Transforming Optimal Tetrahelices Between the Boerdijk–Coxeter Helix and a Planar-Faced Tetrahelix

Abstract

The Boerdijk–Coxeter helix (BC helix, or tetrahelix) is a face-to-face stack of regular tetrahedra forming a helical column. Treating the edges of these tetrahedra as structural members creates an attractive and inherently rigid space frame, and therefore is interesting to architects, mechanical engineers, and roboticists. A formula is developed that matches the visually apparent helices forming the outer rails of the BC helix. This formula is generalized to a formula convenient to designers. Formulae for computing the parameters that give proven edge-length minimax-optimal tetrahelices are given, allowing transformation through a continuum of optimum tetrahelices of varying curvature while maximizing regularity. The endpoints of this continuum are the BC helix and a structure of zero curvature, the equitetrabeam. Only one out of three members in the system change their length to transform the structure into any point in the continuum. Numerically finding the rail angle from the equation for pitch allows optimal tetrahelices of any pitch to be designed. An interactive tool for such design and experimentation is provided. A formula for the inradius of optimal tetrahelices is given. The continuum allows a regular Tetrobot supporting a length change of less than 16% in the BC configuration to untwist into a hexapodal or n-podal robot to use standard gaits.