the Dirac–Motzkin conjecture (which states that any -line arrangement has at least ordinary points) are both simplicial. The dual graph of a line arrangement, in which there is one node per cell and one edge linking any pair of cells that share an edge of the arrangement, is a partial cube, a graph in which the nodes can be labeled by bitvectors in such a way that the graph distance equals the Hamming distance between labels; in the case of a line arrangement, each coordinate of the labeling assigns 0 to nodes on one side of one of the lines and 1 to nodes on the other side. Dual graphs of simplicial arrangements have been used to construct infinite families of 3-regular partial cubes, isomorphic to the graphs of simple zonohedra. It is also of interest to study the extremal numbers of triangular cells in arrangements that may not necessarily be simplicial. In any arrangement, there must be at least triangles; every arrangement that has only triangles must be simple. For Euclidean rather than projective arrangements, the minimum number of triangles is , by Roberts's triangle theorem. The maximum possible number of triangular faces in a simple arrangement is known to be upper bounded by and lower bounded by ; the lower bound is achieved by certain subsets of the diagonals of a regular -gon. For non-simple arrangements the maximum number of triangles is similar but more tightly bounded. The closely related Kobon triangle problem asks for the maximum number of non-overlapping finite triangles in an arrangement in the Euclidean plane, not counting the unbounded faces that might form triangles in the projective plane. For some but not all values of , triangles are possible. Multigrids and rhombus tilings The dual graph of a simple line arrangement may be represented geometrically as a collection of rhombi, one per vertex of the arrangement, with sides perpendicular to the lines that meet at that vertex. These rhombi may be joined together to form a tiling of a convex polygon in the case of an arrangement of finitely many lines, or of the entire plane in the case of a locally finite arrangement with infinitely many lines. This construction is sometimes known as a Klee diagram, after a publication of Rudolf Klee in 1938 that used this technique. Not every rhombus tiling comes from lines in this way, however. investigated special cases of