Periodic Modification of the Boerdijk–Coxeter Helix (tetrahelix)

Garrett Sadler,Richard Clawson,Fang Fang,Klee Irwin

doi:10.3390/math7101001

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”.

Highlights

The Boerdijk–Coxeter helix (BC helix, tetrahelix) [1,2] is an assemblage of regular tetrahedra in a linear, helical fashion (Figure 1a)
(The sequence of faces used while appending, or the sign of the second term in Equation (1), determine the chirality of the helix.) Due to the irrational value of θ, it may be observed that the BC helix has an aperiodic nature, in that the structure has no non-trivial translational or rotational symmetries
It is known that the BC helix exhibits an aperiodic nature such that it possesses no non-trivial translational or rotational symmetries

Summary

Introduction

The Boerdijk–Coxeter helix (BC helix, tetrahelix) [1,2] is an assemblage of regular tetrahedra in a linear, helical fashion (Figure 1a). This assemblage may be obtained by appending faces of tetrahedra together so as to maintain a central axis or, alternatively, R.W. Gray [3] has produced a description of the BC helix by partitioning into 4-tuples the points of R3 given by the sequence (sn )n∈Z sn = (r cos (nθ ), ±r sin (nθ ), nh) ,. Helix; (b) a “5-BC helix” may be obtained by appending and rotating tetrahedra through the angle given by Equation (4) using the same chirality of the underlying helix; (c) a “3-BC helix” may be obtained by appending and rotating tetrahedra through the angle given by Equation (4) using the opposite chirality of the underlying helix

Method of Assembly

Modified BC Helices

The 5-BC Helix

The 3-BC Helix

Conclusions