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Periodic Modification of the Boerdijk–Coxeter Helix (tetrahelix)

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Quantum Gravity Research, Topanga, CA 90290, USA
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Faculty of Health, Engineering and Sciences, University of Southern Queensland, Toowoomba, QLD 4350, Australia
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Author to whom correspondence should be addressed.
Mathematics 2019, 7(10), 1001; https://doi.org/10.3390/math7101001
Received: 5 August 2019 / Revised: 11 October 2019 / Accepted: 18 October 2019 / Published: 22 October 2019
(This article belongs to the Section Mathematics and Computers Science)
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”. View Full-Text
Keywords: helical structure of tetrahedra; boerdijk-coxeter helix; icosahedral aggregates of tetrahedra helical structure of tetrahedra; boerdijk-coxeter helix; icosahedral aggregates of tetrahedra
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MDPI and ACS Style

Sadler, G.; Fang, F.; Clawson, R.; Irwin, K. Periodic Modification of the Boerdijk–Coxeter Helix (tetrahelix). Mathematics 2019, 7, 1001.

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