Calculating the Segmented Helix Formed by Repetitions of Identical Subunits
Channel: Public Invention
Date: 2021-09-07

This is a potentially new result showing that ANY physical object that is chained together using the same rule produces a helix, and the radius, travel, screw angle, and other parameters of this segmented helix can be calculated easily from just knowledge of the faces used to conjoin the objects. Further, we show that there is a "zoo" of exactly 28 Platonic helices that are unique in shape and give the parameters for all of them. This work will be presented Sept. 9th, 2021 at the 2nd IMA Mathematics and Robotics Conference in Manchester, UK (which was postponed and virtualized due to COVID-19). 

The most relevant links are:

1) Interactive website: https://pubinv.github.io/segmented-helixes/index.html
2) Paper as submitted: https://github.com/PubInv/segmented-helixes/blob/master/2ndIMAMathOfRobotics/shconf.pdf
3) Longer paper with full details: https://github.com/PubInv/segmented-helixes/blob/master/doc/StackingHelix.pdf

Here is the abstract:

Abstract. Eric Lord has observed:

In nature, helical structures arise when identical structural sub- units combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant.

A complete version of this paper proves this. If a robot is composed of modular structural subunits that can change their shape or relation, the shape of the robot can change. If they all change in the same way, the robot will be a segmented helix of varying length and curvature. Closed-form expressions are given for the parameters of the segmented helix generated from the intrinsic properties of a chained object and its conjoining rule. The construction of these from the rule for conjoining repeated subunits of arbitrary shape is provided, allowing the complete parameters describing the unique segmented helix generated by arbitrary stackings to be easily calculated. Free-libre open-source interactive software and a website is provided which performs this computation for arbitrary prisms along with interactive 3D visualization[13]. A theorem, proved in a longer version[11], is stated that any chain can be transformed continuously between a toroid-like helix and a maximally-extended helix by varying joint-face normal twist.
Keywords: solid geometry, segmented helix, robotics, chemistry