Boerdijk-Coxeter Helix in the 16-Cell

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This video shows a Boerdijk-Coxeter helix in the 16-cell – a twisted ring of eight regular tetrahedra, in which each tetrahedron has one face in common with the next. The eight vertices of the tetrahedra that make up the helical ring are all located on the surface of the 3-sphere, i.e., they all have a distance of 1 from the center of the 3-sphere (the origin of the euclidean 4D space). The tetrahedra are projected from 4D to 3D using a stereographic projection of the vertices. Stereographic projection was used to ensure that tetrahedra, which are disjoint in 4D, do not cross each other as a result of the projection. Because of the projection, the tetrahedra are severely distorted and do not appear to be regular, even tough in 4D they are. The helical ring rotates throughout the video to show its geometry more clearly. Initially, the tetrahedra are visualized by their faces. After one revolution, the edges of the tetrahedra are faded in. One set of edges is colored cyan. This set of edges forms a helical polygon: a Petrie polygon of the 16-cell. It connects all eight vertices cyclically. Each tetrahedron contains exactly three edges of the Petrie polygon. The cyan helix winds around the Boerdijk-Coxeter ring exactly three times. In addition, two sets of edges each are colored magenta. They connect vertices to each other in steps of two vertices, i.e., they skip every second vertex of the Petrie polygon. Each magenta helix winds around the Boerdijk-Coxeter ring exactly once. The two magenta helices are intertwined: they form a Hopf link. Finally, one set of edges is colored orange. The edges connect vertices to each other in steps of three vertices, i.e., it skips every second and third vertex of the Petrie polygon. The orange helix winds around the Boerdijk-Coxeter ring exactly once and forms a second Petrie polygon of the 16-cell. The three types of helical polygons form the complete set of edges of the tetrahedra. After they have been faded in, the tetrahedra are faded out. Since it is hard to see how the tetrahedra that make up the Boerdijk-Coxeter helix are connected to each other, for the next two revolutions, they are shown individually in the order in which the appear in the helix. In the final revolution, the Boerdijk-Coxeter ring is visualized only by the three types of helical polygons. This video was created based on:

H. S. M. Coxeter: Regular Polytopes, 3rd edition, Dover Publications, New York, 1973, pp. 245,

H. S. M. Coxeter: The Simplicial Helix and the Equation tan⁡nθ=ntan⁡θ, Canadian Mathematical Bulletin, 28(4):385–393, 1985, p. 386.

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CC BY-SA 4.0, as the Wikimedia Commons states it.

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Creative Commons Attribution-Share Alike 4.0

Credit

Carsten Steger

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