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The projective plane over K, denoted PG(2, K) or KP 2, has a set of points consisting of all the 1-dimensional subspaces in K 3. A subset L of the points of PG(2, K ) is a line in PG(2, K ) if there exists a 2-dimensional subspace of K 3 whose set of 1-dimensional subspaces is exactly L .
Another way to put it is that the points of n-dimensional projective space are the 1-dimensional vector subspaces, which may be visualized as the lines through the origin in K n+1. [10] Also the n - (vector) dimensional subspaces of K n+1 represent the (n − 1)- (geometric) dimensional hyperplanes of projective n-space over K, i.e., PG(n, K).
A projective geometry of dimension 1 consists of a single line containing at least 3 points. The geometric construction of arithmetic operations cannot be performed in either of these cases. For dimension 2, there is a rich structure in virtue of the absence of Desargues' Theorem. The Fano plane is the projective plane with the fewest points ...
A two-dimensional complex space – such as the two-dimensional complex coordinate space, the complex projective plane, or a complex surface – has two complex dimensions, which can alternately be represented using four real dimensions. A two-dimensional lattice is an infinite grid of points which can be represented using integer coordinates.
The plane P 1 defines a projective line which is called the line at infinity of P 2. By identifying each point of P 2 with the corresponding projective point, one can thus say that the projective plane is the disjoint union of P 2 and the (projective) line at infinity.
The Fano plane, the projective plane over the field with two elements, is one of the simplest objects in Galois geometry.. Galois geometry (named after the 19th-century French mathematician Évariste Galois) is the branch of finite geometry that is concerned with algebraic and analytic geometry over a finite field (or Galois field). [1]
In mathematics, a plane is a two-dimensional space or flat surface that extends indefinitely. A plane is the two-dimensional analogue of a point (zero dimensions), a line (one dimension) and three-dimensional space. When working exclusively in two-dimensional Euclidean space, the definite article is used, so the Euclidean plane refers to the ...
One common model of the real projective plane is the space of lines in three-dimensional Euclidean space which pass through a particular origin point; in this model, lines through the origin are considered to be the "points" of the projective plane, and planes through the origin are considered to be the "lines" in the projective plane.