My name made the Stella Library discovery credits!


My name made the Stella library discovery credits!

Stella’s creator just came out with a new version of Stella 4d, and a discovery of mine made the built-in library that comes with that software. This is my blog, so I get to brag about that, right? My legal name appears in the small print on the right side, at the end of the first long paragraph. I added the red ellipses to make it easier to find.

You can see the earlier posts related to my discovery of this zonish truncated icosahedron here:

If you’d like to try or buy this software (I recommend Stella 4d over the other options), here’s the link for that:

The Beauty of Uselessness


The Beauty of Uselessness

Given the name of this blog, and the familiarity of the number pictured above, I’m sure you recognize it as the beginning of my favorite number, pi. On various websites, you can find far more digits than are shown above. However, just the digits shown in the top row here are greater in number than that needed for any real-world, practical application. Is pi useful? Is mathematics itself useful? Of course they are . . . but those questions miss the point entirely.

Every teacher who has been in the field for long has heard the complaint, disguised as a question, “What are we ever going to use this for?” Unfortunately, most school systems, as well as teacher-training programs, have chosen to respond to this well-known complaint by repeatedly telling teachers, and teachers-in-training, that it is of extreme importance to show students how the curriculum is “relevant,” and adjust curricula to make them more so. This usually boils down, of course, to trying to convince students that learning is important because, supposedly, education = a better job in the future = more money. Sometime this “equation” works, and sometimes it does not, but it always misses a key point, one that should not be left out, but too often is.

It is a fallacy that learning has to have a practical application to be a worthwhile endeavor. There’s more to life than the fattening of bank accounts. Sadly, many of those making decisions in education do not realize this. Their attempts to reduce education to a strictly utilitarian approach are causing great harm.

This scenario has happened many times: a mathematician discovers an elegant proof to a surprising theorem, a physicist figures out something previously unknown about the nature of reality, or a researcher in another field does something comparable, and someone then asks them, “But what can this be used for?” On occasion, tired of hearing this utilitarian refrain, such researchers give unusually honest responses which surprise and confuse many people — such as, “Someone else might, at some point, find a practical application for this . . . but I sincerely hope that never happens.” Such a response is rarely understood, but it makes the person who says it feel better to vent some of their frustration with those who are obsessed with tawdry, real-world applications for everything.

Many humans — and this is a terrible shame — live almost their entire lives like rats in mazes, running down passages and around corners, chasing tangible rewards — cheese for the rats, or the ability to buy, say, a fancy new car, in the case of the people. People shouldn’t live like lab rats . . . and, unlike lab rats, we don’t have to. People are smart enough to find higher purposes in life. People can, in other words, find, understand, and appreciate beauty — in things which are useless, in the sense that they have no useful applications. We can appreciate things that transcend mere utility, if we choose to do so.

Much of life is utterly banal, for a great many people. They wake up each day, work themselves into exhaustion at horribly boring jobs, go home, numb themselves with television, massive alcohol consumption, or other hollow pursuits, fall asleep, and then get up and repeat the process the next day . . . and then they finally get old and die. Life can be so much more than that, though, and it should be.

The researchers I described earlier aren’t doing what they do for money, or even the potential for fame. Are such things as mathematics and physics useful? Yes, they are, but that isn’t why pure researchers do them. The same can be said for having sex: it’s useful because it produces replacement humans, but that isn’t why most people do it. People have sex, obviously, because they enjoy it. In simpler terms: it’s fun. Most people understand this concept as it relates to sex, but far fewer understand it when it relates to other aspects of life, particularly those of an academic nature.

Academic pursuits are of much greater value when the motivation involved is joy, and the fun involved, rather than avarice. As scrolling through this blog will show you, I enjoy searching for polyhedra which have not been seen before. I certainly don’t expect to get rich from any such discoveries I make in this esoteric branch of geometry, which is itself one subfield, among many, in mathematics. I do it because it is fun. It makes me happy.

Much of life is pure drudgery, but our lives can be enriched by finding joyous escapes from our routines. An excellent way to do that is to learn to appreciate the beauty of uselessness — uselessness of the type that elevates the human spirit, in a way that the pursuit of material goods never can.

This is the approach we should encourage students to have toward education. Learning is far too valuable an activity to be limited, in its purpose, to the pursuit of future wealth. It’s time to change our approach.

By Request: The Compound of Five Rhombic Dodecahedra, with Nets


By Request:  The Compound of Five Rhombic Dodecahedra, with Nets

I’ve been asked by a reader of this blog to post nets for this polyhedral compound. Printing nets with Stella 4d is easy, and I’m happy to post them here, in response to that request. Warning, though: there are many nets needed for this compound.

Each of these smaller images may be enlarged with a single click.

Cuboctahedra 5 net one

Here’s the first net type needed (above). You’ll need thirty copies of this net. The gray parts show, and the white parts are tabs to help put it together. Below is the second type needed, of which you need sixty copies.

Cuboctahedra 5 net two

There’s also a third type of net, and these last two types may need to be rescaled before you print them, to fit the net of the first type, also. You’ll need sixty copies of this third net (below) as well, It’s the mirror-image of the net of the second type.

Cuboctahedra 5

Finally, here’s a non-rotating image of the completed polyhedron, to help with the construction:

Cuboctahedra 5

I recommend using card stock or posterboard, and trying to get as much tape as possible on the inside of the model, making an uncolored version — and then painting it with five different colors of your choice, after the model is assembled. Happy building!

[Software credit:  I used Stella 4d:  Polyhedron Navigator to create all these images. It’s available at Downloading and trying a trial version is free, but you have to buy the fully-functioning version to print nets, or to make these rotating .gif files I post all over this blog.]

The Great Rhombcuboctahedron As a Building-Block


The Great Rhombcuboctahedron As a Building-Block

This solid, also known as the great rhombicuboctahedron, and the truncated icosidodecahedron, can be used to build many other things. In addition to the elongated ring of eight above, for example, there’s this octagonal prism.

Augmented Trunc Cubocta2

Augmented Trunc Cubocta 2

Remember the elongated ring at the top of this post? This pic, directly above, is of a ring of four of those rings.

Augmented Trunc Cubocta3

And, yes, that’s a (non-great) rhombcuboctahedron made of great rhombcuboctahedra. Here it is again, with a different color-scheme.

Augmented Trunc Cubocta4

For the last of these constructions, eight more great rhombcuboctahedra are added to the figure in the two posts above, which is also returned to its original color-configuration. These eight new polyhedra have positions which correspond to the corners of a cube.

augmented rhombcuboctahedron made of great rhombcuboctahedra

Manipulating polyhedra in this manner is easy with Stella 4d, the program I used to do all of this. You may buy it, and/or try a free trial version first, at

My Students’ Painting of the Periodic Table of the Elements


My Students' Painting of the Periodic Table of the Elements

This is my last year teaching at my current school — I’ll be transferring to another school in the same district in the Fall. To create a farewell gift to the school where I have taught for the last three years, I brought a lot of paint and other art supplies from home, bought more when they ran out, and let my students (who are enrolled in Chemistry and Physical Science) paint a large painting of the periodic table on two large wooden boards, each measuring 4′ by 6′. In the Fall, the plan is for the painting to be mounted on the wall of the science wing of my current school, in a location to be chosen by my current department chair, a personal friend of mine.

I think my students did a very good job — better than this picture I took with my cell phone reveals, just due to camera-quality. I am proud of them.

The Royal Octahedron, with Half-a-Dozen Crowns


The Royal Octahedron, with Half-a-Dozen Crowns

Software credit: see to check out Stella 4d, the program I used to make this. A free trial download is available.