User:John Z/Projective plane

The definition of projective plane by incidence properties is something special to two dimensions: in general projective space is defined via linear algebra.

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One can construct projective planes or higher dimensional projective spaces by linear algebra over any division ring - not necessarily commutative. If we use a finite field with pn elements we get a finite projective plane with order pn. The Fano plane is then the plane over the field with two elements, Z2. One can also do the reverse, and construct a coordinate "ring" - a so-called Planar Ternary Ring (not necessarily a genuine ring) corresponding to any projective plane as defined above. Algebraic properties of this "ring" turn out to correspond to geometric incidence properties of the plane. For example, Desargues' theorem corresponds to the coordinate ring being a division ring, while Pappus's theorem corresponds to this ring being commutative. However, the "ring" need not be of this type, and there are many non-Desarguesian projective planes. Alternative, not necessarily associative division rings correspond to Moufang planes. In the case of finite projective planes, the only proof known of the purely geometric statement that Desargues theorem implies Pappus' theorem (the converse being always true and provable geometrically) is through this algebraic route, using Wedderburn's theorem that finite division rings must be commutative.

One can make analogous incidence definitions for projective n-spaces, for n larger than 2. These turn out to not be as interesting as the planar case, as they correspond to classical projective geometry over division rings for a very simple reason: with the extra room to work in, one can prove Desargues theorem geometrically as in its article by using incidence properties in this higher dimensional space and thus the coordinate "ring" must be a division ring.

The plane over the octonions turns out to be an interesting real manifold, which can be used for geometric constructions and understanding of the Exceptional Lie groups.