Yesterday, I played for the first time with GeoMag toys, which I recently purchased. I was quite surprised to have what I believe to be a near-miss to the Johnson solids appear before me, one I’ve never seen, within just a few minutes:
Here’s what it looks like, when viewed from two other angles.
The faces of this three-fold dihedral polyedron are six pentagons, twelve triangles, and nine quadrilaterals. The fact that it has been proven that only 92 Johnson solids exist means that all of these faces cannot be regular. However, the irregularity is so small that I could not detect it in this model.
Next, I used Polydrons to build a net of this near-miss candidate.
What to do next was obvious: remove the “belt” of nine quadrilaterals, creating a net for a second near-miss candidate.
Having constructed this net, I then returned GeoMags to build a 3-d model of this second, “unbelted” near-miss candidate.
I then wondered if I could make a third such solid by removal of the triangles, all of which appeared to be the lateral faces of pyramids.
Could I remove them? Yes, and I did so. Did this create a third near-miss candidate? No. The resulting polyhedron, shown immediately above, is non-convex, and therefore cannot be a near-miss. The faces with dihedral angles greater than 180° are the triangle-pairs found where the pyramids were in the previous model.
With the “belted” and “unbelted” polyhedra before this non-convex non-candidate, the next step is to share them with other polyhedra enthusiasts, get their input regarding the question of whether these are genuine near-misses, and see if these polyhedra have already been found, unknown to me, by someone else.
[Update: please see the next two posts for more on these near-miss candidates.]
RobertLovesPi.net 
Great models – I like the look of Tthe GeoMag construction toy.
I made a couple of near miss Johnson solids using Jovo shapes (similar to Polydron). The pieces fitted together okay but some exhibited bowing, so not an exact fit. When I get some time I will try to reproduce the models you have displayed.
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Thanks! I find the magnetic building systems much easier to work with than Polydrons or Jovotoys — but the latter are ideal for making nets, as I did here (and for classroom use, too). If you do get any GeoMag sets, be sure to get the kind with the stabilizing plastic polygons; they were invaluable when making these models.
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