Discrete Helices

This repo is part of Public Invention, which means it you as a volunteer. Everything we do is free-libre open-source, including everything in this repo.

Devoted to developing the math needed to describe the helix generated by any object which is stacked using the same rule repeatedly, and an interactive website for demonstrating and using this math.

The fact that such a joining of objects ALWAYS produces a helix I call Lord's Observation:

“In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant. A helical structure is thus generated by the repeated action of a screw transformation acting on a subunit.”

I use the term "discrete helix" or "delix" to refer to this object, which coincides with a continuous helix.

Following the normal Public Invention pattern of working on technical subjects, the repo contains a LaTeX paper which is a work-in-progress describing the work. Additionally, the math being developed exists in JavaScript and Mathematica. The Javascript is used to suport an interactive web page that demonstrates the principles, and may eventually be useful as an online "calculator."

The nature of this project (not all Public Invention projects are like this) is that it deserves an academic-style description.

Although I welcome collaboration in all cases, this project is evolving very rapidly; however, the section below "The Platonic Delices" is a project that I probably will not directly undertake. This work will almost certainly result in a publishable paper; for any of you interested in this subject or seeking to obtain a publication, I strongly encourage you to contact me about collaboration on this paper. Although I will not likely undertake it myself, I will happily coach it in my role as Head Coach of Public Invention -- Robert L. Read [email protected]

The Platonic Delices

This is an outline of a new project, based on some recent research. The project is to investigate, leading to a table, some images, and possible 3D software and an academic paper, all of the “Platonic Discrete Helices”. I use the term “Delix/delices” for Discrete Helices.

This project is to use recently developed software, and possibly mathematics and Mathematica, to construct a table describing the helixes generated by all of the Platonic solids, such as the “icosahelix”. It could be done by a programmer with little knowledge of mathematics, but the more math they have, the easier it will be. It probably requires a conversation to fully describe. The project will be fully open-source, like all Public Invention projects.

Here is a longer story about it.

1. The Boerdijk-Coxeter tetrahelix is a famous Platonic Discrete Helix. If you stack Tetrahedra against each other using the same rule, they form a discrete helix (technically, there are 3 such, one of which is a torus, that is, a degenerate helix).

2. During the Public Mathathon held in December, we observed that EVERY repeated rule generated a helix (sometimes degenerate.) This was unexpected, to me.

3. After many hours of research, I found this principle, which I call “Lord’s Observation”, is known, though very rarely articulated. This is the only statement I have found, from a paper from 2002: (https://www.researchgate.net/publication/226976531_Helical_Structures_The_Geometry_of_Protein_Helices_and_Nanotubes)

“In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant. A helical structure is thus generated by the repeated action of a screw transformation acting on a subunit.”

4. Prof. Eric Lord (author of the above paper) has written based on “screw transformations”. However, AFAIK, nobody has ever worked out the math to produce the parameters of the discrete helix from the intrinsic properties of the repeated object itself. That is, given a macroscopic object or a molecule and rules for joining identical copies of such an object, what is the radius, angle of rotation, and travel of the helix which intersects the joint points?

5. But I have now done this, using Mathematica and Javascript. It correctly reproduces the known parameters of the BC-tetrahelix.

6. There have been mentions, in “barely published” papers, of “dodecahelices” and “icosahelices”, and pictures of these objects formed by using the other Platonic solids (other than the tetrahedron.)

7. However, nobody has ever given the properties of these objects precisely and systematically, and have certainly not studied them as a whole.

8. Given that I have computer algorithms that give these properties based on ANY repeated rule, figuring out these properties for the Platonic delices reducing to figuring out:

9. Rules for face attachment (trickier than it looks, for some solids, since there are multiple faces and multiple orientations possible.)

• The face-normals of the Platonic solids,
• The distance between the centroids of the face combinations used by each rule to generate a discrete helix.
• Producing images of these discrete helices.
• Arranging the results in a table, perhaps sorted by pitch.

10. I therefore propose that this investigation to be undertaken to exhaustively enumerate and give the parameters for the Platonic Delices, and I think this will be worth an academic publication, especially if combined with open-source software for their production and viewing in a 3D setting.

11. If the general algorithms are worth 10 Karma Points, I judge this work to be worth about 4 Karma Points. Because:

• People are very interested in Platonic solids, even though from an engineering point of view their helices are not particularly valuable (except BC Helix.)

• This would deepen our understanding and confidence in the algorithms.

• Some of these might have interesting close-form solutions (though you may be forced to give numeric answers.)

• There is a relationship to chemistry, buckyballs, and molecular biology.

This would be somewhat challenging but certainly doable work. It would require: Finding formulae for all the platonic solids (for example, all vertices.)

• Carefully considering all of the rules for how the face may be used (thus there will be a lot more than just one rule for each of the 5 Platonic solids---I would estimate there are 25 regular Platonic delices which can be enumerated).
• Utilizing the math which I have developed, which may not be trivial.
• Utilizing and improving my existing software for interactively rendering 3D shapes in a browser using Three.js, and this software is currently very messy.
• Systematically constructing a table of the results and producing a catalog of images.