I have to acknowledge Normans introduction to vectors as a turning point
in understanding the basis vector idea. But i did need to understand Euclids use of the term "iso" (dual) to appreciate the full significance of the Circular disc{kuklos} in Euclid's treatment of his teaching material.

The importance of the parallel property cannot be over estimated in our stereographic(phorometric) development of geometry. It is not just linear property it is a planar property and it is a circular property.

The circle can be drawn using many rigid instruments, but perhaps the most important one is the right gnomon. The curved gnomon is based on a sector of the kuklos.

Using a right gnomon we can see immediately that the right angle touches the circle in only one point, the point of the circle periphery. We also see that for each point on the circle periphery, there is only one right angle. Thus the circle is a very special form indeed and it is clearly special form of spiral. All spirals that end on the periphery do so at right angles to the point of incidence in the limit or at some angle to that right angle. Curves that are constantly greater than π/4 in this angle spiral outward, curves less than π/4 spiral outwards and inwards and cross themselves, curves at π/2 stabilise into circles, curves greater than π/2 spiral inwards and outwards , etc.The interior of the disc can thus seemingly be a source of spirals which in fact originate due to this boundary condition.

The use of the periphery of the circle as a rotation measure is the standard radian measure of orientation, but as explained in the previous post the subjective process require one point as a goal to set an orientation. One might think that you need another point to start your measure from, and this would be the case in a 3 point system. But in an n point system, choosing the goalpoint esatablishes all the other relative orientations . Thus the goal point is in fact the defining point for radian measure, not the other way round! Thus when i choose a goal point i automatically set all the radian measures relative to it. Therefore my unconscious radian measure changes for each goal point i choose and again this experience i call relativity.

Now the previous post i established the parallel orientations as basis vectors in groups of n =2,3,… across the plane or regions of space.
However these lines of orientation radiate round the point. If instead of lines of radiation we drew concentric circles These circles will represent rotational motion lines and the movement around them i will call twistors.
Now the intersection of the rotational motion lines produces straight or curved lines of intersection resulting in straight or curvilinear vector motions along with rotational twistors.

However, the rotational motion lines are only one of the set of conic section motion lines that surround or shroud a point. The set of conic section curves , surfaces exist as special examples of vorticular shells. These motion surfaces define motion relative to a centre of rotation or 2 centres of rotation. and such surfaces exist for every point.

For a special selection of these vectors produced by rotational motion lines we can define a basis over the whole plane by parallel curvilinear or parallel straight lines.

Thus a more subtle basis may be established using concentric (parallel) cicles of rotational motion.


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