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Finite geometry
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<p>A finite geometry is any <a href="/facts/Geometry/wTQf5ffQ">geometric</a> system that has only a <a href="/facts/Finite_set/za1newlP">finite</a> number of <a href="/facts/Point_(geometry)/6YPAvX2a">points</a>. The familiar <a href="/facts/Euclidean_geometry/V6cTcoCy">Euclidean geometry</a> is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the <a href="/facts/Pixel/UzwBvXBV">pixels</a> are considered to be the points, would be a finite geometry. While there are many systems that could be called finite geometries, attention is mostly paid to the finite <a href="/facts/Projective_space/7MSpA9cM">projective</a> and <a href="/facts/Affine_space/tlvI3oX1">affine spaces</a> because of their regularity and simplicity. Other significant types of finite geometry are finite <a href="/facts/M%C3%B6bius_plane/Yrdwk5st">Möbius or inversive planes</a> and <a href="/facts/Laguerre_plane/e6T1HTBe">Laguerre planes</a>, which are examples of a general type called <a href="/facts/Benz_plane/WflhR3z4">Benz planes</a>, and their higher-dimensional analogs such as higher finite <a href="/facts/Inversive_geometry/VqFmbfDI">inversive geometries</a>. </p><p>Finite geometries may be constructed via <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a>, starting from <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> over a <a href="/facts/Finite_field/UkMGJo3m">finite field</a>; the affine and <a href="/facts/Projective_plane/p6ot8oNv">projective planes</a> so constructed are called <a href="/facts/Galois_geometry/Bs3RgkbC">Galois geometries</a>. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite <a href="/facts/Projective_space/7MSpA9cM">projective space</a> of dimension three or greater is <a href="/facts/Isomorphism/pj8KgkxU">isomorphic</a> to a projective space over a finite field (that is, the projectivization of a vector space over a finite field). However, dimension two has affine and projective planes that are not isomorphic to Galois geometries, namely the <a href="/facts/Non-Desarguesian_plane/G7rlClxR">non-Desarguesian planes</a>. Similar results hold for other kinds of finite geometries. </p>