The diameter of columnar joints forming in cooling basalt and drying starch increases with decreasing growth rate. This observation can be reproduced with a linear-elastic three-dimensional fracture mechanics bifurcation analysis, which has been done for a periodic array of hexagonal columnar joints by considering a bifurcation mode compatible with observations on drying starch. In order to be applicable to basalt columns, the analysis has been carried out with simplified stationary temperature fields. The critical diameter differs from the one derived with a two-dimensional model by a mere factor of 1/2. By taking into account the latent heat released at the solidification front, the results agree fairly well with observed column diameters.