With the help of Tadeusz Dorozinski and Hunter Hughes, my new near-miss (the discovery of which was described here) is now better-understood. The isosceles triangles’ shared bases are about 5% longer than the solid’s other edges, which is within the range generally allowed for near-misses. I have not yet found any mention of this discovery before I found it yesterday, while playing with a broken, plastic d12.
Here is a net for this solid:
Also, here is its dual, as well as a net for the dual.
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.
The astronomical images here were taken by NASA’s New Horizons probe, while the geometry was done using Stella 4d: Polyhedron Navigator, a program you can try for free at http://www.software3d.com/Stella.php.
I predict a narrow electoral victory for Joe Biden, with a wider popular majority. I made this map on the website http://www.270towin.com. It’s free to use.
The faces of this polyhedron are twelve regular pentagons, thirty golden rectangles, and twenty equilateral triangles. I made it using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.
I made this faceted truncated octahedron using Stella 4d, which you can try right here. It has twenty faces: six large squares, six small squares, and eight {6/2} hexagons.
RobertLovesPi.net 