Apollonian Packing of Circles within Ellipses

Round 1

Reviewer 1 Report

A good scientific research.

Introduction is enough and clear.

The goal formulation needs a mach more clear statement of the new original contribution.

Presentation is logical and consistent.

Results are presented graphically in a proper way.

Conclusions could be enlarged with some applications of that theory.

Author Response

Answer to first Reviewer - Requests for two minor revisions

Minor revision \#1.1 The goal formulation needs a much more clear statement of the new original contribution'

We addressed the minor revision request by making changes in the Abstract and Section 1 ``Introduction''. We gave a more definite statement of the new original contribution by rewriting the sentence in the following way:

"The novelty of the proposed approach consists in its applicability to the Apollonian packing of circles within a generic closed convex contour, provided a parametrization of its outer boundary."

Minor revision #1.2 Conclusions could be enlarged with some applications of that theory''

We addressed the request by renaming the `'Results}'' section as ``Results and Discussion'' and expanding the section, by adding the following paragraphs:

"Results attest the good accuracy and reliability of the proposed algorithm for a wide range of values of the eccentricity of the elliptic boundary. Figures show evidence of the recursive nature of the branching sequence of circle filling in the elliptic domain, resulting in a Sierpinski-like fractal morphogenesis [23].The choice of simple highly-symmetrical scaffolding initial configurations, especially in the case of low eccentricity ellipses, e.g., the structure shown in Fig. 6a, fosters a reduction of the number of packing circles for a predefined filling accuracy: this choice would reasonably minimize computational complexity in simulations of electromagnetic scattering of waves by elliptic cross-section cylindrical structures assembled as a juxtaposition of circular cylindrical scatterers. Results might be easily applied to the modeling of physical fractals [24], allowing the numerical study of the circle radius distribution and the relevant fractal dimension [25]. Similarly, Apollonian circle packings within non-circular boundaries could find interesting applications to space filling bearings in connection with the study of tectonics and turbulence modeling [26]. Further analogy may be recognized between the presented structures and covering hierarchical nanostructures [27,28], the tiling of tubular nanostructures or other nanostructure modeling [29-31].

We extended the references list as needed to expand the ``Discussion'' Section. 

Author Response File: Author Response.pdf

Reviewer 2 Report

The paper describes an elegant and useful extension of Circular Apollonian Packing to fill an ellipse with circles. It is well written, covering all the necessary background and details to implement the algorithm. My only suggestions for improvement are about minor points:  

(1)  Title is a bit misleading, it could easily be mistaken, as I did, for packing of ellipses but it's really about filling of ellipses with circles. 

(2)  Novelty could be stated more explicitly and clearly, for the reader's sake. 

(3)  The method starts and ends with symmetrical patterns, is it possible to extend the method to a random start? Some comments in this regards would be helpful to some readers.

The authors stated that their proposed method could be extended to 3D. I eagerly look forward to such an extension, as it would be even more useful in practice.

Author Response

Answer to second Reviewer - Requests for three minor revisions

Minor revision #2.1 Title is a bit misleading, it could easily be mistaken, as I did, for packing of ellipses but it's really about filling of ellipses with circles. 

We addressed the revision request by slightly changing the title ``Apollonian Packing of Ellipses'' to ``Apollonian Packing of Circles within Ellipses'', to avoid ambiguity. The diction ``Circular Apollonian Packing of ellipses'' has been changed into ``Apollonian circle packing within ellipses'' throughout the text, especially in Section 1. Although the proposed algorithm actually achieves a filling of ellipses with circles, the diction ``Apollonian Circle Packing'' is widely used in the literature, as shown in Ref. [3], added on this purpose.

Minor revision #2.2: Novelty could be stated more explicitly and clearly, for the reader's sake. 

We addressed the minor revision request by making changes in the Abstract and Section 1 ``Introduction''. We gave a more definite statement of the new original contribution by rewriting the sentence in the following way:

"The novelty of the proposed approach consists in its applicability to the Apollonian packing of circles within a generic closed convex contour, provided a parametrization of its outer boundary.''

Minor revision \#2.3 The method starts and ends with symmetrical patterns, is it possible to extend the method to a random start? Some comments in this regards would be helpful to some readers.

We addressed the request by rewriting the sentence in paragraph 2.2.2, line 101 as follows:

``The effectiveness of the proposed algorithm is independent from the choice of the scaffolding configuration to which it is applied, even in the case of non-symmetric initial configurations: thus, the choice of the scaffolding configuration within a given ellipse is not univocal, each configuration giving rise to a different circle packing within the same ellipse. Since the determination of the circle filling in a deltoid-like shaped curvilinear triangle may be obtained by means of the Descartes formula, attention must be paid to the filling of the arbelos-like curvilinear triangles. For illustrative purposes, we will focus on scaffolding structures for the Apollonian circle packing within an elliptic domain ${\cal E}$, having major axis $2a$ and minor axis $2b$, ($a,b\in \mathbb{R}$, $a\ge b$) which are symmetric with respect to the ellipse axes and composed by the following $2N-1$ ($N \in \mathbb{N}$, $N>0$) circles...}''

we rewrote the first two sentences in Section 4 ``Conclusions'', as follows:

'`In this paper, an algorithm for the Apollonian circle packing within an ellipse is described. The algorithm has been implemented in a specific code to providesuitable testing and, although, for illustrative puropses, the algorithmhas been applied so as to give rise to simple structures which show a symmetry with respect to the ellipse axes, it is suitable to achieve non-symmetric packings."

Author Response File: Author Response.pdf

Reviewer 3 Report

Apollonian Packing of Ellipses manuscript is valuable and deserves to be published. However, there are some minor sitbacks and the authors wish to have their manuscript improved.

Reference 1 has not been used.

Reference style is not according with the style of the publisher.

The name of Menaechmus is typed wrong (p. 6).

There are very few references about symmetry and the literature should be improved with reference to works covering this issue.

The results (and discussion) section is extremely short. The authors must add there some relevant discussions. For instance their procedure much very well seen as an branching level (much alike subgraphs of pair vertices) and/or the algorithms can be seen much alike covering operations in nanostructure modeling presented as algorithms (Dual, Medial, Stellation, Truncation, Leapfrog, Quadruple and Capra). I would like to see the results section expanded with some relevant discussions.

Author Response

Minor revision #3.1 Reference 1 has not been used.

We corrected the error by adding Ref. [1] at the end of the first sentence in Section 1 ``Introduction'', line 18.

Minor revision #3.2  Reference style is not according with the style of the publisher

We corrected all references in ``References'' section in order to be compliant with the style of the publisher. 

Minor revision #3.3 The name of Menaechmus is typed wrong (p. 6)

We rewrote the title of subsubsection 2.3.1 "The quest of the solution by Menaechmics" to ``The quest of the solution by means of Menaechmics'', to avoid misunderstanding.

Minor revision #3.4 There are very few references about symmetry and the literature should be improved with reference to works covering this issue

The comment, as we understand it, raises a request which is similar to the third question expressed by the second Reviewer. We addressed the request by rewriting the sentence in paragraph 2.2.2, line 101 as follows:

``The effectiveness of the proposed algorithm is independent from the choice of the scaffolding configuration to which it is applied, even in the case of non-symmetric initial configurations: thus, the choice of the scaffolding configuration within a given ellipse is not univocal, each configuration giving rise to a different circle packing within the same ellipse. Since the determination of the circle filling in a deltoid-like shaped curvilinear triangle may be obtained by means of the Descartes formula, attention must be paid to the filling of the \emph{arbelos}-like curvilinear triangles. For illustrative purposes, we will focus on scaffolding structures for the Apollonian circle packing within an elliptic domain ${\cal E}$, having major axis $2a$ and minor axis $2b$, ($a,b\in \mathbb{R}$, $a\ge b$) which are symmetric with respect to the ellipse axes and composed by the following $2N-1$ ($N \in \mathbb{N}$, $N>0$) circles..."

We rewrote the first two sentences in Section 4 ``Conclusions}', as follows:

``In this paper, an algorithm for the Apollonian circle packing within an ellipse is described. The algorithm has been implemented in a specific code to provide suitable testing and, although, for illustrative puropses, the algorithm has been applied so as to give rise to simple structures which show a symmetry with respect to the ellipse axes, it is suitable to achieve non-symmetric packings. `'

Minor revision \#3.5 The results (and discussion) section is extremely short. The authors must add there some relevant discussions. For instance their procedure much very well seen as an branching level (much alike subgraphs of pair vertices) and/or the algorithms can be seen much alike covering operations in nanostructure modeling presented as algorithms (Dual, Medial, Stellation, Truncation, Leapfrog, Quadruple and Capra). I would like to see the results section expanded with some relevant discussions.

We addressed the request by renaming the ``Results'' section as ``Results and Discussion'' and expanding the section, by adding the following paragraphs: 

Results attest the good accuracy and reliability of the proposed algorithm for a wide range of values of the eccentricity of the elliptic boundary.  Figures show evidence of the recursive nature of the branching sequence of circle filling in the elliptic domain, resulting in a Sierpinski-like fractal morphogenesis [23]. The choice of simple highly-symmetrical \emph{scaffolding} initial configurations, especially in the case of low eccentricity ellipses, e.g., the structure shown in Fig. 6a, fosters a reduction of the number of packing circles for a predefined filling accuracy: this choice would reasonably minimize computational complexity in simulations of electromagnetic scattering of waves by elliptic cross-section cylindrical structures assembled as a juxtaposition of circular cylindrical scatterers. Results might be easily applied to the modeling of physical fractals [24], allowing the numerical study of the circle radius distribution and the  relevant fractal dimension [25]. Similarly, Apollonian circle packings within non-circular boundaries could find interesting applications to space filling bearings in connection with the study of tectonics and turbulence modeling [26]. Further analogy may be recognized between the presented structures and covering hierarchical nanostructures [27,28], the tiling of tubular nanostructures or other nanostructure modeling [29-31].

We extended the references list as needed to expand the ``Discussion'' Section.with , he following 11 reerences:

3. Holly, J.E. What Type of Apollonian Circle Packing Will Appear?. Am. Math. Mon. 2021, 128, 611-629.

22. Lawrence J.D. A catalog of special plane curves; Dover Publications: New York, U.S., 1972; pp. 131–134.

23. Bourke, P. An introduction to the Apollonian fractal. Comput. Gr. 2006, 30, 134-136.

24. Varrato, F.; Foffi, G. Apollonian packings as physical fractals. Molecular Physics 2011, 109, 2663-2669.

25. Mauldin R.D.; Urba\'nski, M. Dimension and Measures for a Curvilinear Sierpinski Gasket or Apollonian Packing, Adv. Math. 1998, 136, 26-38.

26. Manna, S.S.; Herrmann, H.J. Precise determination of the fractal dimensions of Apollonian packing and space-filling bearingsJ. Phys. A: Math. Gen. 1991, 24, 481-490.

27. Shao, J.; Ding, Y.; Wang,W.; Mei, X.; Zhai, H.; Tian, H.; Li, X.; Liu, B. Generation of Fully-Covering Hierarchical Micro-Nano- Structures by Nanoimprinting and Modified Laser Swelling. Small 2014, 10, 2595-2601.

28. Son, J.G., Hannon, A.F.; Gotrik, K.W.; Alexander-Katz, A.; Ross, C.A.Hierarchical Nanostructures by Sequential Self-Assembly of Styrene-Dimethylsiloxane Block Copolymers of Different Periods. Adv. Mater. 2011, 23, 634-639.

29. Toffoli, H.;Erkoç, S.; Toffoli, D.Modeling of Nanostructures. in: Handbook of Computational Chemistry; Leszczynski, J., Ed.; Springer: Dordrecht, 2015; pp.1–55. 412

30. Diudea, M. V. Nanostructures: novel architecture. Nova Science Publishers: New York, U.S., 2005.

31. Chandel, V.S.; Wang, G.; Talha, M. Advances in modelling and analysis of nano structures: a review Nanotechnology Reviews 2020, 9, 230-258.

Author Response File: Author Response.pdf