differences in rotational status of a point. Thus 2 or more lines may be collinear, but stiil not be the same line, due to designations of direction, length,axis of rotation,etc, and more particulaly 2 or more points may be coincident but not equal due to rotation direction or status, or weight inherent within a point.
Thus we may proceed to deformation of form in space due to magnification and shear and bend and twist as attributes of motion in space. Motion trace is of singular assistance in these situations to add dimensionality to subjective processing that enhances cogent manipulation of form. Today motion trace is significantly enhanced by video capture of motion.