It is my contention that spheres and circular discs formed a prior philosophy upon which Thales drew and taught many concepts of proportion or Logos Analogos relationships of space. However to demonstrate certain transformations of spatial boundaries as being dual requires the flexibility found in material space as not being created or destroyed. The conservation of matter is a common everyday occurrence that it is taken for granted, but in fact material mediums as all alchemists know do not conserve there shape texture and even volume during and after a chemical reaction.
It was a bold proposal that states that the quantity of matter is conserved in a chemical reaction. While it is not drawn attention to this quantity of matter is an Alchemical concept used and defined by Newton as a measure related to the ratio of motives of material and air, and the measure of the volume of that ratio for the bulk of each object.
Enclosing an object in a sturdy glass container allowed one to assume that all quantity of matter in that volume was isolated. . No matter what was in that volume a relative density was assigned to it via the prescribed calculation. For this reason material was not distinguished and became called Mass.
However careful chemists were able to use the mass as a stepping stone to tabulate pure materials and so characterise them. Then the reacting of several materials together in a sealed container could be shown not to alter this mass characteristic, despite the materials being utterly transformed!
The quantity of matter is part of a chain of concepts Newton established to describe a centripetal/ centrifugal measurement system.. It related every part of an isolated system through these force measures
Conservation of space however has a astrological use, where it is assumed that space is absolute, isolated and still. . With such a conception a boundary change does not affect space. In that regard space passes Tintoretto and out of any dynamic boundary, at least absolute space does. Fractal space behaves differently, and do fractal geometry is different.
Suppose now a circle is squashed, then the space that was inside now stands outside the new topological boundary. However for a ” real” object that space would be deformed within the new boundary , possibly acting to restore the boundary to its initial position.
Using absolute space can I demonstrate that a sector is dual to a half of a rectangle formed by the circle radius and the sector arc length?
It is possible to establish this fairly simply providing absolute space is used, and a specific case involving the arc length equal to the circle diameter is used.
The concept requires also a secure definition of a good or true line sometimes confused with a straight line in other contexts where it is not necessarily the case. This good line must be the line of dual seemeia, or indicators of where arcs cross if drawn from 2 fixed centres , using the dual radii for each centre..
Given this good line a circular sector with arc length equal to I diameter of its circle is placed on the line and rolled so that every point on the arc is brought into contact with only one point on the line.. The centre or vertex of this sector moves one diameter tracing out a rectangle with the radius as the other side.
Conservation of space or absolute space means that the space in this rectangle passes through and or is contained within the arc sector.. Thus the rectangle contains some of all the space swept out by the arc sector. In addition all space has passed symmetrically through this rectangular figure, with space entering and leaving the arc at the same rate.relative to this rectangle.
At the end of this role a symmetrical figure represents the whole process. What we note is this symmetry allows us to half all the spaces in the diagram.the rectangle is created by the sector rolling, thus the whole space in the sector has passed into an through this rectangular space.. We see that there is as much outside this space at the beginning as there is at the end. . This rectangular space therefore has transferred its space into the rectangle . At one stage it was wholly inside the complete rectangle to be. So the space in the rectangle is 2 times the space within the sector. If any space was compressed into it the shape would not be symmetrical . It also took all of its fixed boundary o create this shape and mark out the rectangle..
A legitimate question is how do you know it is transformed into the rectangle? The answer is the arc sector is not transformed into the rectangle . It sweeps through the rectangle, creating the rectangle, thus it’s space places itself everywhere in the rectangle..the rectangle thus must be at least one of the 2 sectors in the symmetric form . If it was just less than 2 the sector would would not extend beyond symmetrically. If more than 2 again it would over extend. Precisely 2 retains symmetry and maintains conservation of space.
It is worth establishing this point with several reasonable or proportional examples, after which we may establish the discovery of Archimedes that an object must displace its own space in order to occupy a different space, or more dynamically an object rolling through space must displace multiples of its own space or multiples of its own space must pass through it.. In the case of a circular disc we may then establish a multiple factor of 1/4 the rectangle it cuts out to quantify its space. In the case of a sphere this reduces to 4/24 of the cuboid it cuts out.
The shapes of space formed when working directly with the circle disc or sphere I call Shunyasutras. The ” rectilineal” shapes are distinct because the good or true line is precisely good or true to itself. This good line is defined however by a property of circles that intersect because they are centred at precisely 2 centres.the point of intersection are called dual points. The dual points that define a good line or a true line are intersections of identical radial displacements from the 2 centres. Shapes of space with a true line fit together along those lines precisely or they don’t.. If they do not the shapes are not dual! It is this powerful notion of duality that is carried through from point to curve to line to shape of space as surface or volume.this notion of duality applies to the Shunyasutras, where a curved arc sector if it is dualed is also a good or true curve!
The straight line however has taken a dominant position in our psyche, despite the fact that it is never found in nature. It is a formal construct that dominates human evaluation schemes. This was not always the case. The circle played a deeper magical role in the past particularly in its rivalry with the spiral.
The absolute empirical dominance of the “spiral” shape and dynamic of space is remarkable when one realises how it is obscured from general metrical perception. In distant times past the circle grew to dominate magical lore and triumphed when it was believed to contain or constrain the spiral. Both these. We’re represented as dragons or serpents
However to measure with the circle always required skill and precise application. The precision of dual points as circle reference points came to dominate the practice of measuring with the circle. The straight lined ruler based on a right angled triangle within a semi circle became an indispensable tool of architecture. The lore of circles that underpins it was soon pushed out of the common thought and left for experts and philosophers to investigate. . Thus I have no intuition of duality of the Shunya ultras, how one shape may be used with a transformed one, and what topological constraints are required to declare duality or fit”.
In the case of arc sectors I can show how the rectangle formed between the arc rolled out ant the diameter contains 2 overlapping arc sectors. The question is does the overlap fit the space mot overlapped?to answer that question I have to employ symmetry observations and exhaustive comparisons, which call upon the notion of ” fit” I have established for the straight line.
The circular gnomon and Lunes play a historical part in the metrication of the Shunyasutras