  Algebraic expressions are found for the effective conductivities of some
infinite tessellations composed of conducting square, triangular, or hexagonal
tiles. A tessellation is further characterized by the number N of different
colors (different tile conductivities) represented. Tiles of a color are
distributed randomly, and constitute an areal fraction 1/N of the tessellation.
The expressions take account of the percolation threshold associated with the
tile type. This generates a series of expressions for the three-color case,
that suggests this approach gives lower bounds for the true effective
conductivities.
