With help from friends on Facebook, I was able to figure out how to make the second of the near-miss candidates in the last post, using* Stella 4d: Polyhedron Navigator*, a program available here. This is quite helpful, for *Stella* has a “measurement mode” than lets me determine just how far off from regularity a given polyhedron is. This is what the “unbelted” polyhedron from the last post looks like, with the pentagons regular:

In this polyhedron, although the pentagons are regular, the triangles are scalene, with angles measuring ~55.35, ~60.81, and ~63.84 degrees. Of the three edge lengths needed for this, the longest is ~9.1% longer than the shortest, and the triangles are definitely non-regular — by visual inspection alone. It is possible to “tidy up” the triangles a bit, but only at the cost of making the pentagons visibly irregular. This is enough to make the call on the “unbelted” near-miss candidate from the last post — it’s a “near near miss,” not a true “near miss.”

All polyhedra in the last post, as it turns out, are related to another near-miss, the discovery of which I had nothing to do with. It has six pentagonal faces, and four which are quadrilaterals. This near-miss may be found here: http://www.mathcurve.com/polyedres/enneaedre/enneaedre.shtml.

[Note: see the next post, also, for more about these polyhedra.]

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## About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. The majority of these things are geometrical. Welcome to my little slice of the Internet.
The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.