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How to stuff 120 tetrahedrons into a single dodecahedron

Consider this: a dodecahedron is a tetrahedron in drag.

Starting with the above dodecahedron, it is easy to imagine a tetrahedron sharing the same center, and four vertices poking through the dodecahedral vertices.

A more interesting way to look at this is as if the dodecahedron actually is a tetrahedron, one with fancy vertices. Each vertex is an identical collection of pieces that represent one fourth of the dodecahedron. This is easier to see if we explode the dodecahedron out to the points of the tetrahedron.

When we morph the four exploded dodecahedral fragments into four tetrahedral fragments we will see the following.

Sucking these back to the center of the tetrahedron will give us a new visual perspective on the dodecahedron.

We can add purple balls to the vertices of the tetrahedron to remind us that we are looking at a four-color dodecahedron melted around a purple tetrahedron.

From here we can set things in motion. Every vertex has an axis through its opposite face. We can rotate the tetrahedron around any of these four axes in 120 degree increments. If we did this with our original purple tetrahedron, to our eye it wouldn’t change, because it has no identifying markings. This is called symmetry.

When an object can be transformed in some way but not fundamentally changed, then that object has an element of symmetry.

There are different kinds of symmetry, like rotation and mirroring already discussed, but we will only need rotation here. When we apply this rotational symmetry to our new, fancy purple tetrahedron we find the following.

We see that by systematically utilizing all of the rotational symmetry in the above tetrahedron we can find twelve possible tetrahedrons occupying the exact same space as our original. In other words, there are twelve discrete combinatorial rotation symmetry options for any tetrahedron. A blueprint for building molecules by specifying one of these rotational options would be an inherently logical system. All that is required is a language that can communicate a specific choice from one molecule to another.

If we consider the mirror symmetry of the tetrahedron, we could produce a mirror twin of each of the above twelve, so then we would have 24 tetrahedrons potentially represented by just this one. As stated, we do not have those twelve mirrors available in this particular system. However, there is really nothing special about any of the four points of the dodecahedron that we have chosen to represent this particular tetrahedron, and we could have therefore just as easily chosen four completely different vertices. If we do so, we generate another tetrahedron, such as the following:

In addition to the original purple tetrahedron we have generated a second, green tetrahedron. The second one shares none of the vertices of the first. In fact, we could do this three more times without using any of the vertices more than once.

We now have five completely separate tetrahedrons located inside our original dodecahedron. Except for the colors we have assigned, all of these tetrahedrons are indistinguishable from the next. A tetrahedron is a tetrahedron, which means that each of these tetrahedrons can be rotated into twelve equivalent options as we did before, creating 60 distinct tetrahedrons. None of these five groups of twelve tetrahedrons share any points; therefore, none of their sixty rotational equivalents overlap either. The group is balanced, and as tetrahedrons they are all interchangeable. The only distinguishing properties are found in the colors we have arbitrarily assigned.

It is mighty convenient that five tetrahedrons with four vertices apiece can perfectly consume the twenty vertices of a single dodecahedron. But this is literally only half the story, because if we go back to the first step in the process, the one where we placed the first tetrahedron, we can see that we actually had two distinct choices. It is true that we could have started with any of the above five tetrahedrons and ended up with the same final formation, but each of the five has a non-equivalent twin called a dual tetrahedron, and the two of them together form a cube.

Every one of the five original tetrahedrons has a dual, so we can repeat the above steps of adding one dual twin tetrahedron for each in the original set. We end up with a configuration of five tetrahedrons that is a dual twin of the original five.

Of course, each of these dual tetrahedrons has twelve rotational equivalents, so we are adding 60 new tetrahedrons to our original 60. We now find a total of 120 distinct tetrahedrons in the dodecahedron. When we combine these dual configurations together and suck the dodecahedron back around it, we see the following.

This suggests that either the points of a dodecahedron provide a logical way to group tetrahedrons, or a tetrahedron provides a logical way to group the points of a dodecahedron – or both. Regardless, I am sure we can find a use for this somewhere in our quest for a logical way to build a molecule. However, with all of the fun we’ve been having adding cool new tetrahedrons to our dodecahedron, we failed to realize that we have created a serious problem. Who can keep all this crap straight? I don’t know about you, but this looks like a jumbled mess to me, so let’s try to clean it up a bit, shall we. Start with the fact that all of the tetrahedrons have duals, and duals make cubes, so we can just as easily view the dodecahedron as five cubes.

That didn’t help much, because although there are only five cubes here, they are still too difficult for lowly humans to comfortably differentiate. Notice, however, that every vertex of the dodecahedron is a composite of exactly two cubes, and the faces form five-color stars. Perhaps we can use this fact to take advantage of the dual tetrahedron cubes. We can add colored balls to the first purple tetrahedron and replace its dual with our melted dodecahedron.

The two dual purple tetrahedrons are now more individually complex, but they are also more identifiable because all eight vertices are in some way marked by one of the other four colors in their global configuration. Now the dual configurations appear as follows.

It’s hard to believe that this new complexity will mitigate our confusion about identifying tetrahedrons in a four-color dodecahedron, but in fact it simplifies the process in fabulous fashion.

We now have strong visual cues to identify the original five tetrahedrons and their five duals. More importantly, each of these ten tetrahedrons has a pattern that will allow us to identify its twelve rotational equivalents. We now have a picture of the dodecahedron that will allow us to identify 120 equivalent alternate representations of that dodecahedron using tetrahedrons. Where could we possibly find a need for such a thing?

Data compression.

The dodecahedron is a natural and phenomenally good compressor of tetrahedrons. In virtually no more space than a single tetrahedron, a dodecahedron can be made to represent 120 unique tetrahedrons. The difference between one tetrahedron and the next is simply its spatial orientation. A simple language describing dodecahedrons allows us to easily and powerfully talk shapes with tetrahedrons.

Look at it this way (you might regret this). Let’s say we go to the store and buy 120 tetrahedral dice and one dodecahedral die. We paint each tetrahedron in one of ten patterns, and then, based on its pattern, we physically orient it in one of twelve ways. Rather than hold it there indefinitely like an idiot, put each one on a tiny little stand so it will be preserved for posterity. We started with 120 identical dice and industriously created a collection of 120 unique, identifiable objects.
Now, invite all of our friends over to marvel at our organizational and painting skills. They will feign interest and patronize our enthusiasm, perhaps recommend a fine Lilly product - Zyprexa, maybe high doses of Prozac. Remain undaunted. We carry our tetrahedrons with us everywhere; discuss them incessantly - even try to build things with them. They are everything to us, but we soon tire of the burden. These damn tetrahedrons are taking up so much space. If only we had a simple way to describe a single one out of the many.

But wait! Suddenly it hits us like a nightmare - the dodecahedron!

If we had a simple language of the dodecahedron we could replace all 120 tetrahedrons in this pesky fanny pack with a single dodecahedron in our pocket. Perhaps a series of colors on the faces would indicate vertices. Knowing the language, we could quickly and accurately orient the dodecahedron according to a prescribed code. Then the dodecahedron would come to literally “mean” the individual tetrahedron and all of its exact angles, just waiting to explode from the dodecahedron.

I am hurt that you even wonder whether I am nutty enough to have attempted the above. Well… I am nutty enough, but at $0.40 apiece, dice are too expensive. How about some pictures?

But this is only the beginning of all the fabulous tetrahedral information stored inside a single dodecahedron. Remember that we found five non-overlapping tetrahedrons originally, and we added five additional tetrahedrons as duals. We now have forty tetrahedral points (10 X 4) but only twenty points in the dodecahedron. This means that each tetrahedral vertex is linked to one other tetrahedron from the dual set. Therefore, every tetrahedron is linked to the four dual tetrahedrons in its five-tetrahedron configuration. These linkages form a network of tetrahedrons. We can walk this network, one link to the next, and get from any of the 120 tetrahedrons to any other in six links or fewer.

Above is a map of just three of the six possible steps in our walk from any given starting place (each sphere represents a tetrahedron, and each color represents a step). A new “best” map must be created with each step. All maps should contain at least six rings, not just the three we have space for here. This is an exceptionally complex relationship between these two shapes, but it is the only way to precisely detail the relationships and distances between tetrahedral walks in a dodecahedron. There is so much complexity here that it is hard to imagine any regularity to this relationship at all. But all of this complexity could be handled by a permutation set of four colors grouped in threes. Where might we find a use for this? The information needed to communicate these relationships in the language of a dodecahedron is tiny compared to the complexity of shapes that can be created by it. It is a system of coherent logic, and any construction blueprint based on it will inherit that logic.

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Last updated August 24, 2003 12:41 PM