Boerdijk–Coxeter helix
Linear stacking of regular tetrahedra that form helices / From Wikipedia, the free encyclopedia
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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. Even in an infinite string of stacked tetrahedra, no two tetrahedra will have the same orientation, because the helical pitch per cell is not a rational fraction of the circle. 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.
CCW and CW turning |
Edges can be colored into 6 groups, 3 main helixes (cyan), with the concave edges forming a slow forward helix (magenta), and two backwards helixes (yellow and orange)[1] |
Buckminster Fuller named it a tetrahelix and considered them with regular and irregular tetrahedral elements.[3]