Knowra Finite geometry Finite geometry Finite geometry studies incidence structures and geometric spaces with finitely many points, often constructed over finite fields. Its objects include finite projective and affine spaces.
Finite field : A field with finitely many elements, whose arithmetic is often used to define finite geometric spaces. Vector spaces over finite fields provide standard coordinates for finite geometries.
Projective plane : A projective geometry in which any two distinct points determine one line and any two distinct lines meet at one point. Finite projective planes are among the basic structures studied in finite geometry.
Fano plane : The projective plane of order two, with seven points, seven lines, and three points on each line. It is the smallest finite projective plane and a compact model of incidence.
Linear code : A set of codewords forming a vector subspace over a finite field, used to detect or correct errors. Geometric point and subspace configurations can define linear codes with useful distance properties.
Incidence structure : A system of points and other objects, such as lines, together with a specified relation of incidence. Many finite geometries are defined by which points lie on which lines or subspaces.
Desarguesian plane : A projective plane coordinatized by a division ring, satisfying Desargues' theorem. Planes built from finite fields are Desarguesian examples.
Projective plane of order three : The finite projective plane with thirteen points and thirteen lines, each line containing four points. It illustrates the point and line counts determined by a finite plane's order.
Projective geometry code : A linear code constructed from incidence relations among subspaces of a finite projective space. It translates finite projective incidence into algebraic error-correcting codes.
Projective space : A geometric space formed from one-dimensional subspaces of a vector space, with points and higher-dimensional subspaces. Its finite-field versions are central examples of finite geometry.
Affine plane : A geometry of points and lines where unique parallels pass through points outside a line and nonparallel lines meet. Finite affine planes can be built from finite fields and converted into projective planes.
Show all 20