for infinite but locally finite arrangements, in which every bounded subset of the plane may be crossed by only finitely many lines, although in this case the unbounded cells may have infinitely many sides. Infinite systems of lines that are not locally finite may have equivalence classes of other shapes; for instance, the system of tangent lines to a unit circle has the interior of the circle as one of its equivalence classes. Complexity of arrangements The study of arrangements was begun by Jakob Steiner, who proved the first bounds on the maximum number of features of different types that an arrangement may have. The most straightforward features to count are the vertices, edges, and cells: An arrangement with lines has at most vertices (a triangular number), one per pair of crossing lines. This maximum is achieved for simple arrangements, those in which each two lines cross at a point distinct from all other crossings. In any arrangement (rotated slightly if necessary to avoid axis-parallel lines) there will be infinite-downward rays, one per line. These rays separate cells of the arrangement that are unbounded in the downward direction. The remaining cells all have a unique bottommost vertex, and each vertex is bottommost for a unique cell, so the number of cells in an arrangement is the number of vertices plus , or at most . These are the numbers of the lazy caterer's sequence. The number of edges of the arrangement is at most , as may be seen either by using the Euler characteristic to calculate it from the numbers of vertices and cells, or by observing that each line is partitioned into at most edges by the other lines. Again, this worst-case bound is achieved for simple arrangements. More complex features go by the names of "zones", "levels", and "many faces": The zone of a line in a line arrangement is the collection of cells having edges belonging to . The zone theorem states that the total number of edges in the cells of a single zone is linear. More precisely, the total number of edges of the cells belonging to a single side of line is at most , and the total number of edges of the cells belonging to both sides of is at most . More generally, the total complexity of the cells of a line arrangement that are intersected by any convex curve is