The situation is improving. There is now a still shot of a particular polytore (cubical crossover q = 0.674013, h = 1.285291, c = 1.0) at http://www.mapleprimes.com/files/8970_polytore2.pdf with an A4-sized planar net for the same at http://www.mapleprimes.com/files/8970_flattore2.pdf
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it's pretty tricky trying to build the wretched thing from a conventional net anyhow! Dividing into (at least) two pieces is recommended, to anyone reckless enough to attempt the feat.
Nonsense! I have one in my hands now, printed from the planar net above, and it folds and unfolds with almost no effort. The key is that you need to start by creasing all the edges in the correct direction: most of them should be mountain folds, but the two long edges of the four big triangles (touching the squares on their third edge) should be valleys. (Maybe the lines should be drawn differently, as is origami-standard.) Fred's polytorus has two types of vertices -- (a) eight that are corners of the original cube, where meet two squares and three triangles, and (b) four more where six triangles come together. In the pdf above, the (a)s are all interior vertices of the net, which means you get a really visceral feeling that they're flat: you watch them go from planar to part of the polyhedron. The planarity of the four (b)s, on the other hand, is not at all obvious here -- the six incident triangles are divided up three and three. Perhaps the model would be improved if the two mostly-interior type-(b) vertices were augmented with a little bit of webbing -- not a full extra copy of the three other triangles that meet there, but just a small stub with (two valley) fold lines, that the counterpart faces can snuggle into when the whole thing is assembled. --Michael -- Forewarned is worth an octopus in the bush.