Dear Brent,

I spent the last 3 days working out conceptual ideas for the hanging sculpture in the atrium. 
I first followed up on the idea of warping a Scherk-Collins saddles-and-holes chain along a simple knot, 
in particular a trefoil or a figure-8 knot.  
Unfortunately, the software, written by an undergraduate student a few years ago 
to warp a straight chain along an arbitrary curve cannot handle this complex geometry. 
Thus I tried to fit manually individual short segments, consisting of only one or two saddles, 
along the desired curve. It is doable, but it is very labor intensive.

Then, following up on our recent phone conversation, it suddenly occurred to me 
that we could also fit a Scherk-Collins chain along the edges of the dodecahedron, 
rather than just a twisted triangular prism. 
And of course we can do that for any of the Platonic solids. 
Believing that simpler is often better, my favorite at the moment is the tetrahedral frame; 
it looks most like an elegant sculpture, rather than like a geometric exercise.

Below please find a more detailed report on four structures that I have worked on so far.

The TREFOIL follows up on the original idea we discussed. 
I tried to make this knot as 3-dimensional as possible. Thus I tried to use the centerline 
of a trefoil formed by a thick cylindrical cord, pulled as tight as possible. 
I then decomposed that shape into 12 best-fitting torus segments of roughly equal arc-length, 
trying to maintain tangent continuity at the junctions. 
Each such segment gets filled with two saddle stories. There are 3 different kinds of torus segments. 
And remembering what we had to do for the "Solar Circle", aka "Millennium Arch", 
this will probably mean 6 different moulds, 2 of which will get used 6 times 
and 4 of which will get used 3 times. 
It will probably take me a few working days to calculate all the torus parameters exactly 
to make the pieces fit together smoothly. What you see in the picture is a first rough approximation.

I also considered the FIGURE-8 KNOT because it is intrinsically more 3-dimensional than the trefoil. 
It seems natural to fit 16 saddle stories along this path, exploiting as much symmetry as possible. 
If we decompose this again into 2-story segments, we would have 8 segments, in 4 identical pairs. 
Thus this structure would probably require 8 different moulds, which all will be used twice. 
I have not done any work yet to decompose this knot into torus segments. 
The fitting work will probably be comparable to what has to be done for the trefoil.

The picture of the DODECAHEDRON gives a first idea what such a structure might look like 
if we place 4-story twisted monkey-saddle chains along each of the 30 edges. 
Figuring out how to connect the flanges gracefully at each vertex will take some doing, 
and I will probably have to obtain, and learn to use, some commercial modeling program 
such as Maya or SolidWorks to be able to do this cleanly. 
This structure naturally decomposes into 60 half-edges, which each could be formed 
from just two different moulds, as in the case of the Solar Circle. 
Assembly would progress by making complete edges first, and then composing those into the dodecahedron. 
This will require some non-trivial jigs to make the overall geometry nice and symmetrical.
However this is my least favorite contender.  I don’t think that this structure is all that promising 
as an elegant sculpture. It will always come across as a dodecahedron with a complicated edge structure. 
The sensual details of the saddles and flanges seem to get suppressed by the overall geometrical structure.

For the TETRAHEDRON I have used curved edges that hug a sphere defined by the four tetrahedral vertices. 
In this initial design I have used quadruped saddles, since they make the joining at the 4 vertices most easy. 
I might be able to make the necessary modifications to the end branches without having to use 
a different kind of modeling program. But monkey saddle edges would also be doable. 
This structure naturally decomposes into 12 half-edges, 
which each could be formed from just two different moulds. 
Assembly would progress by making complete edges first. 
Composing those into the tetrahedron should be relatively straight forward, 
since forming equilateral triangles is much easier than forming regular pentagons. 
This is my favorite sculpture model so far. It has a natural 3-dimensionality; 
it does not primarily look like a figure from a geometry textbook; 
and the details of the Scherk-Collins saddles are able to show themselves 
without being overwhelmed by too much complexity.

Please think about these various possibilities at your leisure. 
I will be gone for the AAAS conference in Ashland and for a mini vacation 
from the 10th through the 21st of June. 
Let’s discuss where to focus our energy and what to do next after I return from my trip.

Best regards

Carlo