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A finite geometry is any geometric system that has only a finite number of points. The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based on the graphics displayed on a computer screen, where the pixels 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 projective and affine spaces because of their regularity and simplicity. Other significant types of finite geometry are finite Möbius or inversive planes and Laguerre planes, which are examples of a general type called Benz planes, and their higher-dimensional analogs such as higher finite inversive geometries. Finite geometries may be constructed via linear algebra, starting from vector spaces over a finite field; the affine and projective planes so constructed are called Galois geometries. Finite geometries can also be defined purely axiomatically. Most common finite geometries are Galois geometries, since any finite projective space of dimension three or greater is isomorphic 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 non-Desarguesian planes. Similar results hold for other kinds of finite geometries. ==Finite affine and projective planes== The following remarks apply only to finite ''planes''. There are two main kinds of finite plane geometry: affine and projective. In an affine plane, the normal sense of parallel lines applies. In a projective plane, by contrast, any two lines intersect at a unique point, so parallel lines do not exist. Both finite affine plane geometry and finite projective plane geometry may be described by fairly simple axioms. An affine plane geometry is a nonempty set ''X'' (whose elements are called "points"), along with a nonempty collection ''L'' of subsets of ''X'' (whose elements are called "lines"), such that: # For every two distinct points, there is exactly one line that contains both points. # Playfair's axiom: Given a line and a point not on , there exists exactly one line containing such that # There exists a set of four points, no three of which belong to the same line. The last axiom ensures that the geometry is not ''trivial'' (either empty or too simple to be of interest, such as a single line with an arbitrary number of points on it), while the first two specify the nature of the geometry. The simplest affine plane contains only four points; it is called the ''affine plane of order'' 2. (The order of an affine plane is the number of points on any line, see below.) Since no three are collinear, any pair of points determines a unique line, and so this plane contains six lines. It corresponds to a tetrahedron where non-intersecting edges are considered "parallel", or a square where not only opposite sides, but also diagonals are considered "parallel". More generally, a finite affine plane of order ''n'' has ''n''2 points and lines; each line contains ''n'' points, and each point is on lines. The affine plane of order 3 is known as the Hesse configuration. A projective plane geometry is a nonempty set ''X'' (whose elements are called "points"), along with a nonempty collection ''L'' of subsets of ''X'' (whose elements are called "lines"), such that: # For every two distinct points, there is exactly one line that contains both points. # The intersection of any two distinct lines contains exactly one point. # There exists a set of four points, no three of which belong to the same line. An examination of the first two axioms shows that they are nearly identical, except that the roles of points and lines have been interchanged. This suggests the principle of duality for projective plane geometries, meaning that any true statement valid in all these geometries remains true if we exchange points for lines and lines for points. The smallest geometry satisfying all three axioms contains seven points. In this simplest of the projective planes, there are also seven lines; each point is on three lines, and each line contains three points. This particular projective plane is sometimes called the Fano plane. If any of the lines is removed from the plane, along with the points on that line, the resulting geometry is the affine plane of order 2. The Fano plane is called the ''projective plane of order'' 2 because it is unique (up to isomorphism). In general, the projective plane of order ''n'' has ''n''2 + ''n'' + 1 points and the same number of lines; each line contains ''n'' + 1 points, and each point is on ''n'' + 1 lines. A permutation of the Fano plane's seven points that carries collinear points (points on the same line) to collinear points is called a collineation of the plane. The full collineation group is of order 168 and is isomorphic to the group PSL(2,7) ≈ PSL(3,2), which in this special case is also isomorphic to the general linear group . 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Finite geometry」の詳細全文を読む スポンサード リンク
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