The Dodecadodecahedron and the Golay Code

Introduction

This is the story about the uniform polyhedron known as the "dodecadodecahedron" and its close relationship with the extended binary linear Golay code of length 24. This page can be considered as a continuation of another page which constructs the Mathieu group M24 from the small cubicuboctahedron. The automorphism group of the Golay code is the Mathieu group, and this explains why these objects are so closely related.

The first object we must consider is a polyhedron. If one is patient enough, then one can make a paper model: This polyhedron is "uniform" because its faces are all regular polygons (in a generalized sense), and each of its vertices is congruent to all of the others. The faces of this polyhedron consist of 12 pentagons and 12 "star" pentagrams. At each of the 30 vertices, one will find two pentagons and two pentagrams meeting alternately. It has 60 edges, where a pentagram and pentagon meet. Using this data and the fact that this represents a compact, connected, and orientable surface, one notices that the genus is 4. Thus, this polyhedron which appears quite solid should be considered as being the surface of a 4-holed torus. Alternately, one may obtain this polyhedron by truncating the great dodecahedron to its edge midpoints. If one does this, then one can clearly see the pentagons emerge from the star vertex figures of the great dodecahedron.

This particular polyhedron comes very close to being regular. Indeed, all of the faces may be described as "pentagonal". The only thing keeping it from being regular is that our particular immersion of it into 3-dimensional space twists half of these 24 pentagons into "pentagrams". The genus-4 polyhedron it represents is regular in a sense only slightly more general than that used to define the usual nine regular polyhedra.

It helps considerably to work with the dodecadodecahedron in this context in the presence of the hyperbolic plane. Our polyhedron, after all, has genus 4, which implies that it is uniformized by the hyperbolic plane. All we have to do is tessellate the hyperbolic plane with regular pentagons in such a way that 4 pentagons meet at every vertex. When we do this, we obtain an infinite regular polyhedron: Figure 2. Universal Cover of the Dodecadodecahedron

One should notice several things about this depiction of the hyperbolic plane. Obviously, it is tessellated into pentagons in such a way that 4 pentagons meet at each vertex. Using Schläfli's notation, this can concisely described as the {5,4} tiling, the meaning of the symbols presumed to be obvious. Next, notice that the same six colors which are present in the paper model are used to color the faces of this tessellation. So far, these are easy observations.

The next observation is a bit more involved. In this regard, it helps to become acquainted with what may be described as an "isochromatic path" on the polyhedron. Choose any pentagon and consider a path originating interior to that pentagon. First we traverse a path which crosses one of the edges, entering into another pentagon. Now, just as you crossed over that edge, consider the furthest vertex from where you are. Thus, after crossing over your first edge, keep going, passing near the center of the pentagon and traversing all the way over to the far vertex. At this vertex, two lines cross each other. Cross over both of those lines. You are now in a pentagon having the same color as where you started. Call a path of this type a "fundamental isochromatic path".

One should practice tracing isochromatic paths on the surface of the paper model of the dodecadodecahedron. One will quickly realize that they are "isochromatic" there as well. In particular, notice that an isochromatic path transports between parallel pentagons/pentagrams having the same color. One should also observe that following 4 fundamental isochromatic paths in succession lands you exactly where you started. In other words, one may say that the fundamental isochromatic path has order 4. (At this stage, one may be able to see the advantage of building a paper model.)

To conclude this section: Notice that we have discussed three geometric objects. The first was the common "polyhedron" known as the dodecadodecahedron. Here we regard this as an immersion of our second object, a compact connected orientable surface of genus 4 which has been tiled with 24 pentagons where 4 pentagons meet at every vertex. Finally, we have the universal covering space of this surface, which we have realized as a tessellation of the hyperbolic plane by pentagons with 4 at every vertex. Abusing terminology extensively, let us refer to any one of these objects as the "dodecadodecahedron". After all, here we are not particularly interested in worrying about immersions of surfaces or universal coverings. the main thing to focus on is this "regular" polyhedron which has 24 pentagons and 4 pentagons coming together at each of the 30 vertices.

The Symmetry Group of the Dodecadodecahedron

The group of orientation-preserving symmetries of the regular dodecahedron, it is well-known by those who know it well, is isomorphic to the group A5 of even permutations on 5 letters, also known as an "alternating" group. As it turns out, the group of orientation-preserving symmetries of the dodecadodecahedron is isomorphic to the "symmetric" group S5 of all 120 permutations on 5 letters. Notice at once that we are not talking about the group of all symmetries of the image of the immersion which we commonly refer to as the dodecadodecahedron. Indeed, this group is none other than the group of all symmetries of the regular dodecahedron, which is isomorphic to the direct product of A5 with the two-element group.

The Golay Code

The extended binary linear Golay code of length 24 is merely a particular 12-dimensional subspace of F224, where F2 denotes the field with 2 elements. Thus, in order to specify it uniquely, it suffices to give one of its spanning sets. Moreover, since we are working over the 2-element field, describing a "vector" in this 24-dimensional space is equivalent to describing a subset of a set with 24 elements, each element of a subset corresponding to a non-zero coordinate and vice versa. Vector addition is then translated into symmetric difference between subsets. Scalar multiplication, as one can guess, is quite trivial for the two-element field: Multiplication by zero annihilates a subset and multiplication by the non-zero element doesn't do anything. Hasten to add that another word for "vector" and "subset" is "codeword".

The dodecadodecahedron provides a means for obtaining interesting spanning sets. The 24-dimensional space is that spanned by all the faces, and we must describe various subsets of the set F of the 24 pentagonal faces. Our spanning sets are all described as orbits of partcular subsets of F under the action of the symmetry group of dodecadodecahedron, namely the symmetric group S5. Naturally, we cannot expect to choose these subsets willy-nilly. The Golay code has some very remarkable properties and we want our subsets to have these properties!

Octads and Dodecads. The first property we should recall is that the Golay code has codewords of only lengths 0, 8, 12, 16, and 24, in the respective quantities 1, 759, 2576, 759, and 1. One can check once again that

1 + 759 + 2576 + 759 + 1 = 4096 = 212,

which is the cardinality of a 12-dimensional vector space over the 2-element field. The codewords of length 8 are called "octads" and the codewords of length 12 are called "(umbral) dodecads".

Hexads. The octads comprise a Steiner system with parameters 5, 8 and 24. This means that the octads each consist of 8 elements of F, and that given any 5 elements of F, there is a unique octad to which they belong. Thus, if one chooses only 4 elements of F, then choosing a fifth element uniquely determines another 3 elements for which all 8 comprise an octad. One can do this for any four elements of F and for any of the remaining 20 elements. Doing this for a fixed set of four elements yields a partition of F into 6 disjoint subsets for which any pair unite to form an octad. Each of these partitions induced by a choice of 4 elements of F is called a "hexad". As it turns out, the 6 sets of 4 mutually parallel faces comprise a hexad.

Facial Octads. Choose a face. The 5 adjoining faces are part of this octad, as are the 3 other faces which are parallel to the initial face. There are 24 facial octads, one for each face. These 24 octads span the code.

Vertical Octads. Choose a vertex. Surrounding this vertex are 4 elements of this octad. The other 4 elements are those faces one arrives at by taking an isochromatic path from one of the 4 initial faces through the chosen vertex. There are 30 vertical octads, one for each vertex.

Equatorial Octads. Here is an observation concerning vertical octads. First, we say that vertices are "opposite" if they are opposite in the 3-dimensional immersion of the dodecadodecahedron. Choose two opposite vertices. Then the corresponding vertical octads are disjoint, and thus there is a third octad, obtained as the complement of their union. There are 15 equatorial octads, one for each reflection plane of symmetry of the icosahedron.

Most of the remaining octads have less symmetry than those described so far. There is another variety of "vertical" octad, and, since there are 30 vertices, there are 30 octads having this shape. Remaining are 3 orbits of 60 octads each and 4 orbits of 120 octads each.

Let's now move on to dodecads. When one talks about dodecads, one runs into the idea of "self-duality". Given a dodecad, which has 12 elements, the complement also has 12 elements. The complement may or may not lie in the same orbit under the action of the symmetry of the dodecadodecahedron. Call a dodecad "self-dual" if it lies in the same orbit as its complement.

Equatorial Dodecads. Choose two faces. The corresponding facial octads are not disjoint, but their sum is a dodecad. This dodecad is self-dual, and there are 12 equatorial dodecads, one for each pair of opposite faces. Alternatively, one may describe equatorial dodecads directly as follows: Choose a pair of opposite faces. These 2 faces, taken together with the 10 faces adjacent to these, comprise one of the equatorial dodecads.

Edgy Dodecads. In order to describe this shape, it helps to refer to the colored {5,4} hyperbolic tiling, which is the universal cover of the dodecadodecahedron. Notice that every bounding two pentagons is part of an hyperbolic line which continues to the circle at infinity. Thus, as one traverses an edge and reaches a vertex, one continues the path by traversing across the perpendicular edge. Adjacent to this line appear infinitely many pentagons. However, recall that we have identified this {5,4} tiling as the dodecadodecahedron, and, one can count, there are only 12 pentagons lying along this line. This is an edgy dodecad. There are 10 dodecads of this shape, one for each line as described above.

Facial Dodecads. Choose any face. This face, being a pentagon, has 5 vertices. One can see a total of 11 faces by looking at all the faces surrounding these 5 vertices. The 12th face of this dodecad is the lone face adjacent to this disc which is invariant by 5-fold rotation about the center of the chosen face. Here's another way to describe the 12th element of a facial dodecad: It is the face that one arrives at by tracing out the inverse of an isochromatic path. Facial dodecads are self-dual, and there are 24 dodecads having this shape, each corresponding to its central face. Notice that one may describe this shape as being a disc.

Facial Dodecads, Alternate Description. Here is an amusing way to describe a facial dodecad, using the paper model seen above. If one is holding the model, oriented so that one of the pentagrams is perpendicular with the line of sight, then the 12 pentagrams/pentagons within sight comprise a facial dodecad. (Again, a paper model is handy.)

There are quite a few more shapes of dodecads, at least 20. All of the others have less symmetry than those described thus far.

Conclusion

It is interesting (?) to note that the only "highly-symmetric" shape of octad or dodecad whose orbit under S5 spans the Golay code is the facial octad.

References

"SPLAG". J. H. Conway and N. J. A. Sloane. Sphere Packings, Lattices and Groups. 3rd ed. Springer-Verlag, New York, 1999.

H. S. M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller. Uniform Polyhedra. Phil. Trans. 246 A, (1954), 401-50.

GAP. Groups, Algorithms, Programming - a System for Computational Discrete Algebra, 2005.