Space-filling curve, it has many variants; i.e.
*Hilbert space-filling curve
*Moore curve
*Peano Curve
*Z-order curve
*Sierpiński curve
The concept you?re allegedly pursuing is that a point in Hausdorff space is isomorphic to a point in a continuous line segment.
I doubt the formulas or theorems would be of any use to you. But there are java applications and other programs that let you simulate them in any number of arbitrary dimensions.
Not sure if you are demonstrating an instance of Self-deprecating humour, pessimism or cynicism with your introduction. Here is word that hopefully matches your criteria for ?actual academic research? as you requested.
Space-filling curves are mathematics concept in the field of topology, not physics as it does not accurately convey the current academic understanding of physical space, as it requires needlessly complicated and convoluted descriptions to describe simple concepts such as the Inverse-square law.
If you propose a theory to describe the properties of space based on the topology of space-filling curves, then it requires formulation of metric tensors for the differential geometer already described by tensor algebra in General Relativity, and objects spatial-temporal trajectory in General and Special Relativity.
The current model utilises Minkowski space?s Lorentzian (Pseudo-Riemannian) manifold with differential topology applied by the Stress?energy tensor, where the scale and momentum of objects distinguish under which mechanics and field theory they operate under.
If you should like to continue to peruse that avenue of thinking then construct a self-consistent geometric model that encompasses and unifies all currently known phenomenology to arbitrary precision and provides testable predictions of yet unknown phenomena. Proof under mathematical rigor demonstrating your equations as necessarily true is a prerequisite to publication.
Failure to do so leaves a superfluous counterproductive interpretation that teaching to others amounts to intellectual dishonesty and fraud in academics and industry.
The topology described by space-filling curves can however be used for a co-ordinate system when the physical constants are set too h->0 & G->0; making Euclid?s postulates applicable by elimination of the Stress?energy tensor, the Planck constant, and ignoring the causality structure rendering a static map. It is highly advised not to plot/map out motion by classical mechanics or a lorentz manifold.
It?s a nice visual aid for teaching mathematics, like the old Armillary spheres. They also both convey the world inaccurately. (Geodesic motion in General relativity superseded the old models of planetary motion.)
Just to clarify, Space-filling curve sculptures make appropriate artistic representation of abstract mathematical structures/concepts; or as visual aids for teaching mathematics, not cosmology.
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Caution! Physical constants and physical laws not confirmed beyond a dozen decimal places.
Theory confirmed arbitrary precisions and demonstrable computationally to a precision 32767 digits.
*Gravitational constant (Gc or G) = 33363?meter^3/(500000000000000?kilogram?second^2)
≈ 6.6726?10^(-11)?meter^3/(kilogram?second^2)
*Planck constant = 13252151?joule?second/20000000000000000000000000000000000000000
13252151/20000000000000000000000000000000000000000 _≈ 6.6260755?10^(-34)?joule?second
66260693/100000000000000000000000000000000000000000 ≈ 6.6260693?10^(-34)?joule?second
Also defined in terms of (_kg*_m^2/_s) kilogram by meter squared per second.
