Investigations into Dodecahedral Space

Remember when I warned you that some of these posts may go into extravagant detail about some strange or obscure hobby that I enjoy? Well, this is one of them. So if you're not particularly interested in the interactions of certain polyhedra in three-dimensional space, you should probably leave now.

Today's topic is the interrelation of cubes and dodecahedra and the examination of said relations with regard to the potential of some kind of dodecahedral space.

For those of you who may not know, I really like polyhedra (especially Archimedian and Catlan Solids). I have models of several on my desk at work -- including an intersection of a snub dodecahedron and a pentagonal hexacontahedron, the patterns for which I calculated and designed myself -- and I thoroughly enjoy tinkering around with a Zome kit my parents got me several years ago. My current obsession is with the relation between dodecahedra and cubes. A dodecahedron has twelve regular pentagons for faces and is the shape for a 12-sided die.

I'm pretty sure you all know what a cube looks like.

Now, a dodecahedron doesn't seem to fit into 3D space very neatly: it's faces are pentagons, which aren't easily described in a square coordinate system and aren't as structurally fundamental as triangles, and neither pentagons nor dodecahedra tessellate (pack together without spaces in between). Cubes, on the other hand, represent the very basis of three-dimensional space. It is therefore fascinating to me that, by drawing a line between two vertices on each face, you can inscribe a cube on the surface of a dodecahedron.

(The struts used for this are unfortunately all the same color, so I covered the ones forming the cube in tin foil.)

The edges of the cube all fall on a unique face of the dodecahedron, and every face of the dodecahedron is occupied by an edge of the cube.

This inscribing of a cube on a dodecahedron allows for easier mapping of the latter into cubic space. And while dodecahedra cannot tessellate, they can be arranged without overlap in a repeating pattern along diagonal cubes:


I found this last bit worth further investigation, since the cube formed inside the dodecahedron is not the only one possible; the inscription process can be used to form 5 unique, rotated cubes.

Here are two, shown in black and silver.

So, if I were to arrange several dodecahedra according to the cubic space defined by the silver struts, then generated the black cubic space, would the black cubes intersect the other dodecahedra the same way as the one in which the first cube was generated? If this were true, I could see it giving rise to a new way of defining space according to the arrangement of dodecahedra rather than the cubic space we all know and love. While perhaps not practical, this would be kind of cool. However, preliminary construction showed that, at least one "layer" out from the original dodecahedron, the two cubic spaces did not converge. This didn't mean convergence was impossible, though; it was still possible that the two spaces converged at some more distant point. However, the increased distance would greatly increase the potential complexity of the "dodecahedral space," as every cube spacing the distance to convergence would itself be able to give rise to a dodecahedron and four other cubic spaces, which would in turn be able generate more cubic spaces before they converged, and so on. Further construction only got more and more flimsy the further I got from the original dodecahedron. This would have to be accomplished with math.

Several maths later...

It turns out the two cubic spaces never converge, leading to infinite complexity and the breakdown of "dodecahedral space." I arrived at this conclusion as follows: If I just took the two original cubes,

and then just looked at the top face of the silver cube, particularly the edges originating at the near-left corner,

If I were to figure out the the coordinates of the vertex of the black cube in silver space (given the cubes had sides of length 1), and then find some multiple of these coordinates that was an integer, it would mean that the two spaces would meet at that point. So taking the near-left corner to be (0,0,0),

An extra silver strut has been added to form the coordinate axes.

I calculated the unit vector of the black strut. If the cubes had edges of length 2 (a change from before, I know, but it makes the math easier), then the end of the black strut has coordinates (phi, 1, 1/phi), where phi is the golden ratio and an irrational number (my favorite irrational number, in case you were wondering). Since the coordinates contain irrational numbers, no multiple of them will ever be an integer, and the two spaces will never converge.


If you're disappointed that nothing came of this after such a long post, well... I kind of was, too. However, uniqueness is an important property in and of itself, so perhaps the knowledge that these two coordinate spaces never converge except in this one dodecahedron could be useful. The first thing that comes to mind is cryptography: is there a way to encode a message on the two cubic spaces so it can only be read at the point where they converge? I don't know.


I wish it were faster and easier to get these descriptions prepared and online; my mind moves too fast and sporadically. In the time that I've been preparing these models and writing this, I've been distracted by modeling the tessellation of rhombic dodecahedra


and experimenting with taking the volumes in a dodecahedra not filled by the inscribed cube and mirroring them on the inside of the cube's faces.


Interestingly, the points in the center mark the vertices of an icosahedron (20-sided die), the dual of a dodecahedron. Is this, combined with the planar golden rectangles of an icosahedron, a clue in the relation between cubes and dodecahedra? I must find out!

Exciting Times

Today is an interesting milestone: it is the last day (at least in the near future) that I will be living by myself. Starting tomorrow, this crazy guy will be moving in with me, and provided we don't kill each other, I think it will be a very good arrangement. Still, I can't help but feel a bit nervous about the change. Not that I have anything against him moving in -- it was I who raised the question of sharing the house in the first place -- and having another person under the roof will greatly reduce the rent, utilities, and boredom. Really, aside from me no longer being able to break into renditions of "Be Prepared" from The Lion King whenever I want, there isn't much reason behind my nervousness; just the issue of change.

A similar thing happened about a year ago after I graduated and got a job. Leaving the dorm of 50-ish people and my three roommates, I moved into an apartment with Hopps. He shared the apartment with me for the summer while he was interning, then he went back to college and I was left alone. It was really difficult adjusting from the crowded dorm to the empty apartment and it took a while, but I eventually got used to it. And now, after a year of living on my own, I'm going to switch back to sharing my residence with another. The change won't be nearly as great as before, but it's still there, and while it's not unsettling in the sense of disturbing, it is a bit of a jolt.

I know that this feeling is temporary and I'm actually looking forward to having a house-mate (is that what you call it?), it's just a change, and I tend to be a creature of habit.

Layout Difficulties

I'm somewhat annoyed at Blogger. I had hoped to set up this blog dividing my posts between three distinct tabs: Life, Projects, and Philosophy. This way I could go into great detail in the latter two pages without clogging things up for people who just want to check in and see how I'm doing. Very neat and clean. Unfortunately, while Blogger does have the option for multiple tabs, I haven't found a way to submit multiple posts for anything other than the Home tab. The only way I know of to completely separate posts of different styles is by creating completely separate blogs, which I don't want to do.

So right now it looks like if you want read about my personal happenings, you'll have to wade through posts detailing exactly how I make my chainmaille armor -- both the internet tutorials I learned from and my improvements on them -- should I ever feel like posting that. The only division will be the labels at the bottom.

If you know how to make Blogger sort posts into separate pages under the same blog, please let me know.

On Men and Monsters

These first few posts are a little late in regard to the events that inspired them, but I just now got this blog up and running and I feel that I should put them somewhere, so here they are.



I am convinced that empathy is one of the highest forms of human reasoning. It combines imagination -- one of the highest intellectual achievements -- with selflessness -- one of the highest reaches of morality. Putting ourselves in the shoes of another, particularly someone who appears vastly different from us, keeps us from becoming completely self-serving automata. I have heard it postulated that self-reflection is a form of (and even gives rise to) consciousness. If this is true, then reflecting on the self of another is even greater. Not only do we recognize that we think and have a being, but that others also have a being unto themselves that is very much like our own -- and similar enough that we can imagine how we would feel if we were in their circumstances.

I am also convinced that the greatest atrocities in human history occur due to a loss of empathy. These tragedies occur when a group in power ceases to view people as people and instead sees them as either tools to be used or obstacles to be eliminated. Offenses like this can range from the obvious repression by totalitarian governments who, at best, see the people they expend as tools "for the greater good," to individual bases of "just using someone for their _____." In all cases, there is a fundamental sense that this is wrong, even if it is efficient and utilitarian, and the argument against such actions boils down to the sentiment, "but they're people, too." This can even be applied to the perpetrators; no one ever sets out to be evil (aside, perhaps, from mental illness). They can certainly become that way, but rather than aiming at being the worst possible human they can imagine, they do so by focusing so much on their own interests and ideals that they forget the reality of the selves of others. In thinking (consciously or not) that they are the only true men, they become monsters.

It is very important to remember this in your interactions with others, particularly those you don't like or disagree with. Every person you interact with is human and everything that comes with it. They are not perfect (just like you), but they also have intellectual processes and emotional responses very similar to your own. Aside from environmental conditions, the differences arise in each person's interests and the way in which they pursue and protect those interests. Now, it is perfectly reasonable to find fault in these set of priorities (I just stated that blind pursuit of misplaced priorities can lead to inhumanity), and I'm certainly not against the destruction of erroneous arguments and points of view, but there's a reason ad hominem attacks are considered fallacious. Never forget that the other person is a person.

The whole issue of the monster-izing effect of the lack of empathy is very prevalent in the United States' current War on Terror. I seriously doubt that those who perpetrated the 9/11 attacks thought of the people in the Twin Towers as the same sort of being as their loved ones. I'm fairly certain that those in terrorist cells fit my earlier description of monsters. And I see a war of and against an "us and them" mentality toward Muslims. Several weeks ago, Osama bin Laden was killed in an operation by Seal Team Six. In the United States, this was met with relief, rejoicing, and sharp criticism toward those who were rejoicing. I would tend to side with the third group. Personally, I felt very little emotion at the news other than a very pragmatic acknowledgment of threat mitigation on one hand and the increased threat of retaliation on the other. There are people who are far more emotionally connected with this war than I am and I cannot pretend to know their feelings, but I do not think that bin Laden's death should be celebrated. I'm not trying to detract from the Seals' accomplishment; they very admirably completed their mission and neutralized a threat, but it was not victory, it was the death of a man. The death of a human being who was working to protect his interests. These interests were wrong and they manner in which he protected them was unjust, but in his mind, there was reason behind his actions.

I know there's a very fine line between celebrating a success and a victory (though, again, not ultimate victory), and celebrating the death of the man Osama bin Laden, and so I am not directing any criticism against those who felt glad when they heard the news. I am, however, critical of those who proverbially danced on his grave, who became so focused on their own jingoism that they viewed him as little more than a one-dimensional enemy, an obstacle with a name.

Was I able to fully empathize with Osama? Absolutely not. Personally, I find it very difficult to empathize with anyone who is more than slightly different from me. But it is an ideal toward which I aim. Was bin Laden a monster? Yes, but he was also a man, and if we forget that we become more like him than we would want to admit.