Periodic Modification of the Boerdijk–Coxeter Helix (tetrahelix)

Abstract

The Boerdijk−Coxeter helix is a helical structure of tetrahedra which possesses no non-trivial translational or rotational symmetries. In this document, we develop a procedure by which this structure is modified to obtain both translational and rotational (upon projection) symmetries along/about its central axis. We show by construction that a helix can be obtained whose shortest period is any whole number of tetrahedra greater than one except six, while a period of six necessarily entails a shorter period. We give explicit examples of two particular forms related to the pentagonal and icosahedral aggregates of tetrahedra as well as Buckminster Fuller’s “jitterbug transformation”.

Keywords

• helical structure of tetrahedra
• boerdijk-coxeter helix
• icosahedral aggregates of tetrahedra