Calculating the Segmented Helix Formed by Repetitions of Identical Subunits thereby Generating a Zoo of Platonic Helices^†

Round 1

Reviewer 1 Report

This paper proved Lord's obervation and, through two methods, computed and cataloged a zoo of platonic helices. The topic is fresh, approach is useful potentially for studying the structure of DNA, and the details are interesting. However there are some parts that are not quite clear and more proper citations are recommended. See below for details:

1. Line 214, it is not clear what the author means by "vector of B" and "a frame of reference constructed by A". A and B are both points, right? Did the author mean a "vector of polyhedron i+1", and a "frame of reference constructed by polyhedron i"?

2. Reference 25 needs to be corrected. "Inc., W.R. Mathematica" to "W.R. Mathematica Inc."

 

Author Response

Thank you for your careful review. I have fixed the two errors that you found, hopefully clarifying the language of the theorem, which was indeed confusing.

1. As requested by Reviewer 1, I:
1. Corrected References #25 to read “Wolfram Research, Inc.” instead of “Inc., Wolfram Research”.
2. To clarify the language of the theorem, I have changed the language as where the reviewer correctly pointed out is confusing (at line 214), to read:

   Informally, i+1 ``looks the same'' to i, no matter what i is chosen, i < N.

   Call a chain of N identical rigid objects conjoined via a rule that

   conjoins Ai+1 to Bi in such a way that every vector

-  of B is always in the same position relative to a frame of reference

-  constructed from A, a periodic chain.

To: 

+  rigidly attached to object i+1 is always in the same position relative to the frame of reference

+  constructed from object i, a periodic chain.

Reviewer 2 Report

The paper is good and can be accepted for publish. More details, please see the attachment file.

Comments for author File: Comments.pdf

Author Response

Thank you for your kind words. I have repaired the missing number on line 626 as you suggest.

Reviewer 3 Report

The paper under review start from an observation of E. Lord: "In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant.”

The author proves the above observation by giving constant-time algorithms for the segmented helix generated from the intrinsic properties of a stacked object and its conjoining rule.

It is shown that any subunit can produce a toroid-like helix or a maximally-extended helix, forming a continuous spectrum based on joint-face normal twist.

The author provides a software which computes, renders, and catalogs an exhaustive “zoo” of 28 uniquely-shaped platonic helices, such as the Boerdijk-Coxeter tetrahelix and various species of helices formed from dodecahedra.

The present paper is interesting and has a high degree of originality.

For these reasons I recommend the publication.

Author Response

Thank you for your review.