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.

Polyhedral cages (p-cages) describe the geometry of some families of artificial protein cages. We identify the p-cages made out of families of equivalent polygonal faces such that the faces of 1 family has 5 neighbours and P_1 edges, while those of the other family have 6 neighbours and P_2 edges. We restrict ourselves to polyhedral cages where the holes are adjacent to at most 4 faces. We characterise all the p-cages with a deformation of the faces, compared to regular polygons, not exceeding 10%.