Boerdijk–Coxeter helix

The Boerdijk–Coxeter helix, named after H. S. M. Coxeter and Arie Hendrick Boerdijk [es], is a linear stacking of regular tetrahedra, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined helices.

There are two chiral forms, with either clockwise or counterclockwise windings.

Unlike any other stacking of Platonic solids, the Boerdijk–Coxeter helix is not rotationally repetitive in 3-dimensional space.

However, modified forms of this helix have been found which are rotationally repetitive,[2] and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the 3-sphere surface of the 600-cell, one of the six regular convex polychora.

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

[3] The coordinates of vertices of Boerdijk–Coxeter helix composed of tetrahedrons with unit edge length can be written in the form where

The collective of such helices in the 600-cell represent a discrete Hopf fibration.

[6] 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.

[7] The collective of edges forms another discrete Hopf fibration of 12 rings with 10 vertices each.

Equilateral square pyramids can also be chained together as a helix, with two vertex configurations, 3.4.3.4 and 3.3.4.3.3.4.

This helix exists as finite ring of 30 pyramids in a 4-dimensional polytope.

And equilateral pentagonal pyramids can be chained with 3 vertex configurations, 3.3.5, 3.5.3.5, and 3.3.3.5.3.3.5: The Art Tower Mito is based on a Boerdijk–Coxeter helix.