The dual tessellation of a regular tessellation has one polygon vertex at the geometric centroid of each regular polygon. Two of these vertices are joined by a line segment iff the original polygons share a polygon edge. Each such line segment crosses the shared polygon edge at its midpoint. This construction specifies the dual uniquely once the original regular tessellation is fixed.
The triangular grid and hexagon tiling are duals of each other, while the square grid is its own dual. For a square grid, the dual is a translation by half a square's side length in each of the two edge directions. Its edges remain parallel to those of the original square grid.
Williams (1979, pp. 37-41) illustrates the dual tessellations of the semiregular tessellations.
