#isotropy

Earlier to Later Heaven: Fugue VII Beyond Descartes - Part 2
A Different Zero

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Mandalic geometry has been formulated in such a manner as to be fully commensurate with Descartes’ coordinate system. Firstly, because it can be.  Beyond that,  because Descartes’ system is known throughout the world, and is endorsed by all conversant in disparate fields of science and mathematics. Moreover, the Cartesian coordinate system is a special case of the mandalic coordinate system,  bearing a relationship to it analogous to that which Newtonian mechanics does to quantum mechanics.

One of the fundamental differences lies in the way the two regard zero locations. Descartes, taking his cue from the Western number line, constructs a coordinate system which envisages a single common origin to all three dimensions, while maintaining between those dimensions a rigid uncompromising distinction. Mandalic geometry views dimension as primary rather than points, lines, or two or three dimensional figures. It does not regard dimensions as intrinsically separate in the manner in which they  exist and relate  to one another.  This allows for a far greater degree of flexibility of what we view as parts of the system, including the possibility of folding each into another,  through different dimensions as well as the same dimension.

For Descartes, zero is the empty location, the no man’s land where positive and negative vectors of each dimension invert or fail to invert.  A negative vector acting on a positive vector or another negative vector will cause inversion.  A positive vector, acting on a negative vector or another positive vector, will not. For mandalic geometry, zeros are that, but more. They are dimension interchange lanes,  and also locations of dimensional amplitudetransition.^[1]

Descartes, influenced still by the number line, proceeds to build a geometric universe based largely on scale. It is an imposing edifice nearly purely divergent,  constructed from three largely independent linear axes of evolutionary zeal.  Taoist cosmology and mandalic coordinates equally eschew an impressive but mundane number line in pursuance of complex twisting and intertwining of parts evolved on the underlying principles of modularity, repetition, reflection, relationship and recursion.^[2]

These are two very different universes of logic.  Descartes’ approach leads to a description of space as being homogenous, isotropic, and fixed while that of mandalic geometry leads alternatively to a spacetime which is inhomogeneous, anisotropic and dynamically variable.^[3] For Descartes space is a background arena,  the theater in which all events transpire.^[4] For mandalic geometry,  space-time is foreground and background both. It is the sole ground which defines the nature of reality.

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Notes

[1] The first,  dimensional interchanges,  occur in the Cartesian coordinate system but are generally neither recognized nor treated as such. Dimensional amplitude transition locations do not occur in Cartesian coordinates,  nor are they found in the simple 3D trigram Cartesian equivalent,  reproduced in the upper diagram above, as they are a manifestation only of compositing of two or more dimensions. They are attributes of all hybrid composite dimensional systems,  for our purposes here, either the 6D/3D hybrid mandalic system of hexagrams,  the 4D/2D hybrid mandalic system of tetragrams,  or the 2D/1D hybrid mandalic system of bigrams.

[2] An important consequence here is that Descartes’ number line-based axes each contain a single zero. When mandalic coordinates are scaled up beyond the basic modular unit, every even number maintains all characteristics of the initial zero, including, most significantly, its multipotentiality. This is a basic axiomatic result of the intermingling, sharing nature of mandalic structure.

[3] It is this variability and dynamism of mandalic coordinates that make the method potentially suitable to mappings of subatomic particles as these are similarly variable and dynamic,  sharing importantly also the ability of exchanges / interchanges among their diverse numbers.

[4] Witness for example how Descartes exploits his newly formed coordinate system to stage, what was then, a cutting-edge geometric exposition of algebra, now referred to as analytic geometry. Mandalic geometry employs coordinates which are pre-invested with the ability to directly impart information regarding spatial transmutations themselves, without requirement of any intermediary.

© 2015 Martin Hauser

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