Interdimensionality

Round 1

Reviewer 1 Report

This paper explores the idea that our observable three-dimensional universe may be embedded within a higher- dimensional space, containing other (not necessarily parallel) ‘universes’ of lower dimensionality. It does so in a general sense, rather than within the context of a specific model, and with special emphasis on the concept of ‘inter- dimensionality’, which is the term the author uses to describe the various possible types of intersection between such lower-dimensional structures. Schematically, he outlines different types of qualitatively different intersectional configurations, each represented by a different sub-figure in the text, and argues that each gives rise to qualitatively different phenomena, from within the reference frames available to each lower-dimensional universe.

Though speculative and, perhaps necessarily, rather vague in places, the paper is well written and stimulating. It is, therefore, appropriate to the volume to which it has been submitted, in memory of the late Prof. Solomon Marcus. I am happy to recommend it for publication in Axioms, but have a number of suggestions - some general and others of a more technical nature - that I would like to author to address before final acceptance.

Firstly, although no specific models are discussed, one cannot read the paper without it immediately bringing to mind various ideas present in the modern literature on theoretical physics. Though I understand that the author’s aim is to provide a more general analysis, unconstrained by the technicalities of a specific model, it would be remiss not to mention the potential connections between his ideas and various paradigms in modern particle physics and cosmology. For example, how do the various scenarios for the ‘intersectionality’ of universes he considers relate to the following models: Holographic scenarios, such as AdS/CFT? D-branes in string theory and the ekpyrotic scenario of string cosmology? (Here, the intersection point between different lower-dimensional universes would appear to be the Big Bang, from our three-dimensional perspective.) Traversable worm holes? And, finally, the Many Worlds interpretation of quantum mechanics? (For a general overview of many such models, see ‘The Multiverse’, B. Carr ed., CUP 2007.) With respect to his comments in Sec. 4.1, on the consistency of observable phenomena with respect to the higher-dimensional bulk space, one cannot help but think of the possible connection between this and the consistent histories approach to Many Worlds models.

Secondly, on a more technical level, the paper could be improved by including fewer undefined terms. It is not necessary to give detailed descriptions of the terms whose definitions can be be found elsewhere in the mathematics, physics, or philosophy of science literature, but it would be helpful, especially to the non-specialist, to give basic explanations of, for example, the following: rule inference problem, halting problem, dimensional shadowing, Menger sponge-like structure, “umklapp”, curved reference frame (Sec. 4) and supercaptivation. Finally, it would be helpful to number the equations and to provide an actual definition of the Hausdorff dimension, d. This is not provided in either Eq. (1) or Eq. (2). After these issues have been addressed, briefly and without expanding the length of the manuscript significantly, the paper can be reconsidered for publication.

Comments for author File: Comments.pdf

Author Response

I kindly thank the Referee for the very clear review of my intentions and goals in writing this paper, as outlined in the first paragraph of the Referee Report.

According to the suggestions of the Referee in the third paragraph, I have added the following paragraph at the end of Section 2:

"As the aim is the provision of a very general analysis that is unconstrained by the technicalities of specific models,
no concrete theory is discussed.
Nevertheless, it might be not too far-fetched to just briefly mention some potential connections between interdimensionality
and various paradigms in modern particle physics and cosmology.
Some of these involve the description of a volume of space as conceptualized by
holographic principles, such as the AdS/CFT correspondence related to D-branes in string theory,
or the ekpyrotic models relying on string theory, branes and extra ``hidden'' dimensions.
Other scenarios in the context of the theory of general relativity involve traversable wormholes
(aka Einstein-Rosen bridges) linking disparate points in spacetime."

I have added the following paragraph at the end of Section 4.1:

"There are similarities between the consistency of observable phenomena regarding the higher-dimensional bulk space
and the consistent histories approach to Many Worlds models~\cite{Carr-2007}.
Both involve multiple ``merging'' paths.

@book{Carr-2007,
  doi = {10.1017/cbo9781107050990},
  url = {https://doi.org/10.1017/cbo9781107050990},
  year = {2007},
  publisher = {Cambridge University Press},
  editor = {Bernard Carr},
  title = {Universe or Multiverse?}
}
"

I have also added brief explanations of the followng terms:

(i) rule inference problem, halting problem

"Rule inference is the process of hypothesizing a general rule or ``law'' from examples or ``phenomena''~\cite{go-67,angluin:83}.
The halting problem is the task to determine,
given an arbitrary computer program and an input,
whether the program will eventually halt or continue to run forever. It is provable unsolvable in general.
As the former rule inference problem can be reduced to the latter halting problem, it is provable unsolvable in general."

(ii) dimensional shadowing:

"Interdimensionality, or, by another naming, dimensional shadowing~\cite{sv4}---the ``emulation'' of a lowerdimensional configuration space
by a fractal subset of a higherdimensional manifold---..."

(iii) Menger sponge:

"... Cantor set or Menger sponge-like structure---fractals obtained by self-similar elimination of proper parts ..."

(iv) umklapp property:

"... is based on its ``umklapp'' property---the sudden change from measure zero to infinity if the dimension parameter
is taken higher or lower than a unique value--- ..."

(v) curved has been changed to " ``bend'' or ``curved'' "

(vi) supercavitation:

"... supercavitation---the formation of vapour bubbles in a liquid caused by flow around an object, allowing almost minimal friction movement inside liquids at nearly sound speeds."

(vii) The equations have been numbered.

(viii) definition of Hausdorff dimension d: I have added a sentence

"That is, the Hausdorff dimension $d$ is the unique dimensional parameter at which the measure
$\mu_\delta$ as a function of the dimensional parameter value $\delta$ smaller or larger than $d$ is infinite or vanishes, respectively."

Author Response File: Author Response.pdf

Reviewer 2 Report

Last paragraph of first section (lines from 34 to 38), should be changed from 1st-singular to 1st-plural:
Line 34: We, therefore, present...
Lilne36: ... We hope that the following ... 

In second section, first formula (between lines 46 and 47) could be more appropriate and more readable. Maybe to not separate into two lines? Suggestion: Put the whole formula into one line.

Same suggestion about formula between lines 47 and 48.

Figure 1. Personal;ly, i dont like free-hand-drawing pictures. If possible, my suggestion is (not obligatory) that authors create this picture using computer.

At the end of second section, after line 73, I suggest that author adds at least one sentence and reference that deals with contemporary applications of fractals. For example:
There are many inovative, usefull and interesting applications of fractal theory today, especially in the field of new materials and nanostructures, that can lead as to a new view of dimensionality, but in other direction - into the innerspace [Ref].

[Ref] V.V.Mitic, et al., Fractal frontiers in microelectronic ceramic materials, Ceramics International, 2019, 45, 7, B (2019), 9679-9685.    

Figure 2. Same comment as with Figure 1.

Figure 3. Same comment as with Figure 1.

Figure 4. Same comment as with Figure 1.

5. section should be changed from 1st-singular to 1st-plural:
Line 126: Let us conclude this...
Line 133: Let us  again point...
Line 134: ...it is our convintion that, ...
Line 136-138: Please, remove the last sentence ("I hope that ... interested in.") This sentence is more like to be part of some letter to editor, and not part of scientific paper. 

If author adds a new reference about fractals, then rest of the references should be renumbered.

Congratulation to author for brave approach.

Author Response

I have revised the manuscript according to the Referee's suggestions as follows:

"Last paragraph of first section (lines from 34 to 38), should be changed from 1st-singular to 1st-plural:
Line 34: We, therefore, present...
Line36: ... We hope that the following ..."

I changed all sentences accordingly.

"In second section, first formula (between lines 46 and 47) could be more appropriate and more readable. Maybe to not separate into two lines? Suggestion: Put the whole formula into one line.

Same suggestion about formula between lines 47 and 48."

I did this; all equations are now in a single line, improving readability.

"Figure 1. Personally, i dont like free-hand-drawing pictures. If possible, my suggestion is (not obligatory) that authors create this picture using computer."

"Figure 2. Same comment as with Figure 1.

Figure 3. Same comment as with Figure 1.

Figure 4. Same comment as with Figure 1."

I understand that cartoonish free-hand-drawing pictures might not be optimal. In this case, I would kindly suggest leaving those figures as they are, because this signifies the very general ideas and possibilities mentioned. I understand that this is a very subjective evaluation, and if necessary and required, would change the drawing style.

"At the end of second section, after line 73, I suggest that author adds at least one sentence and reference that deals with contemporary applications of fractals. For example:
There are many inovative, usefull and interesting applications of fractal theory today, especially in the field of new materials and nanostructures, that can lead as to a new view of dimensionality, but in other direction - into the innerspace [Ref].

[Ref] V.V.Mitic, et al., Fractal frontiers in microelectronic ceramic materials, Ceramics International, 2019, 45, 7, B (2019), 9679-9685."

According to the Referee's suggestion, I have added the following paragraph:

Fractal theory has inspired and evolved into many innovative, useful and interesting applications, especially in new materials and nanostructures.
Such important developments can lead us to new views of, and physical means related to, dimensionality~\cite{Hill-2017,Mitic-2019}.

@article{Hill-2017,
  title = {Cooperative light scattering in any dimension},
  author = {Hill, Tyler and Sanders, Barry C. and Deng, Hui},
  journal = {Physical Review A},
  volume = {95},
  issue = {3},
  pages = {033832},
  numpages = {5},
  year = {2017},
  month = {Mar},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevA.95.033832},
  url = {https://doi.org/10.1103/PhysRevA.95.033832}
}

@article{Mitic-2019,
  doi = {10.1016/j.ceramint.2019.01.020},
  url = {https://doi.org/10.1016/j.ceramint.2019.01.020},
  year = {2019},
  month = may,
  publisher = {Elsevier {BV}},
  volume = {45},
  number = {7},
  pages = {9679-9685},
  author = {Vojislav V. Mitic and Goran Lazovic and Vesna Paunovic and Nenad Cvetkovic and Dejan Jovanovic and Sandra Veljkovic and Branislav Randjelovic and Branislav Vlahovic},
  title = {Fractal frontiers in microelectronic ceramic materials}
}

"5. section should be changed from 1st-singular to 1st-plural:
Line 126: Let us conclude this...
Line 133: Let us  again point...
Line 134: ...it is our convintion that, ..."

I changed all sentences accordingly.

"Line 136-138: Please, remove the last sentence ("I hope that ... interested in.") This sentence is more like to be part of some letter to editor, and not part of scientific paper."

I have removed this sentence.

"If author adds a new reference about fractals, then rest of the references should be renumbered."

References have been renumbered

"Congratulation to author for brave approach."

I very kindly thank the Referee for these encouraging words.

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The author has addressed the points raised in my previous report.

Reviewer 2 Report

No additional comments.

I am satisfied, as reviewer.