Synthetic projective geometry is an axiomatic approach to projective geometry (usually of projective spaces) without use of (algebraic or analytic) coordinate calculations (unlike the wider, modern study of projective and quasiprojective algebraic varieties). In particular, the method does not require that projective space be defined over an algebraically closed ground field, or even a field at all; for example the noncommutative ground division rings are possible as well as combinatorial descriptions of finite projective geometries. In dimension different from only the projective spaces over division rings are possible, while in dimension other cases are possible, in particular non-Desarguesean planes (e.g. planes coming from Hall quasifields).
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