Periodic modification of the Boerdijk-Coxeter helix (tetrahelix)

TL;DR

This paper introduces a method to modify the Boerdijk-Coxeter helix to achieve periodic symmetries, revealing new symmetric tetrahedral structures related to well-known geometric forms.

Contribution

A novel procedure for transforming the aperiodic Boerdijk-Coxeter helix into periodic structures with symmetries, including specific forms linked to pentagonal, icosahedral, and Fuller’s jitterbug transformation.

Findings

Discovery of several distinct periodic tetrahedral structures.

Identification of structures related to pentagonal and icosahedral aggregates.

Connection to Buckminster Fuller's jitterbug transformation.

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 report the finding of several, distinct periodic structures, and focus on two particular forms related to the pentagonal and icosahedral aggregates of tetrahedra as well as Buckminster Fuller's "jitterbug transformation".