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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.
Given an integer α ≥ 1, a tactical configuration satisfying: For every anti-flag (B, m) there are α flags (A, l) such that B I l and A I m, is called a partial geometry. If there are s + 1 points on a line and t + 1 lines through a point, the notation for a partial geometry is pg(s, t, α). If α = 1 these partial geometries are generalized ...
All finite fields of a given order are isomorphic. Finite fields are fundamental in a number of areas of mathematics and computer science , including number theory , algebraic geometry , Galois theory , finite geometry , cryptography and coding theory .
These points and lines cannot exist with this pattern of incidences in Euclidean geometry, but they can be given coordinates using the finite field with two elements. The standard notation for this plane, as a member of a family of projective spaces , is PG(2, 2) .
Although the generic notation of projective geometry is sometimes used, it is more common to denote projective spaces over finite fields by PG(n, q), where n is the "geometric" dimension (see below), and q is the order of the finite field (or Galois field) GF(q), which must be an integer that is a prime or prime power.
Mathematicians can now explain how many people would need to be invited to a party so at least 4 people always know one another. It only took 90 years to solve.
Thus, for every finite projective plane there is an integer N ≥ 2 such that the plane has N 2 + N + 1 points, N 2 + N + 1 lines, N + 1 points on each line, and N + 1 lines through each point. The number N is called the order of the projective plane. The projective plane of order 2 is called the Fano plane. See also the article on finite geometry.
For two-dimensional, plane strain problems the strain-displacement relations are = ; = [+] ; = Repeated differentiation of these relations, in order to remove the displacements and , gives us the two-dimensional compatibility condition for strains