Here’s an icosahedron.
Here is is again, with prisms attached to each face. The prisms are 2.1 times as long as the triangle-edges.
This is a Stewart G3.
The final step is to attach these Stewart G3s to the ends of the long prisms shown above.
I think I’ll stop there. All these polyhedron-manipulations were performed using Stella 4d: Polyhedron Navigator, which you can try for free at http://www.software3d.com/Stella.php.
This is the tetrahedrally-diminished dodecahedron, which can be formed from the Platonic dodecahedron by using faceting. In this case, I used Zometools. If you’d like to look into buying some Zome of your own, the website to visit is http://www.zometool.com.
Stellating polyhedra is one of the things Zome was invented for. The images below are two different rotations of a stellated tetrahedrally-diminished dodecahedron.
In the center of this figure is a regular dodecahedron, but it’s hard to spot. It is then stellated to form a small stellated dodecahedron. Next, its outer vertices are joined by new edges: those of an icosahedron. This also results in the formation of a great dodecahedron. Finally, the icosahedron is stellated to form the great stellated dodecahedron. To take this further, one could connect the outer vertices with new edges: those of a dodecahedron. The entire process can begin again, then, and this could continue without limit, filling all of space.
Here’s a closer view of the interior:
Zometools may be purchased at http://www.zometool.com.
Just as I was about to take this picture of my latest Zome structure, Bandit the Cat slipped in through the all-blue decagonal hole where his tail is still located. He stayed in it until it had been thoroughly inspected, then slipped back out. He did this without causing the slightest bit of damage to the model.
This zonohedron has 242 faces, and is the largest convex polyhedron which can be built with only R0, B0, and Y0 Zome parts. It’s 67 cm tall. If made of all-1 struts, instead of zeroes, it is 1.08 m tall. With all size-2 struts, it is 1.75 m tall. I’ve actually built the “2” version, many times, with students. It requires a lot of support during construction, so that it does not collapse under its own weight. At one point, when teaching at Arkansas Governor’s School, we built one, and got eleven people inside it before it fell. People, unlike cats, generally enter through the multicolored dodecagonal holes.
The Zometool company (http://www.zometool.com) doesn’t make size-3 struts any more, but I still have some left from when they did. This zonohedron made of “3” struts would be 2.84 m tall, which is over nine feet. I may try to built one someday, but not today.
(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.]
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 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, if you’d like to make a prediction-map of your own..
I made this using Stella 4d, which you can try for free right here: http://www.software3d.com/Stella.php.
RobertLovesPi.net 