# Boerdijk-Coxeter Helix in the 16-Cell

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.…

- Kind: Short
- Year: 2025
- Runtime: 1 min
- By: Carsten Steger
- Archive: [Wikimedia Commons](https://commons.wikimedia.org/wiki/File:Boerdijk-Coxeter_Helix_in_the_16-Cell.webm)
- Licence or status: [CC BY-SA 4.0](https://creativecommons.org/licenses/by-sa/4.0) (as the archive states it)
- Credit: Carsten Steger
- Who can watch: Teens profiles and older (the archive gives no rating; placed by Invisible Network from its collection and subjects)

[Watch now](https://invisiblenetwork.ai/watch/archive/commons/165999405?play=1)