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Boerdijk–Coxeter helix

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Title: Boerdijk–Coxeter helix  
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Subject: Tetrahedron, Helix, Harold Scott MacDonald Coxeter, 600-cell, List of polygons, polyhedra and polytopes, Hopf fibration, 16-cell, Grand antiprism, Apeirogon
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Boerdijk–Coxeter helix


The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and A. H. Boerdijk, is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to a single tetrahedron form three intertwined helices. There are two chiral forms, with either clockwise or counterclockwise windings. Contrary to any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation. This is because the helical pitch per cell is not a rational fraction of the circle.

Buckminster Fuller named it a tetrahelix and considered them with regular and irregular tetrahedral elements.[1]

Architecture

See the Art Tower Mito.

Higher dimensional geometry

The 600-cell partitions into 20 rings of 30 tetrahedra, each a Boerdijk–Coxeter helix. When superimposed onto the 3-sphere curvature it becomes periodic, with a period of ten vertices, encompassing all 30 cells. The collective of such helices in the 600-cell represent a discrete Hopf fibration. While in 3 dimensions the edges are helices, in the imposed 3-sphere topology they are geodesics and have no torsion. They spiral around each other naturally due to the Hopf fibration.

See also

Notes

References

  • H.S.M. Coxeter, Regular Complex Polytopes, Cambridge University, 1974.
  • A.H. Boerdijk, Philips Res. Rep. 7 (1952) 30
  • The c-brass structure and the Boerdijk–Coxeter helix, E.A. Lord, S. Ranganathan, 2004, pp. 123–125 [1]
  • Eric A. Lord, Alan Lindsay Mackay, Srinivasa Ranganathan, New geometries for new materials, p 64, sec 4.5 The Boerdijk–Coxeter helix
  • J.F. Sadoc and N. Rivier, Boerdijk-Coxeter helix and biological helices The European Physical Journal B - Condensed Matter and Complex Systems, Volume 12, Number 2, 309-318, [2]

References

  • Chapter 5: Joining polyhedra, 5.36 Tetrahelix p. 53

External links

  • Boerdijk-Coxeter helix animation
  • http://www.rwgrayprojects.com/rbfnotes/helix/helix01.html


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