Combinatorics in glass

Glasses have a complicated structure (a random, regular graph of degree 4), but very simple elementary excitations (decoupled tunnelling modes between two valleys degenerate in energy, or nearly so). These facts can be explained by two straightforward applications of combinatorics. Tunnelling modes arise directly from the local invariance of the structure of the glass, expressible as a fibre bundle. The degenerate valleys are classes of odd permutations of the edges of the graph (legs of the tetrapods which are the local reference frames in silicate glasses), about odd circuits and tunnelling is imposed by gauge invariance. ructure of glass can be obtained by successive decurving (by decoration) of an ideal structure in curved space, polytope {5,3,3}. This decurving can be done at random, both in the sequential order and in space. The structure is then described algebraically as the Perron-Frobenius eigenvector of a sequence of decurving operations. Disorder is represented by the commutator of decurving operations at different points. Glasses constitute therefore a general, permanent representation of combinatorics and randomness ensures that the combinatorics is unencumbered by the constraints of adjustment found in crystals.