of an arrangement to be the connected components of the points not belonging to any line, the edges to be the connected components of sets of points belonging to a single line, and the vertices to be points where two or more lines cross. A line arrangement in the projective plane differs from its Euclidean counterpart in that the two Euclidean rays at either end of a line are replaced by a single edge in the projective plane that connects the leftmost and rightmost vertices on that line, and in that pairs of unbounded Euclidean cells are replaced in the projective plane by single cells that are crossed by the projective line at infinity. Due to projective duality, many statements about the combinatorial properties of points in the plane may be more easily understood in an equivalent dual form about arrangements of lines. For instance, the Sylvester–Gallai theorem, stating that any non-collinear set of points in the plane has an ordinary line containing exactly two points, transforms under projective duality to the statement that any projective arrangement of finitely many lines with more than one vertex has an ordinary point, a vertex where only two lines cross. The earliest known proof of the Sylvester–Gallai theorem, by , uses the Euler characteristic to show that such a vertex must always exist. Triangles in arrangements An arrangement of lines in the projective plane is said to be simplicial if every cell of the arrangement is bounded by exactly three edges. Simplicial arrangements were first studied by Melchior. Three infinite families of simplicial line arrangements are known: A near-pencil consisting of lines through a single point, together with a single additional line that does not go through the same point, The family of lines formed by the sides of a regular polygon together with its axes of symmetry, and The sides and axes of symmetry of an even regular polygon, together with the line at infinity. Additionally there are many other examples of sporadic simplicial arrangements that do not fit into any known infinite family. As Branko Grünbaum writes, simplicial arrangements "appear as examples or counterexamples in many contexts of combinatorial geometry and its applications." For instance, use simplicial arrangements to construct counterexamples to a conjecture on the relation between the degree of a set of differential equations and the number of invariant lines the equations may have. The two known counterexamples to