Selections from the Second Hundred Stellations of the Rhombicosidodecahedron

This survey began in the last post, with selections from the first hundred stellations of this Archimedean solid. In this survey of the second hundred stellations, the first one I find noteworthy enough for inclusion here is the 102nd stellation.

Rhombicosidodeca 102nd stellation

A similar figure is the 111th stellation:

Rhombicosidodeca 111th stellation

There followed a long “desert” when I did not find any that really “grabbed” me . . . and then I came to the 174th stellation.

Rhombicosidodeca 174th stellation

The fact that it is monocolored, the way I had Stella 4d set, told me immediately that this stellation (the one above) has only one face-type. There are twenty of these faces; they are each equilateral hexagons which “circumscibe,” in a way, the triangular faces of an icosahedron. For this reason, I suspect this is also one of the stellations of the icosahedron; I’m making a mental note to do exactly that.

I also make a second virtual model of the 174th stellation of the rhombicosidodecahedron, with the faces colored in such a way as to make the interpenetrating equilateral hexagons more obvious.

Stellated Rhombicosidodeca

After that interesting stellation, the next one to caught my attention is the 179th stellation.

Rhombicosidodeca 179th stellation

Next of note, the 182nd stellation is similar to the icosahedron/dodecahedron compound, but with the dodecaheron larger than it is in that compound, so that edges, one from each component polyhedron, do not intersect, but are instead skew. Another way to view it is that the dodecahedron is encasing the icoahedron, but with enough room left for portions of the icosahedron to protrude from the faces of the “dodecahedral cage.”

Rhombicosidodeca 182nd stellation

Next is the 183rd stellation.

Rhombicosidodeca 183rd stellation

Here is the 187th stellation, which is quite similar to the last one shown. The pulsating effect, first seen in the last post above, is an accident, and not discovered until after these images were already made, using Stella 4d, which may be tried here. Why didn’t I re-create the .gifs? Simple: I don’t feel like taking the ~10 minutes each to do so.

Rhombicosidodeca 187th stellation

The 190th stellation may also be viewed as a dodecahedron, augmented with variations of pentagonal pyramids on each face:

Rhombicosidodeca 190th stellation

Next, the 191st stellation:

Rhombicosidodeca 191st stellation

And, after that, the 192nd stellation.

Rhombicosidodeca 192nd stellation

The next stellation which grabbed by attention: the 198th.

Rhombicosidodeca 198th stellation

Finally, I’ll close this set of highlights from this part of the rhombicosidodecahedron’s stellation-series with that solid’s 199th stellation.

Rhombicosidodeca 199th stellation

Selections from the First Hundred Stellations of the Rhombicosidodecahedron

Since shortly after I learned of their existence, I have found the rhombicosidodecahedron to be the most attractive of the Archimedean solids. That’s a personal aesthetic statement, of course, not a mathematical one.

Rhombicosidodeca

This solid has a long stellation-series. With Stella 4d, the program I used to make these images, it’s easy to simply scroll through them. The stellation of this polyhedron follows these stellation-diagrams; I used Stella 4d to make them as well. You may research, try, or buy this program at this website. The first of these stellation-diagrams is for the planes of the twelve pentagonal faces.

Rhombicosidodeca -StelDiag for twelve face-planes

For the planes of the twenty triangular faces, this is the stellation-diagram:

Rhombicosidodeca -StelDiag for twenty face-planes-StelDiagFinally, there are the the planes of the thirty square faces.

Rhombicosidodeca -StelDiag for thirty face-planes-StelDiag

The following survey of the first hundred stellations is not intended to be exhaustive; I’m including all those I find worthy of inclusion on subjective aesthetic grounds. The first stellation shown here is actually the 25th stellation of the rhombicosidodecahedron:

Rhombicosidodeca 25th stellation

Next, the 30th stellation:

Rhombicosidodeca 30th stellation

The next one is the 33rd stellation.

Rhombicosidodeca 33rd stellation

And next, the 38th stellation.

Rhombicosidodeca 38th stellation

Here is the 46th stellation:

Rhombicosidodeca 46th stellation

And the 48th stellation:

Rhombicosidodeca 48th stellation

Next, the 58th stellation:

Rhombicosidodeca 58th stellation

And now, the 62nd stellation.

Rhombicosidodeca 62nd stellation

Next is the 85th stellation; it’s also a compound of an icosahedron (blue), and a yellow polyhedron I have not yet identified, except as the nth stellation of something. This I know: I have seen the yellow polyhedron before. If you happen to know what it is, the identify it in a comment.

Rhombicosidodeca 85th stellation might also be a compound

The next stellation shown is the next one in the series, the 86th. It demonstrates a phenomenon I have observed, but cannot explain, and that is the tendency, in sequences of stellations, to have a large number of similar stellations in a row, followed by a sudden, much more extreme change in appearance, from one stellation to the next, as seen here. It’s a phenomenon which I would like to better understand.

Rhombicosidodeca 86th stellation

To be continued, with selections from the next hundred stellations….

A Cluster of Twenty Great Icosahedra, Excavated from the Faces of a Central Icosahedron, Along with Its Dual

Augmented Icosa its excavated with great icosas

These twenty great icosahedra were excavated from the faces of a central icosahedron, which is concealed in the figure’s center. These excavations exceed the limits of the central icosahedron, resulting in each great icosahedron protruding in a direction opposite that of the face from which it is excavated. In a certain sense, then, the figure above has negative volume.

To make this, I used software called Stella 4d: Polyhedron Navigator. It can be researched, bought, or tried for free here.

Also, here is the dual of the polyhedral cluster above, made with the same program.

Augmented Icosa its excavated with great icosas the dual