Rotation is the most surprising action, especially when decentralized.
Within rotation is paralleism and parallax and synchronism and origami. The notion of vectors is encapsulated by rotation. Some experimentation i am doing is showing how rotation has generalised "number line" concepts into spatial vector concepts, and preserved the notion of combination.
Euclid when defining "multiplication" by arithmos used a gnomon, the essence of rotation,to place the combining monads. Thus 2+3 in vectors rotates, generating area volume and form with form relationships as it does.
Any 2 centres of rotation can rotate in "parallel", "synchronous", equiangular rate of spin, etc providing a dynamic basis fro relationships which are periodic and stable; providing the notion of reference frame; providiing the notion of transformational geometries, and finally the notion of sympathetic vibration and resonamce.
In the previous post i set out some formulae for rotation vectors. The consequence of iterating the yin yang formula is quite dramatic in Quasz. It also highlights the point that sign , although a useful shorthand is very misleading spatially. Vectors completely replace the need for sign, and vectors are of course combinations of the trig ratios /functions. Combinatorics needs to be emphasised over addition and subtraction which are specific forms of combination or aggregation, as i have called it while searching for the meaning of it. And this meaning is to be found in Kombinationslehre.
In any case the combination of 2 and 3 has to be specified by object and by vector, and the resultant is a spatial arrangement, or rather a selection of spatial arrangements, from which we choose as we need.
The compass multivector network thus specifies an arrangement of objects in space, and if these objects are contiguous or continuous then a single larger object id described, but if not then an arrangement of objects is described which may represent anything from a dust cloud to a solar system and beyond.
The rigidity of the relationships in the compass multivector network describes the additional attributes of the larger body with regard to flexibility fluidity and phase state etc.
The conservation of energy and mass principles may be described in terms of compass multivector networks, and the growth of a leaf and a child may similarly be described.
The data required to process these changes is huge, but computational tools not only make it possible, but feasible, and it is regularly achieved through modern computing platforms.
How do we teach this stuff? Surprisingly Euclid set out a method 2300 years ago, based on a platonic redaction of Pythagorean philosophy which is more than equal to the task. It has been called Stoikeioon ever since. Roughly translated it means "Data set out in an orderly fashion for teaching". The Data is about space, that is our subjective interaction with space..
Many geniuses have learned a thing or two by studying it.
The path notion arises in the contex of semeia as a dynamic referenc to a moving semeion. As a consequence it relates also to moving subjective vectors and thus a path can be described by a set of subjective vector p + p(). The set of subjective vectors can be seen as a set of dynamic forms p+p()+∂(p+p()), and a path may in some circumstances be describable by a function of subjectiv vectors Ω(s) where S is the subjective processing centre, which is also the hub for all subjective compass vector networks, which would be used in describing a path.
A path Ω = ∫∂(p + p()) which is a continuous or contiguous combination of all the transformations in the subjective vectors from p + p() to p' + p'().
The path notion is very similar to the trace ofa moving semeion, or a gramme, but a path is a subjective experience.
The circular path or the path around a closed object are special forms of a path, as are straight lines