In fish retina, four kinds of photoreceptor cells (or cones) are two-dimensionally arranged in a very regular manner, forming cone mosaics. Mosaic pattern differs between species--two typical patterns are "row mosaic" and "square mosaic", exemplified by the cone mosaics in zebrafish and in medaka, respectively. In this paper, we study a cell-rearrangement model. Cells with pre-fixed fate exchange their locations between nearest neighbors and form regular mosaic patterns spontaneously, if the adhesive force between nearest neighbors and between next-nearest neighbors depend on their cell types in an appropriate manner. The same model can produce both row and square mosaic patterns. However, if the cell-cell interaction is restricted to nearest neighbors only, the square mosaic (medaka pattern) cannot be generated, showing the importance of interaction between next-nearest neighbors. In determining whether row mosaic (zebrafish pattern) or square mosaic (medaka pattern) is to be formed, two shape factors are very important, which control the way adhesions in different geometric relations are combined. We also developed theoretical analysis of the parameter ranges for the row mosaic and the square mosaic to have higher total adhesion than alternative spatial patterns.