Graph Theoretic Analyses of Tessellations of Five Aperiodic Polykite Unitiles

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

please revise it as I suggested. 

Comments for author File: Comments.pdf

Comments on the Quality of English Language

Can be improved. 

Author Response

Responses to reviewer number 1:

Thank you for highlighting six aspects of our work:

• Extending Blumeyer's 2 × 2 table on chirality and periodicity
• Using Heesch numbers and corona structures provide further insights into

tiling patterns.

• Analyzing the distribution of polykite unitiles and their flipped versions (anti-

polykites) with Voronoi tessellations and their Delaunay triangulations.

• Also, providing unique Hamiltonian cycle encodings for polykite unitiles.
• Applying Pitteway violation analyses to aperiodic tessellations.
• Furthermore, this study contributes to a better understanding of aperiodic tessellations and provide useful perspectives on self-assembling structures with potential applications in biomimetic materials, nanotechnology, and synthetic biology.

We also appreciate the incredible close attention to details throughout the article.

We addressed most of your minor points without comment as they were simple improvements.  

The abstract, introduction, and conclusion have both been modified to address your concerns. Namely, we revised the introduction … by adding (1) more related results from literature, (2) clearing the statement of research questions, (3) explaining the structure of the remaining sections.

 

We have added more references to existing graph theory literature on tilings and planar graphs to better situate the context of our research.

 

We have extended the legends for a number of our figures and tables.

 

Unfortunately, our Ka-me software cannot be generalized to systematically compute Heesch numbers?

 

We have addressed the connection of Heesch numbers which deal more globally with our local junctions described with our Hamiltonian cycle of vertex degrees.

Some errors that were noted were simply due to converting a Word document to a pdf. For example, Table 2.1 was broken and so one row was incomplete.

However, to address some of your more major concerns, we highlight several specific additions:

First, we have added a section to show how our classification of junctions of tiles in a tessellation is similar to prior work on Penrose tessellations although this prior work showed the configurations they did not add the details of degrees or vertices and the angles involved in each configuration. We now include an addition figure that has appeared frequently in the literature and an original table  

Second, we note the extension of the Dodd theorem on Hamiltonian cycles of Dürer nets to identify polykites. We have described theorems and  propositions and added remarks that justify our extension.

Defense of our title: We believe that our title is analogous in detail to three papers with short, general titles that we cite Shutov, A.V., Maleev, A.V. “Penrose tilings as model sets” (Shutov and Maleev, 2015 []; "Aperiodic monotiles: from geometry to groups"  Coulbois et al., 2024 []; and, "Classical dimers on Penrose tilings" (Flicker, Felix, Simon, and Parameswaran , 2020 []) and that we feel our title is sufficiently detailed such that readers will be able to infer what our article is about. The other two reviewers did not raise any objections to our title. Therefore, we would prefer not to change it other than to add the word “five” to increase specificity.

Re your question about the periodic homochiral example via hares. Could this be

generalized to other unitiles?  We will explore your question but will not hazard a conjecture within this manuscript. Obviously, our research has focused on aperiodic rather than periodic so this is not primary to us.

 

Polykites are polyforms  obtained from a regular triangular grid superposed on a regular hexagonal grid (its dual). A deltoidal trihexagonal lattice is the dual of rhombitrihexagonal tiling or a Diana tiling (Sloane, 2020 []).

Neither vertex 24333 in the turtle polykite nor vertex 22423 in the red squirrel polykite would be invisible on their outer hull of the unitile if we did not consider that these vertices are relevant when we include the kite edges of the polykite. Such vertices are sometimes referred to as “quasivertices” (e.g., Kaplan, 2021 []).

The way in which one determines rules for which prototiles join one another are often referred to as “matching conditions” (e.g., Kaplan, 2021 []).

Could the Ka-me tool be generalized to systematically compute Heesch numbers? (NO)

Also, you suggest to change or remove the reference of Math Stack Exchange since it does not have peer reviewed publications. We realize this, but we feel that it and two other citations to web sites are appropriate to include because we think that readers will find them helpful even though they are not cited in peer reviewed publications.

Again, we appreciate your close reading and helpful advice.

Reviewer 2 Report

Comments and Suggestions for Authors

see attached review with markup

Comments for author File: Comments.pdf

Author Response

Reviewer number 2

Re your recommendation to change the classification of our manuscript from a research paper to an expository paper is fine by us.

Although we make many new observations and feel that we have shown important applications of graph theory that have not been applied to these sorts of aperiodic tessellations, the primary purpose of our work is to develop insights on how we could build a self-assembling polyhedra with an aperiodic tessellation of a single type of polykite unitile.

Since we submitted the article, co-author Biswas was able to present our work at a conference in Chicago, Illinois, where Professor Reidun Twarock was one of the keynote speakers. Twarock has been the leading expert on the mathematics of quasicrystal patterns on viral capsids. She showed exceptional interest on the originality of our approach. Thus, her affirmation has encouraged us to continue to pursue our approach in our quest to produce a self-assembling aperidiodic tessellation of 3D monotiles to form a polyhedron model of quasicrytaline viral capsids.

Re: What bothers me, and is the main point of revision beyond outlining the purpose and scope of the presentation to make clear the expository nature, is that reference is made with adjacent work freely without sufficiently articulated intention. For example, the authors refer to viral capsids several times but no formal description of them is provided. Doing so might help to explain why the intended program of describing their structure in terms of (3-dimensional) tiles is feasible. We have added references to other attempts to build self-assembling models of quasicrystalline tiling of viral capsids and explain the importance of such models to the development of viral capsid models of drug delivery carriers and their advantages.

Re: “The connections with ethnomathematics are curiosities but do not lead to any significant insights that suggest the five types of tiles considered here are in some way fundamental (compared to other potential bases for tessellations).” We have added a section on how the graph theoretic approach can both be used in ethnomathematics education that are different than previous work with Penrose tiles or periodic tessellations of deformed triangles, squares, and hexagons. The question of whether previous cultures have constructed aperiodic tessellations such as  in Islamic tiling remain of research interest in ethnomathematics. We have added references to connect to both education and research.

RE: “The concept of corona number is presented but there is very little analysis of what the number or the Heesch numbers can tell us about complexity of an aperiodic tiling or whether it has any connection with the use of Hamiltonian cycles outlined in the first section.” We have addressed the connection of Heesch numbers which deal more globally with our local junctions described with our Hamiltonian cycle of vertex degrees.

Re: “Much the same goes for the last section on Voronoi diagrams (where there also appear to be some erroneous labelings of the figures relative to what is stated in the text).” We have elaborated this section.

Re: “I have some specific suggestions in the markup. Apologies if they are not all concrete regarding how to address them.” We have tried to address them.

RE: Again, the main thing is to rewrite the introduction to make it clear that the paper does not contain explicit mathematical results, but rather outlines some aspects of aperiodic tilings and makes some speculative observations on their connections to adjacent research areas.” The introduction has been re-written to better situate our work around our fundamental goal to produce biomedical models of aperiodic tessellations of polykite monotiles to self-assemble into a polyhedron.

Again, we appreciate your close reading and helpful advice.

Reviewer 3 Report

Comments and Suggestions for Authors

The paper is suitable for publication after minor revisions aimed at clarifying the distinction between proven results and conjectural observations, improving the prominence of recent references, and tightening some of the longer passages for readability.

Comments for author File: Comments.pdf

Author Response

Reviewer number 3

We have clarifyied the distinction between proven results and conjectural observations, improvied the prominence of recent references, and tightened some of the longer passages for readability.

Re the appearance of references, we have done a major re-organization of reference ordering and have added many additional references to more recent literature.

We revised the introduction by adding (1) more related results from literature, (2) improving the statement of research questions, and (3) explaining the structure of the remaining sections.

We have tried to state more clearly why this graph-theoretic route is preferable to traditional geometric analysis for moving from 2D tessellations to 3D closed tessellations.

We have added a section on how the graph theoretic approach can both be used in ethnomathematics education that are different than previous work with Penrose tiles or periodic tessellations of deformed triangles, squares, and hexagons. The question of whether previous cultures have constructed aperiodic tessellations such as in Islamic tiling remain of research interest in ethnomathematics. We have added references to connect to both education and research.

We discuss periodic homochiral tessellations of hares in more detail. Since we are primarily interested in aperiodic tessellations, we don’t believe that question is primary to our work.

Some claims in the conclusion which the reviewer felt were overextended have been removed and we now focus more on two new challenges which may be helpful in addressing our primary goal to produce a self-assembling 3D polyhedron with an aperiodic homochiral  tessellation composed of one polykite type.

Re the request to: “to mention other standard references such as Mann (2004) and Adams (2022), in addition to Kaplan (2021). We already had cited them in the context of coronas. We added an addition Kaplan (2009, 2021 re-published) citation.

Re: “Adding short recap paragraphs at the end of major sections would also help to reinforce the key points for the reader.” Done.

Again, we appreciate your close reading and helpful advice.

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

This is a review of a revision of the original.  The revision addresses to different extents concerns brought up in the initial review.  To address these concerns the revision is substantially longer than the original. However, most of the added material is explanatory commentary that provides context for statements, tables and figures laid out in the initial submission that were sorely lacking context.  In this regard the revision is a substantial improvement that is more coherent than the original and addresses the primary concerns.
 
There is still room for improvement of the introductory material. Not much is needed here but a few sentences are needed earlier than the outline provided in lines 74—84 to signal that the current work outlines different perspectives on aperiodic tesellations. Without this, the second half of the first paragraph (lines 43-51) and subsequent paragraphs which were added make it sound like building models for viral capsids is actually part of the present work. A few sentences about the perspectives introduced in the present work can help to signal that there is a bigger project of modeling and determining “matching rules” for 3-D tiles that each of the perspectives outlined here will serve.

For this reason I recommend a second, but minor, revision.

Author Response

Response to Reviewer number 2

Re your recommendation to change the classification of our manuscript from a research paper to an expository paper is fine by us.

Although we make many new observations and feel that we have shown important applications of graph theory that have not been applied to these sorts of aperiodic tessellations, the primary purpose of our work is to develop insights on how we could build a self-assembling polyhedra with an aperiodic tessellation of a single type of polykite unitile.

Since we submitted the article, co-author Biswas was able to present our work at a conference in Chicago, Illinois, where Professor Reidun Twarock was one of the keynote speakers. Twarock has been the leading expert on the mathematics of quasicrystal patterns on viral capsids. She showed exceptional interest on the originality of our approach. Thus, her affirmation has encouraged us to continue to pursue our approach in our quest to produce a self-assembling aperidiodic tessellation of 3D monotiles to form a polyhedron model of quasicrytaline viral capsids.

Re: "What bothers me, and is the main point of revision beyond outlining the purpose and scope of the presentation to make clear the expository nature, is that reference is made with adjacent work freely without sufficiently articulated intention. For example, the authors refer to viral capsids several times but no formal description of them is provided. Doing so might help to explain why the intended program of describing their structure in terms of (3-dimensional) tiles is feasible."----- We have added references to other attempts to build self-assembling models of quasicrystalline tiling of viral capsids and explain the importance of such models to the development of viral capsid models of drug delivery carriers and their advantages.

Re: “The connections with ethnomathematics are curiosities but do not lead to any significant insights that suggest the five types of tiles considered here are in some way fundamental (compared to other potential bases for tessellations).” ----- We have added a section on how the graph theoretic approach can both be used in ethnomathematics education that are different than previous work with Penrose tiles or periodic tessellations of deformed triangles, squares, and hexagons. The question of whether previous cultures have constructed aperiodic tessellations such as  in Islamic tiling remain of research interest in ethnomathematics. We have added references to connect to both education and research.

RE: “The concept of corona number is presented but there is very little analysis of what the number or the Heesch numbers can tell us about complexity of an aperiodic tiling or whether it has any connection with the use of Hamiltonian cycles outlined in the first section.” ---- We have addressed the connection of Heesch numbers which deal more globally with our local junctions described with our Hamiltonian cycle of vertex degrees.

Re: “Much the same goes for the last section on Voronoi diagrams (where there also appear to be some erroneous labelings of the figures relative to what is stated in the text).” ---- We have elaborated this section.

Re: “I have some specific suggestions in the markup. Apologies if they are not all concrete regarding how to address them.” ---- We have tried to address them.

RE: Again, the main thing is to rewrite the introduction to make it clear that the paper does not contain explicit mathematical results, but rather outlines some aspects of aperiodic tilings and makes some speculative observations on their connections to adjacent research areas.” -----The introduction has been re-written to better situate our work around our fundamental goal to produce biomedical models of aperiodic tessellations of polykite monotiles to self-assemble into a polyhedron.

Again, we appreciate your close reading and helpful advice.