A Rhombic Triacontahedron, Made of Icosidodecahedra

It turns out that it is possible to use multiple icosidodecahedra as building blocks to build that polyhedron’s dual, the rhombic triacontahedron. This construction appears below, in four different coloring-schemes.

Augmented IcosidodecaAugmented Icosidodeca2Augmented Icosidodeca3Augmented Icosidodeca4

These rotating virtual models were constructed using Stella 4d, a program available at http://www.software3d.com/Stella.php.

A Rhombicosidodecahedron, Made of Rhombicosidodecahedra

This “metarhombicosidodecahedron” took a long time to build, using Stella 4d, which you can find at http://www.software3d.com/Stella.php — so, when I finished it, I made five different versions of it, by altering the coloring settings. I hope you like it.

Augmented Rhombicosidodeca

Augmented Rhombicosidodeca2

Augmented Rhombicosidodeca4

Augmented Rhombicosidodeca5

Augmented Rhombicosidodeca6

A Collection of Unusual Polyhedra

In the post directly before this one, the third image was an icosahedral cluster of icosahedra. Curious about what its convex hull would look like, I made it, and thereby saw the first polyhedron I have encountered which has 68 triangular faces.

68 triangles Convex hull

Still curious, I next examined this polyhedron’s dual. The result was an unusual 36-faced polyhedron, with a dozen irregular heptagons, and two different sets of a dozen irregular pentagons.

dual of 68 triangles Convex hull -- this dual has 36 faces including 12 heptagons and 12 each of two types of pentagon

Stella 4d (the program I used to make all these images), which is available at http://www.software3d.com/Stella.php, has a “try to make faces regular” function, and I tried to use it on this 36-faced polyhedron. When making the faces regular is not possible, as was the case this time, it sometimes produce surprising results — and this turned out to be one of these times.

dual of the 68-triangle polyhedron after 'try to make faces regular' used

The next thing I did was to examine the dual of this latest polyhedron. The result, a cluster of tetrahedra and triangles, was completely unexpected.

dual of the dual of the 68-triangle polyhedron after 'try to make faces regular' used

The next alteration I performed was to create the convex hull of this cluster of triangles and tetrahedra.

Convex hull of that triangular mess

Having seen that, I wanted to see its dual, so I made it. It turned out to have a dozen faces which are kites, plus another dozen which are irregular pentagons.

dual of the Convex hull of that triangular mess 12 kites and 12 irregular pentagons

Next, I tried the “try to make faces regular” function again — and, once more, was surprised by the result.

dozen kites and dozen pentagons after 'try to make faces regular' used

Out of curiosity, I then created this latest polyhedron’s convex hull. It turned out to have four faces which are equilateral triangles, a dozen other faces which are isosceles triangles, and a dozen faces which are irregular pentagons.

Convex hull Z

Next, I created the dual of this polyhedron, and it turns out to have faces which, while not identical, can be described the same way: four equilateral triangles, a dozen other isosceles triangles, and a dozen irregular pentagons — again. To find such similarity between a polyhedron and its dual is quite uncommon.

dual of Convex hull Z

I next attempted the “try to make faces regular” function, once more. Stella 4d, this time, was able to make the pentagons regular, and the triangles which were already regular stayed that way, as well. However, to accomplish this, the twelve other isosceles triangles not only changed shape a bit, but also shifted their orientation inward, making the overall result a non-convex polyhedron.

TTMFR

Having a non-convex polyhedron on my hands, the next step was obvious: create its convex hull. One more, I saw a polyhedron with faces which were four equilateral triangles, a dozen other isosceles triangles, and a dozen regular pentagons.

Convex hull

I then created the dual of this polyhedron, and, again, found myself looking at a polyhedron with, as faces, a dozen irregular pentagons, a dozen identical isosceles triangles, and four regular triangles. However, the arrangement of these faces was noticeably different than before.

latest Convex hull

Given this difference in face-arrangement, I decided, once more, to use the “try to make faces regular” function of Stella 4d. The results were, as before, unexpected.

TTMFRA

Next, I created this latest polyhedron’s dual.

TTMFRA dual

At no point in this particular “polyhedral journey,” as I call them, had I used stellation — so I decided to make that my next step. After stellating this last polyhedron 109 times, I found this:

109 stellationsTTMFRA dual

I then created the dual of this polyhedron. The result, unexpectedly, had a cuboctahedral appearance.

Faceted Dual

A single stellation of this latest polyhedron radically altered its appearance.

stellation Faceted Dual

My next step was to create the dual of this polyhedron.

dual Faceted Stellated Poly

This seemed like a good place to stop, and so I did.

Three Polyhedral Clusters of Icosahedra

In the last post on this blog, there were three images, and the first of these was a rotating icosahedron, rendered in three face-colors. After making it, I decided to see what I could build, using these tri-colored icosahedra as building blocks. Augmenting the central icosahedron’s red and blue faces with identical icosahedra creates this cubic cluster of nine icosahedra:

cube of icosahedra

If, on the other hand, this augmentation is performed only on the blue faces of the central icosahedron, the result is a tetrahedral cluster of five icosahedra:

5 icosa

The next augmentation I performed started with this tetrahedral cluster of five icosahedra, and added twelve more of these icosahedra, one on each of the blue faces of the four outer icosahedra. The result is a cluster of 17 icosahedra, with an overall icosahedral shape.

icosa made of icosa

All of these images were made using Stella 4d, which is available at http://www.software3d.com/Stella.php.

On Icosahedra, and Pyritohedral Symmetry

Icosa pyrito & two tets

In this icosahedron, the four blue faces are positioned in such a way as to demonstrate tetrahedral symmetry. The same is true of the four red faces. The remaining twelve faces demonstrate pyritohedral symmetry, which is much less well-known. It was these twelve faces that I once distorted to form what I named the “golden icosahedron” (right here: https://robertlovespi.wordpress.com/2013/02/08/the-golden-icosahedron/), but, at that point, I had not yet learned the term for this unusual symmetry-type.

To most people, the most familiar object with pyritohedral symmetry is a volleyball. Here is a diagram of a volleyball’s seams, found on Wikipedia.

Volleyball_seams_diagram

Besides the golden icosahedron I found, back in 2013, there is another, better-known, alteration of the icosahedron which has pyritohedral symmetry, and it is called Jessen’s icosahedron. Here’s what it looks like, in this image, which I found at http://en.wikipedia.org/wiki/Jessen%27s_icosahedron.

Jessen_icosahedron

The rotating icosahedron at the top of this post was made using Stella 4d, a program which may be purchased, or tried for free (as a trial version) at http://www.software3d.com/Stella.php.

A Space-Filling Arrangement of Polyhedra Using Truncated Cubes, Rhombcuboctahedra, Cubes, and Octagonal Prisms

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms

This image above has only one polyhedron-type hidden from view, in the center:  a red truncated cube. Next, more of this pattern I just found will be added.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 2

The next step will be to add another layer of blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 3And now, more yellow cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 4This was an accidental discovery I made, just messing around with Stella 4d, a program you may try for yourself at http://www.software3d.com/Stella.php. The next cells added will be red truncated cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 5

Next up, I’ll add a set of pink rhombcuboctahedra.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 6The next set of polyhedra added: some yellow cubes, and blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 7Now I’ll add more of the red truncated cubes.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 8At this point, more yellow cubes are needed.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 9The next polyhedra added will be pink rhombcuboctahedra.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 10

And now, more of the blue octagonal prisms.

space filling attempt with RCO and cubes and truncated cubes and octagonal prisms 11

As long as this pattern is followed, this may be continued without limit, filling space, without leaving any gaps.

“How are you today?”

how are you

At least in this part of the world, “How are you today?” — or variations thereof — is commonly used as a way to start conversations, as a bit of “small talk.” The odd part of this social convention is that, when people ask this, they usually don’t really want to hear an honest answer — or, indeed, any answer at all that isn’t part of the standard “small talk” script.

The usual answer (“Fine, thanks,” or something like it) is yet another empty phrase — more small talk. Unusual answers, though, have great potential for fun. I first encountered this idea in a class I took, many years before, where the teacher told us that his habit was to answer, instead, with an upbeat, “Getting better!” I’ve tried this, and the facial expressions often seen, in response, are indeed quite entertaining. Small talk is annoying — to me, anyway — but disrupting it, by simply deviating from the usual script, can be a lot of fun.

Here are some other possible answers, but this game is probably most fun if you make up your own.

  • “I’m glad you asked. Actually, my feet hurt. Do you know why?”
  • “Well, I’d feel a lot better if I hadn’t just blown my whole budget for the week on chocolate. It tasted good when I ate it all for breakfast this morning, though!”
  • “Hopefully, I’ll be able to answer your question in a few minutes. Say, where’s the nearest restroom?”
  • “Terrible. My beloved pet cricket just died.”
  • “I’m hoping it gets better soon. Could you recommend a good mechanic nearby, as well as a chiropractor?”
  • “I’m feeling great! There is nothing like a couple of extra-strength placebos to start the day!”
  • “I’m okay now, but I’m not looking forward to this afternoon at all. You have heard about the giant asteroid heading straight for us, right? It’s supposed to hit somewhere near downtown, at about four o’clock.”
  • “Well, I’m broke. May I borrow fifty bucks until next month?”

While I do greatly value honesty, I obviously exclude jokes from the category of lies. Also, suggestions for other funny responses, in comments, would be much appreciated.