Boerdijk–Coxeter helix

Coxeter helices from regular tetrahedra<table><tbody><tr><td></td></tr><tr><td>Right-handed and left-handed helices</td></tr><tr><td><p>Edges can be colored into three groups, one helix (cyan) connecting every vertex, two helices (magenta) connecting vertices in steps of two vertices, and three helices (orange) connecting vertices in steps of three vertices (see the video's Wikimedia Commons page for a detailed description of its contents).</p></td></tr></tbody></table>

<p>The Boerdijk–Coxeter helix, named after <a href="/facts/H._S._M._Coxeter/D2l5jWGG">H. S. M. Coxeter</a> and Arie Hendrick Boerdijk [es], is a linear stacking of regular <a href="/facts/Tetrahedron/7mkSrl6j">tetrahedra</a>, arranged so that the edges of the complex that belong to only one tetrahedron form three intertwined <a href="/facts/Helix/BqC4AKwL">helices</a>. There are two <a href="/facts/Chirality_(mathematics)/FtKNlHTb">chiral</a> forms, with either right-handed or left-handed windings. Unlike any other stacking of <a href="/facts/Platonic_solids/jraREXFD">Platonic solids</a>, 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, and in 4-dimensional space this helix repeats in rings of exactly 30 tetrahedral cells that tessellate the <a href="/facts/3-sphere/u67nImpY">3-sphere</a> surface of the <a href="/facts/600-cell/tNBtDUe9">600-cell</a>, one of the six regular convex <a href="/facts/4-polytope/mY8vBpec">polychora</a>.
</p><p><a href="/facts/Buckminster_Fuller/fLvx87hc">Buckminster Fuller</a> named it a <i>tetrahelix</i> and considered them with regular and irregular tetrahedral elements.
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Boerdijk–Coxeter helix open-in-new

Boerdijk–Coxeter helix