tetrated | RobertLovesPi.net

I had a strange mishap recently with a member of my large collection of polyhedral dice. This hollow d12 fell apart, into two panels of six pentagons each.

I held them together at vertices, rotating one of the panels slightly.

Those gaps aren’t rhombi, because their four vertices are noncoplanar. Instead of rhombi, therefore, I’m filling the gaps with pairs of isosceles triangles. I’m going to request help from experts to find the edge length ratio for these isosceles triangles, but I know it isn’t 1:1, since all 92 of the Johnson solids have been found.

I think this particular near-miss may have been found and posted before in a Facebook group devoted to polyhedra, as a magnetic ball-and-stick model, but I don’t think it was named at that time. The name “ditrated dodecahedron” is derived from “tetrated dodecahedron,” which you can read about right here. The tetrated dodecahedron has four panels of pentagons rotated away from the center, while the ditrated dodecahedron has only two panels. The latter’s faces are twelve regular pentagons, and ten isosceles triangles.

I’m going to post this in that Facebook group where I think this near-miss to the Johnson solids may have been seen before, in an effort to spread the discovery-credit around anywhere it has been earned. I’d also like to have a Stella 4d model of this solid, and for that, again, I need the help of experts. Once I know more about this near-miss, I’ll post part two. [Update: part two is right here.]

The Hextrated Pentagonal Icositetrahedron

Years ago, I split a dodecahedron into four panels of pentagons, rotated the pentagon-panels and moved them outward from the center, and did so just the right amount to create gaps that could be filled with triangles. Thus was named the tetrated dodecahedron, which you can read more about here: https://en.wikipedia.org/wiki/Tetrated_dodecahedron

The choice of word “tetrated” was somewhat unfortunate, for tetration already exists in mathematics, as a means of expressing very large numbers, and which I shall not explain here. I didn’t learn this until much later, though, and by that time, it was too late to turn “tetrate” into something else. It had come to mean an operation one does on a polyhedron: break it into four multi-face panels, move them out and rotate them just enough, and fill the resulting gaps with triangles.

As such, “tetrate” can, in the geometrical sense, be modified for differing numbers of panels of multiple faces from a polyhedron. Consider the pentagonal icositetrahedron, the dual of the snub cube. Here, it has been split into six panels, and then each panel moved out from the center and rotated, with triangles filling the gaps. The triangles differ between color-groups slightly, but are close to equilateral, except for the ones shown in green, which simply are equilateral.

(Image created with Stella 4d — software you can try yourself at http://www.software3d.com/Stella.php.)