Tetrahedral Structures ala Bit-Player

The wonderful blog entry found here by Brian Hayes at Bit-Player was inspirational and led me to an area that I had long thought barren! I recently got my son some of the Geomag sticks and ball bearings and playing with them I got the impression that they were fun, but the models that I could make were limited. Well I was SO wrong. In his blog entry Brian Hayes introduced me to the tetrahelix, but also issued an irresistible challenge concerning a bridge that he built. He said "The result (below) is no longer a helix at all but a weird sort of bridge with an arched spine and two zigzag rails. If you extend the arc further, will the two ends meet to form a closed loop? I don’t have enough Geomags to answer that question." I just couldn't resist. I modelled it on Mathematica and was eventually led to a notation for a chain of attached tetrahedrons. I started by labelling his original figure

I started with arbitrarily numbering the vertices of the tetrahedron on the end.  From this, I could specify the next tet by writing down the vertex that was reflected through the plane of the other three.  For example, in his picture above, 1 was reflected to 1'.  I continued this forward to get the sequence:

{1,4,3,2,4,1,3,4,2,1,4,3,2,4,1,3,4,2,1,4,3,2,4}

The great thing is this repeats!  The unit is {1,4,3,2,4,1,3,4,2}.  The natural questions now just flood in with this notation.  For example, I can do random walks using this sequence.  Also, I believe the fact that this structure curves is due to the over abundance of the number 4. For example, the unit {1,2,3,4} leads to the tetrahelix (as do all of the permutations of this unit) which makes me believe that there is definitely some deeper math going on here.  For example I believe an even permutation leads to a right-handed helix and an odd permutation leads to a left-handed helix.  Anyway, I found with 95 steps using the above unit, the ring comes close, but does not close as you can see below.

This made me wonder if any units can actually close after repeating.  I think not, but I haven't checked yet.  I imagine that if you are dealing with units, an integral multiple of the unit would have to match up with the full circle.  But then, you could meander away and then come back, or you could have a non-repeating number that closed on itself.

Anyway here is a short cdf where you can explore units up to 6. 

I have not worried about intersection, and if you relaxed the requirement that consecutive numbers aren't equal (this just flips a tet back to where it started) you can make any tetrahedral structure, you just label the tetrahedra sequentially. When you get to the end of a limb, you reflect the sequence till you get back to the main line.  This is not the best way of doing things, but it seems to be complete.  Anyway, many more things to explore!  I think I have to invest in a few more Geomag sets.

Effective Mass Surface of an Ellipsoid

Here are the ellipsoidal effective mass surfaces.

$E = E_0 \pm \frac{\hbar^2 k_{r}^2}{2 m_e} f(\theta, \phi)$

Band Warping