Half Dimensions #3

Exploratory thoughts.

Prima facie - none of these appear to entail each other but the first set of topics appears to lend additional credibility to my original idea here.

Related, Precursors

1. Fuzzy Spacetimes - treatments of Space Time Points as Operators and resultant Manifolds become Fuzzy.
   • https://link.springer.com/chapter/10.1007/1-4020-3786-4_5
   • https://arxiv.org/abs/2303.16326
2. Fuzzy Topological Spaces - concerns the degree of Set Theoretic Membership within the context of a Topological Space.
   • https://www.sciencedirect.com/science/article/pii/0022247X68900577
3. Mereological Gunk - that Spatio-Temporal Regions don't have boundaries (Mereological Fusion) or that Wholes have no bottom-most Primitive Parts (Atoms).
   • https://marcsandersfoundation.org/wp-content/uploads/2019/11/paper_Russell.pdf

Centric Theses

1. The Totality of Presence vs. Holographic Shadowing - a Point in a Space is represented with Scalars of all Coordinates (or quantity relationships with another Point that's respectively represented with Scalars of all Coordinates - Vector).

First Construction - a geometric object is Fully Present in some Spatial Dimensions but not in others.

i. Let X be a geometric object.

ii. Let S be a Space having N Dimensions.

iii. X doesn't exist in S.

iv. Let S* be a Space having N+1 Dimensions.

v. X is Shadowed in S* if X exists in S* but not S.

Second Construction - a geometric object is not Fully Present in some Space.

i. X is Fully Present in an N-Space S if X has exactly N-Many Scalar Coordinates.

2. Dimensional "Unity".

   • That a Dimension can, or should, be completely represented by a single Number Line (whether the Reals or Naturals, etc.).
   • Regards the representation of a Dimension.
   • Consider a Discontinous Line - when we plot a Function it might be Discontinuous, but the background Spatial Dimension over which the Line is superimposed never is. Models of a Spatial Dimension in which the background Dimension can itself be Discontinuous.

3. Non-Integered Space Time Metrics - not the representation of a Dimension but the quantity of Dimensions.

   • Where the quantity of Dimensions is not an Integer.
   • This was the original idea described here.

4. Seperable, Sharply Bounded Space Time Dimensions vs Overlapment - that Dimensions are distinct and not overlapping. Consider if they weren't.

   • Suppose the same Scalar Coordinate for one Dimension was simultaneously a Scalar Coordinate for another.
   • Consider if a Dimension where treated as a Spacetime itself (or a multitude of converging Lines).