In both the FCC and HCP arrangements each sphere has twelve neighbors. For every sphere there is one gap surrounded by six spheres (octahedral) and two smaller gaps surrounded by four spheres (tetrahedral). The distances to the centers of these gaps from the centers of the surrounding spheres is for the tetrahedral, and for the octahedral, when the sphere radius is 1. Relative to a reference layer with positioning A, two more positionings B and C are possible. Every sequence of A, B, and C without immediate repetition of the same one is possible and gives an equally dense packing for spheres of a given radius. The most regular ones are FCC = ABC ABC ABC... (every third layer is the same) HCP = AB AB AB AB... (every other layer is the same). There is an uncountably infinite number of disordered arrangements of planes (e.g. ABCACBABABAC...) that are sometimes collectively referred to as "Barlow packings", after crystallographer William Barlow. In close-packing, the center-to-center spacing of spheres in the xy plane is a simple honeycomb-like tessellation with a pitch (distance between sphere centers) of one sphere diameter. The distance between sphere centers, projected on the z (vertical) axis, is: where d is the diameter of a sphere; this follows from the tetrahedral arrangement of close-packed spheres. The coordination number of HCP and FCC is 12 and their atomic packing factors (APFs) are equal to the number mentioned above, 0.74. Lattice generation When forming any sphere-packing lattice, the first fact to notice is that whenever two spheres touch a straight line may be drawn from the center of one sphere to the center of the other intersecting the point of contact. The distance between the centers along the shortest path namely that straight line will therefore be r1 + r2 where r1 is the radius of the first sphere and r2 is the radius of the second. In close packing all of the spheres share a common radius, r. Therefore, two centers would simply have a distance 2r. Simple HCP lattice To form an A-B-A-B-... hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres' centers. Suppose, the goal is to fill a box with spheres according to HCP. The box would be placed on the x-y-z coordinate space. First form a row of spheres. The centers will all lie on a straight line. Their x-coordinate