Read, R.L. Calculating the Segmented Helix Formed by Repetitions of Identical Subunits thereby Generating a Zoo of Platonic Helices. Mathematics2022, 10, 2533.
Read, R.L. Calculating the Segmented Helix Formed by Repetitions of Identical Subunits thereby Generating a Zoo of Platonic Helices. Mathematics 2022, 10, 2533.
Read, R.L. Calculating the Segmented Helix Formed by Repetitions of Identical Subunits thereby Generating a Zoo of Platonic Helices. Mathematics2022, 10, 2533.
Read, R.L. Calculating the Segmented Helix Formed by Repetitions of Identical Subunits thereby Generating a Zoo of Platonic Helices. Mathematics 2022, 10, 2533.
Abstract
Eric Lord has observed: “In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant.” This paper proves Lord’s Observation. Constant-time algorithms are given for the segmented helix generated from the intrinsic properties of a stacked object and its conjoining rule. Standard results from screw theory and previous work are combined with corollaries of Lord’s observation to allow calculations of segmented helices from either transformation matrices or four known consecutive points. The construction of these from the intrinsic properties of 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 . We prove that any subunit can produce a toroid-like helix or a maximally-extended helix, forming a continuous spectrum based on joint-face normal twist. This software, website and paper, taken together, compute, render, and catalog an exhaustive “zoo” of 28 uniquely-shaped platonic helices, such as the Boerdijk-Coxeter tetrahelix and various species of helices formed from dodecahedra.
Computer Science and Mathematics, Geometry and Topology
Copyright:
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
