We wish to\int_{S}|Z-f(Z)|^{2},Sthe unit square, over all choices off(Z)which take just n values. For large n it is shown thatf(Z)approaches that function which on each face of a "hexagonal mesh" has the value of the centroid of that face. This problem in the design of quantizers for a pulse-code modulation (PCM) system arose in which pairs of adjacent samples are quantized in an effort to capitalize on the correlation between samples and on the geometry of the plane.