Selections from the Second Hundred Stellations of the Rhombicosidodecahedron

Posted on 31 May 2015 by RobertLovesPi

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.

A similar figure is the 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.

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.

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

Next is the 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.

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

Next, the 191st stellation:

And, after that, the 192nd stellation.

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

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

Selections from the First Hundred Stellations of the Rhombicosidodecahedron

Posted on 31 May 2015 by RobertLovesPi

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.

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.

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

Finally, there are the the planes of the thirty square faces.

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:

Next, the 30th stellation:

The next one is the 33rd stellation.

And next, the 38th stellation.

Here is the 46th stellation:

And the 48th stellation:

Next, the 58th stellation:

And now, the 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.

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.

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

A Great Icosahedron, Augmented with Twenty Icosahedra

Posted on 27 May 2015 by RobertLovesPi

The polyhedral clusters above and below use different coloring-schemes, but are otherwise identical. Invisible, in the center, is a great icosahedron. Each of its faces has been augmented by a (Platonic) icosahedron.

Both images were created using Stella 4d, software you can try here.

Selections from the Stellation-Series of the Great Rhombicosidodecahedron

Posted on 25 May 2015 by RobertLovesPi

The great rhombicosidodecahedron, also known as the truncated icosidodecahedron, has a long and complex stellation series. Here are some highlights from that series, chosen using aesthetic, rather than mathematical, criteria.

All these virtual models were made using Stella 4d, which you can try and/or buy here.

Two Polyhedral Compounds: the Icosidodecahedron with the Truncated Cube, and the Rhombic Triacontahedron with the Triakis Octahedron

Posted on 22 May 2015 by RobertLovesPi

These two compounds, above and below, are duals. Also, in each of them, one polyhedron with icosidodecahedral symmetry is combined with a second polyhedron with cuboctahedral symmetry to form a compound with pyritohedral symmetry: the symmetry of a standard volleyball.

A program called Stella 4d was used to make these compounds, and create these images. It may be purchased, or tried for free, at this website.

Sixteen Polyhedra with Cuboctahedral Symmetry

Posted on 16 May 2015 by RobertLovesPi

I made these using Stella 4d, a computer program available at this website.

Two Compounds with Pyritohedral Symmetry: the Icosidodecahedron / Truncated Octahedron Compound, and the Rhombic Triacontahedron / Tetrakis Cube Compound

Posted on 12 May 2015 by RobertLovesPi

Stella 4d, a program you can try here, was used to create these two compounds. Both have pyritohedral symmetry: the symmetry of a standard volleyball. The two compounds are also duals.

A Large Collection of Polyhedra with Icosidodecahedral Symmetry, Some of Them Chiral

Posted on 8 May 2015 by RobertLovesPi

I made these using Stella 4d, available here.

A Dodecahedron with Four Symetrically-Truncated Vertices

Posted on 8 May 2015 by RobertLovesPi

Dodecahedra have icosahedral (also called icosidodecahedral) symmetry. In the figure above, this symmetry is changed to tetrahedral, by truncation of four vertices with positions corresponding to the vertices (or, instead, faces) of a tetrahedron. The interchangeability of vertices and faces for the tetrahedron is related to the fact that the tetrahedron is self-dual.

[Image created using Stella 4d, available here.]

Selections from the Stellation-Series of the Icosidodecahedron

Posted on 4 May 2015 by RobertLovesPi

The icosidodecahedron has a long and interesting stellation-series, and you can see the whole thing using Stella 4d, the program I used to make the rotating .gifs here. Rather than keep the scale the same in each frame, I set the program to make the polyhedron as large as possible, while still fitting in the image-box. This creates the illusion that the polyhedra below are “breathing.”

The polyhedron above is the 20th stellation of the icosidodecahedron — the one that appeared as the sole image in the last post here, but with completely different colors. The next one shown is the 31st stellation.