Rotation for me is the fundamental motion.
The experience of rotation is perhaps traditionally not the first notion of a philosophical analysis, and consequently analyses go down a straight road. The special nature of the notion of straight is obvious but it is ignored culturally and philosophically. Our experience is that everything demonstrates curvature. The substrate or manifold of our experience is arbitrarily curved. I call it fractal curvature, which means curvature at all scales with "almost similarities" involved. This is the reason why the straight stands out!
So curvature is complex. it is made more complex by curvature being experienced as a continuum and or as discrete regional experiences. The identification of the sphere and the vortex, and their connection through the torus provides distinctive curve forms that can be used to deconstruct curvature.
The analysis of the Sphere began Aeons ago, But the analysis of the vortex seems to be traceable back to the ancient Greeks.
The modern presentation of greek ideas i am referring to Newton, who deconstructed the notion of the sphere and the Vortex into 8 principles.
The first principle is the centre of a spherical region being a "dual" region from any 2 (dual) regions on the spherical surface region that is identifiable by its uniqueness which is that it is the same dual region for any 2 regions on the surface.
by iteration we can refine these regions to a mysterious entity which we imagine as a infinitesimal region and we give it the name "point".
We are then able to understand extension as being the spherical surface that surrounds a specific point taken as a reference point. This means that extension is essentially defined as spherical surfaces from a centre.
Now the intersection of 2 spheres(dual) to define a plane of dual points requires first the intersection of 3 arbitrary spheres whose centres lie on the intersection points(dual of 2 spheres. These intersection points form a curved line which we will call a circle. By taking any 3 arbitrary dual points on this circles as centres of arbitrary spheres we can construct a pair of unique riple points of intersection. These triple points for all arbitrary triples of spheres form a set of triple points in space which we call a straight line.
using the straight line we may construct a circular plane with the initial dual points of intersection and isolate that triple point in that plane as the centre of the circle..
With this construction i can now define the radius of a sphere and a circle, using straight lines.
I can now redefine extension and the properties of a sphere in terms of these radii.
This early or fundamental construction of the straight lines of so called geometry are not explicitly stated. Therefore, every notion of symmetry and similarity and congruence, including proportion that derives from the sphere is "lost" to the modern mind.
Newton went further in describing the dynamical situation . The circle was defined dynamically by competing motives, centripetal and centrifugal and tangential and circular. The Idea is Archimedian who described the more general spiral as tangential, circular and centrifugal force dynamics, that is there is a net force centrifugally and circularly and thus tangentially. Consequently an inward spiral is described as a net centripetal force dynamic, involving circular and tangential forces.
Newton's general law of motion requires equal and opposite reaction forces, but this applies to static and dynamic equilibria. Clearly in the dynamic situation some forces are not balanced by equal ones, but his fourth law of motion discussing inertial motions places the dynamical description in the same system of the staic one, where the equal and opposite forces now become subsumed inertial ones.
Newtons general analysis of motion is therefore based on the 8 inertial forces centripetal and centrifugal, clockwise and anticlockwise circularity, tangential clockwise and anticlockwise, and finally resultant and anti resultant forces.
In the dynamic system these motives are represented by lines which are fluents. some will clearly be instantaneous. the notion of constancy is therefore derived from the notion of instantaneous motion.
Although the notion of straight line motion is propounded as Newtonian , i have to point out that the notion of "right" defined as straight refers to the definition of a straight line by dual or triple points. It is therefore a common greek notion of a constructed line! it is consequently not a natural law but a construction used to define a method of measurement.
Clearly no moving object ever moves in a straight line unless forced to.
For Newton this was a statement of a well known Mechanical principle from which philosophical analysis could proceed. The point of Newton's laws was to set the mechanical situation on a sound footing. His first 4 laws introduce a curvilineal, fluent space., which we call an inertial frame.
The modelling of rotation is based on this system. We use 2 functions principally the sine (trigonometric) ones and the logarithmic (exponential) ones.
In both cases we model the centrifugal and centripetal forces only. By plotting these two force or displacement patterns on orthogonal axes we mark outthe point of a quarter turn. if the axes are not orthogonal we mark out an arc of some circle or curve. To make the functions model a full circle we have to set ou some rules for constructing a circle of points. this usually means stipulating ranges and sequences, decision points (or flip points)and whaaaaat values are specified.
The values of any function are specified. The word function derives its meaning historically from the prussian notion of function in society, Tables of values had a specified function. Such tables formed the basis of function theory. The sine function is specified up to π/2 from comparison of lenths in right triangles. To model rotation it was necessary to specify the triangles be in a circle drawn from the centre. With this construction rules of arcs and sign were specified. A rotation is usually specified from -π to π, these two valuesbeing identical. The decision to switch π to -π+∂the angle or arc is ∂ bigger than π is what keeps the whole ball rolling.
The sine function just happens to be defined on the circle, but so can certain other functions, namely logarithms and exponentials. We can therefore specify these two functions as centripetal and centrifugal axial motions. By a similar set of specifications we can modell a rotation
the difference between these models and the circular force is profound. Our models are not complete and are non Newtonian. We often miss out the circular force, claiming it is the same s the tangential because Newton said so. Newton did not say this. in fact he explicitly goes against this line of reasoning in his Lemmae. His argument is that the circular force cannot be equal to the tangential force because this would mean he could demonstrate that a curve becomes discontinuous if it loses its curvature. His point was that the ratios of the bounding quantities approached equality, not that the curve and the lines become one and the same!
Our analysis and construction of curvature therefore is often at odds with reality. We often attempt to dismiss curved force or force of curvature as an artefact. In point of fact the only artefacts are the radial lines and the tangents introduced to help us get a better apprehension of curved motion.
Thus it is straightforward to establish a Grassmann Ausdehnungs Groesse for curved motion , and i have made some naive attempts in the Twistor research i have posted last year.