These were all derived in various ways from the polyhedra seen in the last two posts. The rest are smaller at first, but each can be enlarged with a single click of your mouse. Each of them has icosidodecahedral symmetry.
I used Stella 4d to make these images, and you can find that program at http://www.software3d.com/Stella.php.
This is the dual of the polyhedron seen as the second image in the last post on this blog. If colored differently, so that only parallel faces have the same color, it looks like this (click to enlarge):
I used Stella 4d to make these images, and you can find that program at http://www.software3d.com/Stella.php.
Each pair is a different color. Because these decagons intersect in space, but do not meet at edges, they do not form a true polyhedron. They are merely a symmetrical configuration of twelve decagons in space, surrounding a central point.
I made this out a “true polyhedron” by hiding all the other faces from view. Before the hiding and recoloring of faces, this looked this way (you can click on it to enlarge it):
I used Stella 4d to make these images, and you can find that program at http://www.software3d.com/Stella.php.
Click on the smaller pictures, if you wish to enlarge them, one at a time.
Those last two were duals of each other. The next two are as well.
These next two are duals, as are the pair that follows them.
I’ll finish with one more dual pair.
All of these were made using Stella 4d: Polyhedron Navigator, which is available at http://www.software3d.com/Stella.php.
Created using Stella 4d, software available at http://www.software3d.com/Stella.php.
I created this using Stella 4d: Polyhedron Navigator, a program you can find athttp://www.software3d.com/Stella.php.
To make this, I started with the dual of the great rhombicosidodecahedron, a polyhedron known as the dysdyakis triacontahedron. I then augmented half of its faces with tall prisms (thereby creating the chirality in this polyhedron), and took the convex hull of the result. The sixty red triangles are the tops of the augmentation-prisms.
A stellation of the above polyhedron, and a color-change, produced this result, also chiral. It may be enlarged with a click.
These polyhedra were created using Stella 4d, a program which you may buy — or try for free, as a trial download — at http://www.software3d.com/Stella.php.
I created this using Stella 4d: Polyhedron Navigator, a program you can find at http://www.software3d.com/Stella.php.
If someone had asked me if it were possible to form a symmetric polyhedra out of irregular triangles and octagons, using exactly sixty of one type each, I would have guessed that it were not possible. Why does it work here? Part of the reason is that each triangle borders three octagons, and each octagon borders three triangles — a necessary, but not sufficient, condition. This is a partial truncation of an isomorph of the pentagonal hexacontahedron, the dual of the snub dodecahedron. As such, no surprise — it’s chiral.
This was made while stumbling about in the wilderness of the infinite number of possible polyhedra using Stella 4d: Polyhedron Navigator. You can get it here: http://www.software3d.com/Stella.php.
To make this polyhedron using Stella 4d: Polyhedron Navigator (a program which is available at this website), I started with an icosidodecahedron, augmented all faces with prisms of height 1.6 times greater than their bases’ edge length, and then took the convex hull of the result. I’m proposing it as a candidate for the loosely defined group of polyhedra called near-misses to the 92 Johnson solids: convex polyhedra which are almost, but not quite, Johnson solids, due to slight irregularity in some of their faces.
In this case, the pentagons and green triangles are regular, and have the same edge length. The blue triangles, however, are isosceles, with vertex angles of ~67.6687 degrees. The yellow almost-squares are actually rectangles, with edges next to blue triangles which are ~2.536% longer than the edges next to pentagons or green triangles.
I stumbled upon this design earlier today, while simply exploring polyhedra more-or-less randomly, using Stella. Below is the prototype I found at that time, which I merely made a .gif of, but did not perform measurements on.
In this prototype, the most significant difference I can detect is in the yellow faces, which are trapezoids, rather than rectangles, since the pentagon edge-length is slightly longer than that of the green triangles.
Stella has a “try to make faces regular” function built-in to try to help improve upon polyhedra such as these, but here’s what happens when that function is used on the first polyhedron shown above:
Behold! It worked — all of the faces are perfectly regular. However, that caused another problem to appear, and you can see it most easily by looking at the blue triangle-pairs: this polyhedron is slightly non-convex. It’s also easily described as a truncated dodecahedron, with each of the twelve decagonal faces augmented by a pentagonal rotunda.
I’ll show this to some other people who are polyhedron-experts, and will update this post with what I find after I’ve talked to them. My questions for them, as usual in such situations, are two in number:
1. Has this polyhedron been found before?
2. Is it close enough to regularity to qualify for “near-miss” status?
If it hasn’t been found before, but is judged unworthy of “near-miss” status, it will at least join the newly-described group I call “near near-misses” — polyhedra which don’t quite qualify for near-miss status, by visual inspection. Obviously, this new group’s definition is even more “fuzzy” than that of the near-misses, but there is a need for such a category, nonetheless.
[Update: Robert Webb, who wrote Stella 4d (and is not the blogger here, despite our sharing a first name), has seen this before, so it isn’t an original discovery of mine. He doesn’t accept it as a near-miss on the grounds that it naturally “wants” to be non-convex, as seen in the last of the three images in this post, and I agree with his reasoning. I’m therefore considering this to be a “near-near-miss.”]
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