On 8/11/09, Fred lunnon <fred.lunnon@gmail.com> wrote:
... Jim Propp asked for a polyhedral origami torus, that is one properly embedded in 3-space, and having vertices where the face angles always add to 2\pi. The following construction gives a pencil of examples with 20 faces (4 square + 16 triangular), 12 vertices, 32 edges. ...
Maybe it should be emphasised that (for 0 < q < 1) these polyhedra are =properly= embedded in 3-space --- there are no touching, overlapping or self-intersecting faces, edges or vertices --- the interior is open and connected. The only regrettable feature is that the corners and some edges are re-entrant: easily seen to be unavoidable if they are to be "flat" (developable). A rather attractive robotics and graphics project would be to animate the folding of the max h(q) case, starting from a flat sheet: the "map" is easy to construct. I'm trying not to think about it ... Regarding the relation between height h and radius q of the interior vertices: by symmetry, it's obvious that the implicit constraint f(h, q) = 0 say must involve only h^2. In fact, for every instance I've examined, if q is rational, then h^2 is also apparently rational. Surely this must mean that h^2 = g(q), where it seems reasonable to conjecture that g(q) might be polynomial. I haven't tried to deduce this from the problem; however, solving equations for its coefficients doesn't seem to work. Any ideas, anybody? The values I have (guessed) so far are h(-sqrt2) = h(+sqrt2) = 0; h^2(0) = h^2(1) = 3/2, h^2(1/2) = h^2(2/3) = 119/72, h^2(-1) = 11/18, h^2(-1/2) = 231/200, h^2(1/3) = 731/450, h^2(-4/3) = 59/450, h^2(-2/3) = 287/288, h^2(-1/3) = 1139/882, h^2(3/4) = 1311/800, h^2(1/4) = 2511/1568. The plot of f(h, q) = 0, or h^2 = g(q) if such it be, is a quartic-like oval, symmetric about the h-axis, and flattened on the positive side of the q-axis. It's noteworthy that on the other side, the extremum (q, h) = (-sqrt2, 0) occurs just at the place where the (by now) self-intersecting polyhedron has everted so far that, as q increases further, it is just about to become properly embedded once more --- but cannot, as h becomes imaginary! Incidentally, I tried using Plouffe's inverter for these computations --- are you there, Simon? On the home page http://pi.lacim.uqam.ca/ for "Histoiry" read "History"; for "Tables de constantes" read "Tables of Constants". It didn't cope too well with some cases, I have to say: for example, it failed to recognise that 0.9965277777777785 ~ 287/288; and it claimed 1.285604051711792 ~ 1/12*(-238)^(1/2), nearly correct apart from a mysterious intrusive factor of \iota! Fred Lunnon