area of research. The Cayley plane (OP2), a projective plane over the octonions, is one of these because the octonions do not form a division ring. Conversely, given a planar ternary ring (R, T), a projective plane can be constructed (see below). The relationship is not one to one. A projective plane may be associated with several non-isomorphic planar ternary rings. The ternary operator T can be used to produce two binary operators on the set R, by: a + b = T(a, 1, b), and a ⋅ b = T(a, b, 0). The ternary operator is linear if . When the set of coordinates of a projective plane actually form a ring, a linear ternary operator may be defined in this way, using the ring operations on the right, to produce a planar ternary ring. Algebraic properties of this planar ternary coordinate ring turn out to correspond to geometric incidence properties of the plane. For example, Desargues' theorem corresponds to the coordinate ring being obtained from a division ring, while Pappus's theorem corresponds to this ring being obtained from a commutative field. A projective plane satisfying Pappus's theorem universally is called a Pappian plane. Alternative, not necessarily associative, division algebras like the octonions correspond to Moufang planes. There is no known purely geometric proof of the purely geometric statement that Desargues' theorem implies Pappus' theorem in a finite projective plane (finite Desarguesian planes are Pappian). (The converse is true in any projective plane and is provable geometrically, but finiteness is essential in this statement as there are infinite Desarguesian planes which are not Pappian.) The most common proof uses coordinates in a division ring and Wedderburn's theorem that finite division rings must be commutative; give a proof that uses only more "elementary" algebraic facts about division rings. To describe a finite projective plane of order N(≥ 2) using non-homogeneous coordinates and a planar ternary ring: Let one point be labelled (∞). Label N points, (r) where r = 0, ..., (N − 1). Label N2 points, (r, c) where r, c = 0, ..., (N − 1). On these points, construct the following lines: One line [∞] = { (∞), (0), ..., (N − 1)} N lines [c] = {(∞), (c, 0), ..., (c, N − 1)}, where c = 0, ..., (N − 1) N2 lines [r, c] = {(r) and the points (x, T(x, r, c)) },