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Symmetry and Information

Symmetry provides a lovely foundation for information, but symmetry alone will not produce information. Think of information as part of a process that allows for counting and chosing variations. Consider a tetrahedron sitting on a table.

We can count the number of different ways that the tetrahedron can look exactly like this. It has four equal faces to place on the table, leaving three equal faces to rotate forward, so there are twelve ways that the tetrahedron can be rotated yet appear unchanged. If the tetrahedron remains in its ideal form, perfectly symmetric, then no information is produced by these rotational equivalents. There is no criterion on which to count and chose from actual instances of the rotated solid. For information to be created through the rotational symmetry of this solid it must somehow have its symmetry broken.

If we imagine twelve copies of the same tetrahedron sitting on the table, but with each vertex identified in some fashion, then the twelve rotational equivalents can serve as units of information.

For instance, the rotations of this single solid might serve as twelve letters of an alphabet, or twelve digits in a number system.

This is exactly the process used to create information with Code World. If none of the symmetry gets broken, there is no way to distinguish one conformation from another, and there is little but decorative use for the device. However, one of the most powerful aspects of Code World is to realize exactly how the symmetry gets broken. We start with the root form of all five perfect solids, the tetrahedron, and we employ the same simple pattern on both the globe and the glider.

Since the tetrahedron is dual to itself, the pattern technique can be applied either to faces or vertices.

The single tetrahedron pattern is then doubled, inverted and mirrored to produce the pattern for the octahedron...

...and the cube.

In all of these illustrations purple is used as the base color, but the same pattern can be re-arranged for each of the four other colors. It is difficult to grasp in two dimensions, but from an information standpoint, the patterns on the cube and octahedron are identical. This pattern is then incorporated from all five colors to bring us to the next informative level.

The icosahedron can be thought of as an extension of the pattern on the octahedron, but the dodecahedron is more easily thought of as five cubes.

Again, the pattern on the dodecahedron is informatively equivalent to the icosahedron.

Note how the tetrahedron lays out the basic pattern for all five solids. This pattern is doubled to create the equivalent patterns of a cube and an octahedron. The octahedron is a doubling of faces, whereas the cube is a doubling of vertices. Both are dual tetrahedrons. Elements of these two are multiplied and combined to form an equivalent pattern in an icosahedron and a dodecahedron. The icosahedron is a combination of five octahedrons, and the dodecahedron is a combination of five cubes. A shrewd reader will note that this seems to give us too many faces and vertices, but remember that the original dual tetrahedrons in the cube are mirrored inversions...

...and they overlap.

So the five dual tetrahedrons in an icosahedron and a dodecahedron share dual elements symmetrically.

Although the physical appearances of these five solids seem unrelated...

...this illusion is due to our natural fixation with the shape of their faces. From an information standpoint the five platonic solids are subsets of the same spatial relationships, which boils down to placing points equidistant on the surface of a sphere, and nothing is more symmetrical than a sphere.

Now consider how nature takes advantage of symmetry breaking whenever information is called for. At the very heart of the process is the Pauli exclusion principle, which says that no two electrons can be alike. But at higher levels symmetry can emerge again in a more complex system. Take salt, for instance, there is a virtually perfect symmetry to its cubic arrangement of atoms.

There is no useful information in a system of such perfect symmetry. In contrast, life requires a system that is bursting with information. The method at this level therefore is to start with maximum symmetry and then break it in the most useful way.

It is difficult to see how one symmetry breaking can be more useful than another, but Code World provides a fabulous example at a very basic level of how this is so. There are virtually an infinite number of ways to decorate a tetrahedron and a dodecahedron to completely break their collective symmetries, but a precious few are much more efficient and useful than others. (See prototypes.)

Life is a highly complex optimization, so we can be sure that it has selected a remarkable technique for breaking symmetry in some context, and in fact It has not failed to delight us in this regard. The information structure of the genetic code is an ingenious system for squeezing every last drop of information and utility out of every sequence of nucleotides. The combination of symmetry and symmetry breaking achieves a remarkable feat of logically tying together six seemingly random reading frames. This symmetry breaking technique can only be fully appreciated when one starts from a perspective of symmetry and its role in information.

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Last updated August 24, 2003 12:41 PM