an isomorphism is called a correlation. If a correlation exists then the projective plane C is self-dual. In the special case that the projective plane is of the PG(2, K) type, with K a division ring, a duality is called a reciprocity. These planes are always self-dual. By the fundamental theorem of projective geometry a reciprocity is the composition of an automorphic function of K and a homography. If the automorphism involved is the identity, then the reciprocity is called a projective correlation. A correlation of order two (an involution) is called a polarity. If a correlation φ is not a polarity then φ2 is a nontrivial collineation. Finite projective planes It can be shown that a projective plane has the same number of lines as it has points (infinite or finite). Thus, for every finite projective plane there is an integer N ≥ 2 such that the plane has N2 + N + 1 points, N2 + N + 1 lines, N + 1 points on each line, and N + 1 lines through each point. The number N is called the order of the projective plane. The projective plane of order 2 is called the Fano plane. See also the article on finite geometry. Using the vector space construction with finite fields there exists a projective plane of order , for each prime power pn. In fact, for all known finite projective planes, the order N is a prime power. The existence of finite projective planes of other orders is an open question. The only general restriction known on the order is the Bruck–Ryser–Chowla theorem that if the order N is congruent to 1 or 2 mod 4, it must be the sum of two squares. This rules out . The next case has been ruled out by massive computer calculations. Nothing more is known; in particular, the question of whether there exists a finite projective plane of order is still open. Another longstanding open problem is whether there exist finite projective planes of prime order which are not finite field planes (equivalently, whether there exists a non-Desarguesian projective plane of prime order). A projective plane of order N is a Steiner system (see Steiner system). Conversely, one can prove that all Steiner systems of this form () are projective planes. The number of mutually orthogonal Latin squares of order N is at most . exist if and