# Logarithmic Rotation in the Roots of the Unit Circle

I have dodged around it long enough: the conjugation of space establishes a factorisation of conjugates that are mutually independent. Therefore they may serve as elements in any reference frame, that is as independent vectors or magnitudes or portions that measure space either as proportions or as comparasisons. We learn to reference a point in space by using independent measures. Such measures themselves may be conjuncted. The consequence of such a conjunction is unimaginable!

However, let us say that the reference tool is consistent of independent measures which are portions of a sphere. Then these portions when conjuncted will form a sphere. The portions are themselves differentials of the sphere. The conjunctions, combinatorially are integrals of the sphere. Uuch integrals therefore form the sphere piece wise sequentially. Developing the sphere in this fashion develops a spherical motion which is unusual but progressive. The conjunction of these integrals therefore represents growth of spherical attributes, from lineal rotation to surface chimatics, to probability density distributions, to solid space deformations.

On the other hand the differentials of the sphere provide opportunity to determine maximal and minimal differential changes which will equate to maximal and minimal proportions or portion measure for a metron within a given conjugacy system.
The differentials of the sphere form the basis of the Lagrangian for a spherical system, each distinguished portion being a distinguished differential. The conjunction of such differentials gives rise to the notion of a chain rule for the differential of the whole spherical space. Such a differential has no meaning beyond a maximal or minimal space, thus such a conjunction is suited to principles of least action or principles of maximal action. For instance surface tension may be addressed in this way, or the maximal pressure a sphere may endure before exploding!

What I am conjuring here is what Grassmann called the Verknupfunste form of any form. Such a state for a form allows the fundamentals of differentials, integrals, portions, proportions and ratios to be inherent within any form, and to be congruent to any form dynamically in a moment of Enstehenden, that is substantiation, or instanciation. Such a moment is capable of 2 descriptions: procreative instanciation or Erzeugung, or piece wise linking of constituents until completion brings the moment of Instanciation or Enstehenden.
It is merely presumed that one is continuous while the other is discrete and combinatorial, that one form is whole and entire or Steten while the other is discretely joined or Verknüpften. In any case it serves us to use one terminology as an analogous labelling of the other, and so to naively formulate the Akte or actions in these simpler terms, replacing this description by the more complex discrete and combinatorial one only when necessary.

The Lagrangian therefore begins as a identification of the complex integral and differential formula with the naive forms. Then we manipulate the forms as desired, and finally unpack the formulae accordingly and solve. Grassmann's analytical method based on Lagrange and Laplace means that forms become labels for complexes of formulae. If the parameters are thus constructed, deforming the form deforms the formulaic complex and leads to the solutions required.

Thus conjugation leads, on metrication, to an inevitable logarithmic description of space by metron. Therefore we can only sensibly work locally and or choose a sensible scale to make our results "Archimedian", that is always commensurable, never infinite.

Today infinite processes are routinely dealt with, but in an unintelligible way, because the conception is weak. The strongest conception we have is based on metrication. Even finite but very large is problematical,

Thus choosing the right scale/ metron or Maasstab is an essential part of Grassmann's analytical method.

It occurs to me also, that if Vielenfach is Grassmann's early conception of the Mannigfältigkeit that Riemann proposed, then in fact we have a direct translation of Euclid's pollapleisios into Vielenfach and then into which means that there is a direct link between the Arithmoi and the Manifold concepts.

Returning now to the conjugation of the periphery of a disc, each conjugate represents a portion or a differential of the periphery, and introducing a matron leads to a logarithmic conjugation of the periphery. The conjunction of these independent parts or intervals leads to a logarithmic progression. This logarithmic progression takes one in a motion around the periphery. We therefore have a conjunction of a metron conjugate that logarithmically is equivalent to rotation
This is the fundamental mathematical model of rotating space. By conjuncting independent portions of the circles periphery we can describe changes in rotational state around specific axes or axles.

This of course is not "true" rotation. We are merely using a cyclical property of the conjugate of a periphery to model distinguished states. Only by utilising a consistent frame of reference, a consistent formalism and a consistent process can we encode rotation in this way.

Suddenly our notation, our coordinate choices, out conjugation, the proportions, the integrals and differentials become clay in our "hand" to model space consistently and elegantly by encoding and decoding in a particular way.
The nitty gritty is complex, or as complex as we want it to be! So over time we have developed less complex terminology to label each part of the process, and in addition, the really complex iterative part we can hand off to a processor, which can perform this logarithmic cycling very easily very quickly and mechanically. By other "magic" we can flash these up onto a screen and demonstrate animated rotation!
There are other ways we can animate rotation and that is using the trig ratios. The Euler Cotes identity therefore is a rotation identity. It says rotation may be achieved either by logarithmic conjugacy or by trigonometric parametrisation. The insertion of i into the identity is to distinguish the parameters, with the driver parameter thus being emphasised as ø.

This is but one use of the conjugacy and adjugacy that is the fundamental property of the children of Shunya