I recently hosted a Mathathon, and was surprised to see that repetitively stacking patterns of tetrahedra always produces a helical structure (though sometimes a degenerate one.)
After some thought it became apparent that any repetitive construction of the same transformation or the same physical object in the same face-to-face orientation will produce a helix. This is sometimes called a screw transformation.
After some considerable google research, I discovered that this was known, but not widely articulated, being mentioned only in a paper from 2002 by Eric Lord, AFAIK:
In nature, helical structures arise when identical structural subunits combine sequentially, the orientational and translational relation between each unit and its predecessor remaining constant. A helical structure is thus generated by the repeated action of a screw transformation acting on a subunit.
I am seeking a formula for the radius and pitch (or curvature) of the helix that runs through the face intersections based on intrinsic properties of the object which is being stacked together. In other words, if I told you I had identical blocks of wood with flat faces cut at particular arbitrary angles and I always glued them together at the same spot at the same angle relative to each other, what will the radius and pitch of the helix that intersects all the joint points (A glued to B) be?
All my current research (which is very sloppy and preliminary) is available at GitHub and licensed under Creative Commons by Attribution Share-Alike.
I can solve this problem using Mathematica (you can find my sloppy notes in progress after end of the document of the LaTeX file here (in the comments at the end of the file.)
It is perhaps not too hard to see that by using linear algebra and Mathematica you can compute 3 joints put together, then use the fact that angle bisectors from the joints intersect the axis of the helix to compute two points nearest points on the helix using the formula for closest points of two skew lines (those being the angle bisectors.) This is more like an algorithm than a formula; that is, given a description of the object you can find the radius and pitch, but it tells you nothing about designing objects to achieve a given radius and pitch, as a molecular biologist, structural engineer, or robotocist might with to do.
My question is:
Is there a known formula for this helix based on the intrinsic properties (length and face angle cuts) of the stacked object?
If there is not, I intend to develop one if possible, and I welcome collaboration.