Tidally Locked Binary Icosidodecahedra

binary icosidodecahedra

I’ve been trying to figure out for over a year how to make images like the one above, without having holes in the two polyhedra, facing each other. At last, that puzzle of polyhedral manipulation using Stella 4d (software available at this website) has been solved: use augmentation followed by faceting, rather than augmentation followed by simply hiding faces.

Three Different Depictions of the Compound of Five Cubes

The most common depiction of the compound of five cubes uses solid cubes, each of a different color:

Cubes 5

This isn’t the only way to display this compound, though. If the faces of the cubes are hidden, then the interior structure of the compound can be seen. An edges-only depiction, still keeping a separate color for each cube, looks like this:

Cubes 5 edges

If these thin edges are then thickened into cylinders, that makes a third way to depict this polyhedral compound. It creates a minor problem, though: edges-as-cylinders looks awful without vertices shown as well, and the best way I have found to depict vertices, in this situation, is with spheres. With vertices shown as spheres, however, a sixth color, only for the vertex-spheres, is needed. Why? Because each vertex is shared by six edges: three from a cube of one color, and three from a second cube, of a different color.

Cubes 5 thick edges

Finally, here are all three versions, side-by-side for comparison, and with the motion stopped.


All images in this post were created using Stella 4d: Polyhedron Navigator, software you may try for free at this website.

My New Middle Initial and Name: A Mathematical Welcome-Back Gift from My Alma Mater


I just had a middle initial assigned to me, and then later, with help, figured out what that initial stood for. With apologies for the length of this rambling story, here’s an explanation for how such crazy things happened.

I graduated from high school in 1985, and then graduated college, for the first time, with a B.A. (in history, of all things), in 1992. My alma mater is the University of Arkansas at Little Rock, or UALR, whose website at http://www.ualr.edu is the source for the logo at the center of the image above.

Later, I transferred to another university, became certified to teach several subjects other than history, got my first master’s degree from there (also in history) in 1996, and then quit seeking degrees, but still added certification areas and collected salary-boosting graduate hours, until 2005. In 2005, the last time I took a college class (also at UALR), I suddenly realized, in horror, that I’d been going to college, off and on, for twenty years. That, I immediately decided, was enough, and so I stopped — and stayed stopped, for the past ten years.

Now it’s 2015, and I’ve changed my mind about attending college — again. I’ve been admitted to a new graduate program, back at UALR, to seek a second master’s degree — one in a major (gifted and talented education) more appropriate for my career, teaching (primarily) mathematics, and the “hard” sciences, for the past twenty years. After a ten-year break from taking classes, I’ll be enrolled again in August.

As part of the process to get ready for this, UALR assigned an e-mail address to me, which they do, automatically, using an algorithm which uses a person’s first and middle initial, as well as the person’s legal last name. With me, this posed a problem, because I don’t have a middle name.

UALR has a solution for this: they assigned a middle initial to me, as part of my new e-mail address: “X.” Since I was not consulted about this, I didn’t have a clue what the “X” even stands for, and mentioned this fact on Facebook, where several of my friends suggested various new middle names I could use.

With thanks, also, to my friend John, who suggested it, I’m going with “Variable” for my new middle name — the name which is represented by the “X” in my new, full name.

I’ve even made this new middle initial part of my name, as displayed on Facebook. If that, plus the e-mail address I now have at UALR, plus this blog-post, don’t make this official, well, what possibly could?

Various Views of Three Different Polyhedral Compounds: Those of (1) Five Cuboctahedra, (2) Five of Its Dual, the Rhombic Dodecahedron, and (3) Ten Components — Five Each, of Both Polyhedra.

Polyhedral compounds differ in the amount of effort needed to understand their internal structure, as well as the way the compounds’ components are assembled, relative to each other. This compound, the compound of five cuboctahedra, and those related to it, offer challenges not offered by all polyhedral compounds, especially those which are well-known.


The image above (made with Stella 4d, as are others in this post — software available here) is colored in the traditional style for compounds: each of the five cuboctahedra is assigned a color of its own. There’s a problem with this, however, and it is related to the triangular faces, due to the fact that these faces appear in coplanar pairs, each from a different component of the compound.


The yellow regions above are from a triangular face of the yellow component, while the blue regions are from a blue triangular face. The equilateral triangle in the center, being part of both the yellow and blue components, must be assigned a “compromise color” — in this case, green. The necessity of such compromise-colors can make understanding the compound by examination of an image more difficult than it with with, say, the compound of five cubes (not shown, but you can see it here, if you wish). Therefore, I decided to look at this another way: coloring each face of the five-cuboctahedra compound by face type, instead of by component.


Another helpful view may be created by simply hiding all the faces, revealing internal structure which was previously obscured.


Since the dual of the cuboctahedron is the rhombic dodecahedron, the dual of the compound above is the compound of five rhombic dodecahedra, shown, first, colored by giving each component a different color.

RD 5 colored by component

A problem with this view is that most of what’s “going on” (in the way the compound is assembled) cannot be seen — it’s hidden inside the figure. An option which helped above (with the five-cuboctahedra compound), coloring by face type, is not nearly as helpful here:

RD 5 colored by face type

Why wasn’t it helpful? Simple: all sixty faces are of the same type. It can be made more attractive by putting Stella 4d into “rainbow color” mode, but I cannot claim that helps with comprehension of the compound.

RD 5 colored rainbow

With this compound, what’s really needed is a “ball-and-stick” model, with the faces hidden to reveal the compound’s inner structure.

RD 5 colored hollow

Since the two five-part compounds above are duals, they can also be combined to form a ten-part compound: that of five cuboctahedra and five rhombic dodecahedra. In the first image below, each of the ten components is assigned its own color.

Compound of 5 Cuboctahedra and dual colored by component

In this ten-part compound, the coloring-problem caused in the first image in this post, coplanar and overlapping triangles of different colors, vanishes, for those regions of overlap are hidden in the ten-part compound’s interior. This is one reason why this coloring-scheme is the one I find the most helpful, for this ten-part compound (unlike the two five-part compounds above). However, so that readers may make this choice for themselves, two other versions are shown below, starting with coloring by face type.

Compound of 5 Cuboctahedra and dual colored by face typet

Finally, the hollow version of this ten-part compound. This is only a personal opinion, but I do not find this image quite as helpful as was the case with the five-part compounds described above.

Compound of 5 Cuboctahedra and dual colored rainbow

Which of these images do you find most illuminating? As always, comments are welcome.

Teaching in Central Arkansas? Here’s a Key to Acronyms in Common Use in Our Profession.

For the full list, please follow this link.

Reading Acronym Soup

This site was compiled by teachers. While we have strived for accuracy, we cannot guarantee that this alphabetized list is free from error. It is also not intended to represent the viewpoints of our employers, nor any other organizations. Its purpose is simply to help teachers, especially those new to the profession in our area, navigate education-related “acronym soup.” Suggestions for additions, corrections, or clarifications are welcome — simply leave them in a comment, below.

  • ACSIP: Arkansas Consolidated School Improvement Plan
  • ACT: American College Testing
  • ACTAAP: Arkansas Comprehensive Testing, Assessment, and Accountability Program
  • ADE: Arkansas Department of Education
  • AEA: Arkansas Education Association (state affiliate of the NEA, and our professional organization, which you can join here)
  • ADHE: Arkansas Department of Higher Education
  • AESOP: Automated Educational Substitute Operator (their website is used to report teacher absences and request substitute teachers)
  • AGS: Arkansas Governor’s School
  • AIMSS (also called Arkansas AIMS…

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