Please Register or Log in to view the hidden image! This is an image of the three regular tilings of Euclidean 2-space (i.e. the plane). The images in the bottom row are called flags. Each is the smallest number of tiles that contains a vertex which is transitive (has the same number of edges as all the vertices in the tiling). Consider the dual tiling of each flag. The first, a triangular tiling has a hexagonal dual tiling. The second, a hexagonal tiling has a triangular dual tiling. The last, square tiling is self dual. Which more roughly stated is a hexagon of triangles, a triangle of hexagons, and a square of squares. If you have a tiling of the 3-plane with cubes, the flag is the smallest cube of cubes which is also self dual.