Adam I. Gerard

Half Dimensions

So, I got to thinking one day recently if the following [concept] makes much sense:

"We think of 0D (a point), 1D (1-dimension, a line), we think of 2D (a flat plane), etc. What about 1.1D, 0.4444...D, or 2.5D? What about (a number of) R-Space Dimensions rather than N-Space Dimensions?" (Note: I'm referring to the number/numeric quantity of dimensions not the representation of a dimension using the Reals.)

So, in continuing to break-down time and space concepts - why the choice of 1,2,3,4,... dimensional spaces and/or space-times?

Apparently this is a relic of ancient geometry and its insistence on rational numbers!

R-Space Dimensions?

So, why not 1.1 dimensions or 2.5 dimensions?

We use the ordinals to delimit N-space (Minkowski, Hilbert, Hyperbolic, Euclidean) why not the Reals and/or Complex numbers?

Conceptually, what is the difference between 2 and 2.5 dimensions, if possible? What would this mean?

Would this have an impact on our understanding of space-time? Topology?

Formal Clarifications

I mentioned an idea involving dimensions that were non-integer (e.g. - indexing dimensions to the Reals rather than to the Ordinals or positive Integers).

Is this trivial? I don’t think so. One of the implicit dogmas of our conceptions of space and time is that each of these spatial and temporal dimensions is sharply bounded:

  1. What if these lived in gradations on the interval [0,1] or [1,2]?
  2. 0 ⟶ 0.1111 ⟶ 1 ⟶ 1.1 ⟶ 1.3442 ⟶ 1.567 ⟶ 2 and so on.
  3. The idea that there is something “between” dimensions 0,1 and 1,2 and so on. The reigning dogma has it that there is no overlap: the first, second, third,..., and the nth dimensions are related by way of Cartesian Multiplication.
  4. Furthermore, there are non-trivial properties that emerge merely from indexing some set to some transfinite set with cardinality greater than aleph null (for example, moving from the Positive Integers to the Reals allows one to use the calculus of infinitesimals which otherwise does not apply - it's undefined for the Integers alone).

That is to say, there are novel mathematical properties of interest that emerge from even this subtle alteration in the conception of space and time.

To be clear:

  1. I am not advocating the use of the Reals for representing a dimension (Minkowski space).
  2. I am advocating indexing spatial and temporal dimensions against the Reals (regardless of how the dimensions themselves are considered).
  3. I am not solely advocating indexing dimensions against the Reals (e.g. - representing 4N with the notation {1, 1.1, 1.2, 1.3}).

To my knowledge, both the representation and underlying time and space concepts would need to be fundamentally revised to accomplish the above. It is insufficient to merely use a different index set. The first piece, the mathematical representation is described above. The second involves the idea of sharply bounded dimensions in the first place (whether we call it dimension A, B, 1.1, 1, etc.). Moreover, that a point, say, in such an n-space is represented by an n-tuple.

Gerard Space (sketch):

  1. A dimension, D, is defined as (one-to-one with) a Real number line.
  2. We index D to the Reals (e.g - 1.1R, 4.1332R, etc.).
  3. We note that the theory of infinitesimals may apply between dimensions.
  4. Otherwise, standard Topological notions apply at each dimension in {0,1,2,3,4,5,...,n}.
  5. TBD

Formal Comments

Descartes (and his coordinate system) uses Cartesian multiples/products: R ✕ … ✕ R to create order pairs or n-tuples which then correspond to dimensions in a space.

In breaking-down time and space concepts, we eventually run up against dogmas surrounding how we represent such notions - as ordinals 0,1,2,3,4,...

Einstein's greatest contribution (arguably) was to challenge the idea that Space was distinct (in mathematical treatments and ontologically) from Time. That Spacetime was the proper notion (not Three Spatial Dimensions plus Time as a separate thing) along with the attendant notion of frames of reference.

Again, I challenge: why not 1.1 dimensions or 2.5 dimensions? We use the ordinals to delimit N-space (as used in Minkowski, Hilbert, Hyperbolic, and Euclidean spaces). Why not the Reals and/or Complex numbers? (And, I am not referring here to the set of index.)

Traditionally, points within a space are conceived as an n-Tuple (as a product of Cartesian Multiplication). How does one represent the Cartesian multiple of N^1.1111?

Mathematical Definitions and Precursors

A fine, pure mathematics, definition exists here. Note that N-Space Dimensions are presupposed in every such definition.

Descartes is taken to be a father of Infinitesimal Calculus. Also, Cartesian Multiplication (the Cartesian Coordinate System).

Briefly, Cartesian multiples are: R ✕ … ✕ R (multiplied n-times) to create (ordered) n-tuples which are then taken to correspond to a point, region, or location within some space so-represented and/or defined. Each spatial dimension is related to the other (so, the 1st dimension does not interact solely with the 0th and the 2nd dimension but also the 3rd, 4th, etc.) but do not meaningfully overlap.

The idea of spaces whose dimensions are restricted in their interactions solely to "adjacent" dimensions is also an interesting notion.

This concept is also implicit in using Hausdorff dimensions or Minkowski dimensions (with respect to the underlying space) although the Hausdorff measure can be non-integer-valued and is relevant in comparing fractals rather than specifying an underlying spatial dimension concept itself:

“[The Hausdorff measure is also called the fractal dimension] and is a fine-tuning of this definition that allows notions of objects with dimensions other than integers. Fractals are objects whose Hausdorff dimension is different from their topological dimension.”

Graphics Rendering

A related notion from industry can be found in the GPU rendering technique called 2.5D which enables a way to better represent 3D items in 2D (which, I will note, is in accordance with the Holographic Principle and apparently allows for cool things like better isometric perspectives in games).

Note that the technique above maps 3D spaces into 2D spaces (so the underlying spaces are N-Spaces, not R-Spaces).

Regarding Space and Time

This continues my sustained attack on Kantian concepts (out of admiration for the man, to attempt to demolish the limits he perceived to constrain knowledge).

I see Kant as a visionary philosopher who posited the chief dogma of the scientific age (which is representationalism) and whose core distinctions remain at the center of nearly every current philosophical and scientific challenge of interest (AI, tokening, realism, language and its limits, semantics, intention, mind, robotics, sentience/consciousness, etc.).