I recently rediscovered a shape I hadn’t encountered in decades: the Boerdijk–Coxeter helix also known as a tetrahelix.
The helix part of the name is quite apparent, but the tetra is less so. It turns out that the helix in the main image above can be thought of as a bunch of regular tetrahedra glued together. The first 3 “links” in the chain of the tetrahelix are shown below.
In the pictures above, three tetrahedra are shown. If you imagine gluing them along their white faces as they are positioned in the picture on the left above, you will get a shortened version of the tetrahelix in the main image of this blog. The famous inventor and architect Buckminster Fuller studied tetetrahelices, and they continue to come up in architecture as well as science.
If you think the tetrahelix is beautiful, you are not alone. There is center for the arts in Mito, Japan based on its shape. More recently, architects are looking into using the tetrahelix as a basis for some cutting edge work. Something this beautiful often occurs in nature as a molecule or some other structural element, and this is certainly true of the tetrahelix.
A really interesting property of the tetrahelix is that it is rotationally non-periodic. Put in another way, no two vertices (corners) of the tetrahelix fit on the same line that’s parallel to its axis. See, for example, the white line below.
It goes through the vertex marked by the blue star, but not through any other vertex. You may think that if you make the tetrahelix long enough, eventually it will pass through another vertex. Not so. That’s because each rotation angle between each pair of tetrahedra is an irrational multiple of a full revolution. In a way, one could look at the tetrahelix as a physical embodiment of what it means to be a rational number.
Another delightful property of the tetrahelix is it can be made by from strips of triangles that are either 1, 2, or 3 triangles wide just by wrapping the strip around itself and connecting it. You can see it in the video below!
Want to see what happens to the tetrahelix when the equilateral triangles are replaced by scalene ones? It’s in this blog. Happy spiraling!