The Greatly Augmented Rhombicosidodecahedron

Greatly Augmented Rhombicosidodeca

I call this variant on the rhombicosidocahedron “greatly augmented” because it was formed by augmenting each pentagonal face of a central rhombicosidodecahedron with great dodecahedra, while each triangular face is augmented with great icosahedra. It was made using Stella 4d, which may be found here.

Icosidodecahedra, Icosahedra, and Dodecahedra

If one starts with a single icosidodecahedron, and then augments its pentagonal faces with dodecahedra, and its trianguar faces with icosahedra, this is the result.

Augmented Icosidodeca

This figure has gaps in it where two pentagons and two triangles meet around a vertex. If one puts icosidodecahedra in those gaps, this is the resulting figure.

rEAugmented Icosidodeca

Next, once again, the pentagonal faces are augmented with dodecahedra, and the triangular faces with icosahedra.

rerEAugmented Icosidodeca

These virtual polyhedral models were all built using Stella 4d, available at http://www.software3d.com/Stella.php.

An Icosahedron, Augmented with Twenty Triangular Cupolae, Together with Its Dual

Icosahedron Augmented with Triangular Cupolae

After making the above polyhedron using Stella 4d (a program you can try for free at www.software3d.com/Stella.php), I checked its dual, which is shown below. I was surprised at its appearance, for it resembles a stellated polyhedron, even though it was created by a completely different process.

Icosahedron Augmented with Triangular Cupolae's Dual

An Octahedron, Augmented with Eight Great Icosahedra, and the Dual of this Augmented Polyhedral Cluster

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An Octahedron, Augmented with Eight Great Icosahedra, and the Dual of this Augmented Polyhedral Cluster

I made these using Stella 4d, which you can try at http://www.software3d.com/Stella.php. Here is its dual, also:

GrIcosadual-Augmented Octa

 

 

The Convex Hull of a Prism-Augmented Icosidodecahedron As a (Possibly) New Near-Miss Candidate

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The Convex Hull of a Prism-Augmented Icosidodecahedron As a New Near-Miss Candidate

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.

NM1

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:

NMNC

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.”]

A Rhombcuboctahedron, with Its Square Faces Augmented By Hexacontakaitriacosioigonal Prisms, Together with Two Views of the Convex Hull of That Augmented Polyhedron

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A Rhombcuboctahedron, with Its Square Faces Augmented By Hexacontakaitriacosioigonal Prisms

The eighteen regular prisms (whose bases each have 360 sides) augmenting the square faces of the rhombcuboctahedron hidden in the center, above, can be oriented in more than one way. I simply chose the orientation I liked best.

After that, I took the convex hull of the figure above, just to see what would happen. With each different face type having a different color, it looks like this (click to enlarge either or both images below, if you wish):

Convex hull

I then chose a different color-scheme. Instead of giving faces of each type a different color, I colored the faces by their number of sides. This led to a more pleasing result:

Convex hugll

The things that look like rounded yellow rectangles are an illusion; polyhedra don’t have curved faces. They are actually numerous thin, adjacent, near-coplanar rectangles with the same color.

All three images were created with Stella 4d, software available at http://www.software3d.com/Stella.php.