Tag Archives: near miss

A Near-Miss to the Johnson Solids, Which I’m Naming the Ditrated Dodecahedron, Part Two

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(If you haven’t yet read part one, I strongly recommend reading it first.)

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.

These images were generated using Stella 4d, which you can try for free at http://www.software3d.com/Stella.php.

[Update: I am now convinced that I am not the first person to find this near-miss. On the other hand, I don’t know who that first person actually is.]

A Near-Miss to the Johnson Solids, Which I’m Naming the Ditrated Dodecahedron, Part One

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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.]

A Near-Miss to a Ring of Twelve Heptagonal Antiprisms

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Twelve heptagonal antiprisms, meeting at triangular faces, can make a ring with only a small gap remaining. It’s a “near miss,” made using Stella 4d, which you can try for free right here.