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Tiling a plane with a dodecahedron

This is not such an easy thing to do, and one must wonder why we should even try. It is slightly more pragmatic when one considers that Code World is a game based on a dodecahedron. This is not the first such game. In fact, a famous mathematician named Sir William Rowan Hamilton marketed a game based on forming a closed loop using the vertices of a dodecahedron. He created a wooden game board that accepted pegs attached to a string. The puzzle solution looked something like this:

This is a Hamiltonian circuit in a branch of mathematics known as graph theory. Unfortunately, Code World cannot use this approach for several reasons. The regions of the graph must be continuous, but the graph above will not tile a plane because it is a pentagon, and pentagons do not tessellate, or form periodic tilings. It is part of what is known as the crystallography restriction.

A good example of a solid that will tile a plane is the tetrahedron.

This is a form of graph known as a colored graph. Since Code World connects nodes in a directional manner, it also can be referred to as a network. In the case of the tetrahedron above we could take the information of a single tile and periodically repeat it indefinitely to cover an entire surface.

In this case we have made a clear distinction between the regions of the graph that stand for tetrahedral faces and the ones that stand for vertices. However, since the tetrahedron is dual to itself, we could morph the two regions to the point where they are equal.

Can we do the same sorts of things with a dodecahedron?

Technically the answer is no, but there is a way to simulate the logic of a dodecahedral network in two-dimensions. I discovered it by playing around with information theory, and imagining the vertices of a dodecahedron as the cycles of a sine wave. If we examine this object closely, we can define three distinct zones. The equator presents a continuous band of oscillating vertices. The two poles interlace with the equator with offset bands of oscillating vertices that demonstrate exactly half the frequency of the equator.

We can match the nodes of the two poles and the equator. When we do, we notice that there are vacant nodes.

Since the shape is symmetrical, we can fill the vacant nodes by inverting and repeating the shape and merging multiple copies as follows.

Connecting the adjacent nodes will complete the tile, and that, my friends, is how you “tessellate” a dodecahedron.

We can now fill a plane with an infinite number of periodic repeats.

More interesting still to a discussion of DNA, the repeating tile can be folded around to mate with itself on the sides, and it can be extended longitudinally as long as we like. In other words, it is a sequential tessellation in one dimension.

This is isomorphic with the structure of DNA's double helix.

This seems to be nothing more than a symbolic coincidence, but it does provide some helpful insight regarding the genetic code. Consider first that every sequence of nucleotides can be thought of as a sequence of dodecahedrons, so we might imagine decoding a sequence by clipping it out twelve nucleotides at a time and overlaying it on a decoding key. This means that a continuous Hamiltonian circuit can connect all of the tiles in a sequence of nucleotides and decoding keys.

In this case I have put the entire circuit on a single tile, and I have highlighted nodes to stand for codons. In an actual, nearly random nucleotide sequence it might be more informative to depict the nucleotide sequence like this.

Now we can easily visualize a significant question: what happens when the sequence shifts?

Decoding is predicated on a three-face reference frame of the circuit, and when it shifts it will shift all of the codons. A large sequence of nucleotides can easily adopt any one of these three frames. More curious, the coding sequence known as a sense strand has a complimentary mate known as the anti-sense strand, and there are three more frames on it. So every nucleotide sequence must be looked at from six different angles from a decoding standpoint. Empiric evidence from genome studies shows the fingerprints of all six reading frames in nucleotide sequences. This means that, from a random sequence of nucleotides, preference is given to sequences that match existing reading frames in one of the other five frames. There must be some type of better than random utility in the connection between these frames.

How could you do that?

How could you devise an encryption scheme that maintains a logical relationship between six reading frames, all of which might decode into completely different sequences?

It would require a phenomenal amount of complex symmetry, and It can be done within the symmetry of a dodecahedron. The actual structure of the genetic code merely reinforces the notion, because it is tailor made for a dodecahedron, and laid out perfectly to accomodate these shifts. Of course life had billions of years to find this particular solution.

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Last updated August 24, 2003 12:40 PM