Projective plane - Citizendia

See real projective plane and complex projective plane, for the cases met as manifolds of respective dimension 2 and 4

In mathematics, a projective plane has two possible definitions, one of them coming from linear algebra, and another (which is more general) coming from axiomatic and finite geometry. Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points" In Mathematics, the complex projective plane, usually denoted CP 2 is the two-dimensional Complex projective space. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Linear algebra is the branch of Mathematics concerned with A finite geometry is any geometric system that has only a finite number of points. The first definition quickly produces planes that are homogeneous spaces for some of the classical groups, including the real projective plane $\mathbb{P}^2$ . In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group The classical Lie groups are four infinite families of Lie groups closely related to the symmetries of Euclidean spaces There is a certain leeway in using the term The second is suitable for an exhaustive study of the simple incidence properties of plane geometry. In Geometry, the Relations of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L' and 'intersects' (as in 'line L1 In Mathematics, plane geometry may mean geometry of a plane, geometry of the Euclidean plane, or sometimes

1 Visualising the real projective plane
2 Combinatorial definition
- 2.1 Examples
- 2.2 Properties
3 Linear algebra definition
4 Generalized coordinates
5 Higher dimensions
6 Degenerate planes
7 Connection with Latin squares
8 Construction of projective planes of prime order
9 Small orders
10 See also
11 References

Visualising the real projective plane

In the projective plane $\mathbb{P}^2$ , a point $x$ is represented by the homogeneous coordinate $(x 1, x 2, x 3)$ . In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations If we think of $(x 1, x 2, x 3)$ as a point in real space $\mathbb{R}^3$ with the third value of the homogeneous coordinate as a value in the $z$ direction, then $\mathbb{P}^2$ can be visualized as $\mathbb{R}^3.$

Points, rays, lines, and planes

A line in $\mathbb{P}^2$ can be represented by the equation $a x + b y + c = 0$ . If we treat $a$ , $b$ and $c$ as the column vector $\mathbf{l}$ and $x$ , $y$ , $1$ as the column vector $\mathbf{x}$ then the equation above can be written in matrix form as:

$\mathbf{x}^T\mathbf{l}=0$ or $\mathbf{l}^T\mathbf{x}=0$

Or using vector notation

$\mathbf{x}\cdot\mathbf{l}=0$ or $\mathbf{l}\cdot\mathbf{x}=0$

$k(\mathbf{x}^T\mathbf{l})=0$ sweeps out a plane that goes through zero in $\mathbb{R}^3$ and

$k(\mathbf{x})$ sweeps out a ray ( a ray goes through zero).

The plane and ray are subspaces in $\mathbb{R}^3$ . In Linear algebra, an Euclidean subspace (or subspace of R n) is a set of vectors that is closed under addition A subspace always goes through zero.

Ideal points

In $\mathbb{P}^2$ the equation of a line is $a x + b y + c = 0$ and this equation can represent a line on any plane parallel to the $x, y$ plane by multiplying the equation by $k$ .

If $z = 1$ we have a normalized homogeneous coordinate. All points that have $z = 1$ create a plane. Let's pretend we are looking at that plane and there are two parallel lines drawn on the plane. From where we are standing we can see only so much of the plane (the area outlined in red). If we walk away from the plane along the z axis we can see more of the plane. In our field of view original points have moved. We can reflect this movement by dividing the homogeneous coordinate by a constant. In the image to the right we have divided by 2 so the $z$ value now becomes 0. 5. If we walk far enough away what we are looking at becomes a point in the distance. As we walk away we see more and more of the parallel lines. The lines will meet at a line at infinity ( a line that goes through zero on the plane at $z = 0$ ). Lines on the plane when $z = 0$ are ideal points. The plane at $z = 0$ is the line at infinity.

The homogeneous point $(0,0,0)$ is where all the real points go when you're looking at the plane from an infinite distance, a line on the $z = 0$ plane is where parallel lines intersect.

Duality

In the equation $\mathbf{x}^T\mathbf{l}=0$ there are two column vectors. In Linear algebra, a column vector or column matrix is an m × 1 matrix, i You can keep either constant and vary the other. If we keep the point constant $\mathbf{x}$ and vary the coefficients $\mathbf{l}$ we create new lines that go through the point. If we keep the coefficients constant and vary the points that satisfy the equation we create a line. We look upon $x$ as a point because the axes we are using are $x, y$ and $z$ . If we instead plotted the coefficients using axis marked $a, b, c$ points would become lines and lines would become points. If you prove something with the data plotted on axis marked $x, y$ and $z$ the same argument can be used for the data plotted on axis marked $a, b$ and $c$ . That is duality.

Lines joining points and intersection of lines (using duality)

The equation $\mathbf{x}^T\mathbf{l}=0$ calculates the inner product of two column vectors. In Mathematics, the dot product, also known as the scalar product, is an operation which takes two vectors over the Real numbers R The inner product of two vectors is zero if the vectors are orthogonal. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i To find the line between the points $\mathbf{x}_1$ and $\mathbf{x}_2$ you must find the column vector $\mathbf{l}$ that satisfies the equations $\mathbf{x}_1^T\mathbf{l}=0$ and $\mathbf{x}_2^T\mathbf{l}=0$ , that is we must find a column vector $\mathbf{l}$ that is orthogonal to $\mathbf{x}_1$ and $\mathbf{x}_2$ . The cross product will find such a vector. In Mathematics, the cross product is a Binary operation on two vectors in a three-dimensional Euclidean space that results in another vector which The line joining two points is given by the equation $\mathbf{x}_1 \times \mathbf{x}_2$ .

To find the intersection of two lines you look to duality. If you plot $\mathbf{l}$ in the coefficient space you get rays. To find the point $\mathbf{x}$ that is orthogonal to the two rays you find the cross product. That is $\mathbf{l}_1 \times \mathbf{l}_2$ .

Projective transformation

A projective Transformation in $\mathbb{P}^2$ space is an invertible mapping of points in $\mathbb{P}^2$ to points in $\mathbb{P}^2$ that maps lines to lines. A $\mathbb{P}^2$ projectivity has the equation: $\mathbf{x'}=\mathbf{Hx}$ . Where $\mathbf H$ is an invertible $3\times3$ matrix. This is, a projectivity is any conceivable invertible linear transform of homogeneous coordinates.

Combinatorial definition

According to the more general, combinatorial definition, a projective plane consists of a set of lines and a set of points, and a relation between points and lines called incidence, having the following properties:

Given any two distinct points, there is exactly one line incident with both of them.
Given any two distinct lines, there is exactly one point incident with both of them.
There are four points such that no line is incident with more than two of them.

The second condition means that there are no parallel lines. The last condition simply excludes some degenerate cases (see below).

Examples

A projective plane is an abstract mathematical concept, so the "lines" need not be anything resembling ordinary lines, nor need the "points" resemble ordinary points. The most common projective plane is the real projective plane, which is a topological surface with surprising geometric properties; after that is the complex projective plane of algebraic geometry, a topological four-dimensional manifold. Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points" In Mathematics, specifically in Topology, a surface is a Two-dimensional Manifold. In Mathematics, the complex projective plane, usually denoted CP 2 is the two-dimensional Complex projective space. Algebraic geometry is a branch of Mathematics which as the name suggests combines techniques of Abstract algebra, especially Commutative algebra, with A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be For any field K, there is a projective plane with three homogeneous coordinates in K, which can also be thought of in terms of a three-dimensional vector space V over K, 'points' being one-dimensional subspaces and 'lines' two-dimensional subspaces. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added

The Fano plane

The smallest possible projective plane is the Fano plane. In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each It has only seven points and seven lines. (See also finite geometry. A finite geometry is any geometric system that has only a finite number of points. ) In the figure at right, the seven points are shown as small black balls, and the seven lines are shown as six line segments and a circle. However, one could equivalently consider the balls to be the "lines" and the line segments and circle to be the "points" — this is an example of the duality of projective planes: if the lines and points are interchanged, the result is still a projective plane. In the Geometry of the Projective plane, duality refers to geometric transformations that replace points by lines and lines by points while preserving A permutation of the seven points that carries collinear points (points on the same line) to collinear points is called a "symmetry" of the plane. In Geometry, the Relations of incidence are those such as 'lies on' between points and lines (as in 'point P lies on line L' and 'intersects' (as in 'line L1 Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or

Properties

It can be shown that a projective plane has the same number of lines as it has points. This number can be infinite (as for the real projective plane) or finite (as for the Fano plane). A finite projective plane has

n² + n + 1 points,

where n is an integer called the order of the projective plane. The integers (from the Latin integer, literally "untouched" hence "whole" the word entire comes from the same origin but via French (The Fano plane therefore has order 2. ) There exists a finite projective plane of order n, if n is a prime power, and for all known finite projective planes, the order n is a prime power. In Mathematics, a prime power is a Positive integer power of a Prime number. The existence of finite projective planes of other orders is an open question. The only general restriction known on the order is the Bruck-Ryser-Chowla theorem that if the order n is congruent to 1 or 2 mod 4, it must be the sum of two squares. The Bruck – Chowla – Ryser theorem is a result on the Combinatorics of Block designs It states that if a ( v, b This rules out n = 6. The next case n = 10 has been ruled out by massive computer calculations, and there is nothing more known, in particular n = 12 is still open. There is a projective plane of order n if and only if there is an affine plane of order n. In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. When there is only one affine plane of order n there is only one projective plane of order n, but the converse is not true. A projective plane of order n has n + 1 points on every line, and n + 1 lines passing through every point, and is therefore a Steiner S(2, n + 1, n² + n + 1) system (see Steiner system). In combinatorial Mathematics, a Steiner system (named after Jakob Steiner) is a type of Block design. Conversely, one can prove that all Steiner systems of this form (n ≥ 2) are projective planes.

Linear algebra definition

One can construct projective planes (or higher dimensional projective spaces) by linear algebra over any division ring—not necessarily commutative. In Mathematics a projective space is a set of elements constructed from a vector space such that a distinct element of the projective space consists of all non-zero vectors which In Mathematics, commutativity is the ability to change the order of something without changing the end result See for example quaternionic projective space. In Mathematics, quaternionic projective space is an extension of the ideas of Real projective space and Complex projective space, to the case where coordinates If we use a finite field with pⁿ elements we get a finite projective plane with order pⁿ. In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements The Fano plane is then the plane over the field with two elements, Z₂. In Finite geometry, the Fano plane (after Gino Fano) is the Projective plane with the least number of points and lines 7 each

The plane over the octonions turns out to be an interesting real manifold, which can be used for geometric constructions and understanding of the exceptional Lie groups. In Mathematics, the octonions are a nonassociative extension of the Quaternions Their 8-dimensional Normed division algebra over the Real In Mathematics, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected Normal subgroups

Generalized coordinates

One can construct a coordinate "ring"—a so-called planar ternary ring (not a genuine ring) corresponding to any projective plane in the combinatorial definition. In Mathematics, a planar ternary ring (PTR or ternary field is an Algebraic structure (RT where R is a non-empty set and Algebraic properties of this "ring" turn out to correspond to geometric incidence properties of the plane. For example, Desargues' theorem corresponds to the coordinate ring's being obtained from a division ring, while Pappus's theorem corresponds to this ring's being obtained from a commutative field. In Projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states In a Projective space, two Triangles In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible Pappus's hexagon theorem (attributed to Pappus of Alexandria) states that given one set of Collinear points A, B, C, and another In Mathematics, commutativity is the ability to change the order of something without changing the end result However, the "ring" need not be of these types, and there are many non-Desarguesian projective planes. Alternative, not necessarily associative, division rings like the octonions correspond to Moufang planes. In Abstract algebra, an alternative algebra is an algebra in which multiplication need not be Associative, only alternative. In Mathematics, associativity is a property that a Binary operation can have Ruth Moufang (1905&ndash1977 was a German Mathematician. Born on January 10th 1905 in Darmstadt Germany to the German chemist Dr In the case of finite projective planes, the only proof known of the purely geometric statement that Desargues' theorem then implies Pappus' theorem (the converse being always true and provable geometrically) is through this algebraic route, using Wedderburn's theorem that finite division rings must be commutative. In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible

Higher dimensions

It is possible to make analogous incidence definitions for higher dimensional projective geometries, with dimension larger than 2. These turn out to not be as interesting as (or one might say, they are better behaved than in) the planar case, as they are to the classical projective spaces over division rings. In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible The reason is that with the extra room to work in, one can prove Desargues' theorem geometrically as in its article by using incidence properties in this higher dimensional space; thus the coordinate "ring" must be a division ring. In Projective geometry, Desargues' theorem, named in honor of Gérard Desargues, states In a Projective space, two Triangles In Abstract algebra, a division ring, also called a skew field, is a ring in which division is possible

Degenerate planes

Degenerate planes do not fulfill the third condition above. There are two families of degenerate planes.

1) For any number of points P₁, . . . , P_n, and lines L₁, . . . , L_m,

L₁ = { P₁, P₂, . . . , P_n}

L₂ = { P₁ }

L₃ = { P₁ }

. . .

L_m = { P₁ }

2) For any number of points P₁, . . . , P_n, and lines L₁, . . . , L_n, (same number of points as lines)

L₁ = { P₂, P₃, . . . , P_n }

L₂ = { P₁, P₂ }

L₃ = { P₁, P₃ }

. . .

L_n = { P₁, P_n }

Connection with Latin squares

A projective plane of order n (n ≥ 2) exists if and only if there is an affine plane of this order. The number of mutually orthogonal latin squares of order n is at most n − 1. A Graeco-Latin square or Euler square of order n over two sets S and T, each consisting of n symbols is an n × It turns out n − 1 is possible if and only if there is an affine plane of this order.

Construction of projective planes of prime order

Method 1

This is the standard construction using homogeneous coordinates over a finite field. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations In Abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements

Method 2

To construct a projective plane of order N (N prime), proceed as follows:

Create one point P

Create N points, which we will label P(c) : c = 0, . . . , (N − 1)

Create N² points, which we will label P(r, c) : r, c = 0, . . . , (N − 1)

On these points, construct the following lines:

One line L = { P, P(0), . . . , P(N − 1)}

N lines L(c) = {P, P(0,c), . . . , P(N − 1,c)} : c = 0, . . . , (N − 1)

N² lines L(r, c): P(c) and the points P((r + ci) mod N, i), where i = 0, . . , N − 1 : r, c = 0, . . . , (N − 1)

Note that the expression

(r + ci) mod N

will pass once through each value as i varies from 0 to N − 1, but only if is N is prime.

By this construction, we have two degenerate planes: one point incident with one line (for N = 0) and a triangle consisting of three points and three lines (for N = 1). Every plane constructed with prime N (N > 1) fulfills all three conditions above.

For example, for N=2:

One line L = { P, P(0), P(1)}

2 lines L(c) = {P, P(0,c), P(1,c)} : c = 0, 1

4 lines L(r, c): P(c) and the points P((r + ci) mod 2, i), where i = 0, 1 : r, c = 0, 1

Small orders

While the classification of all projective planes is far from done, here are some results for some orders :

2 : all isomorphic with PG(2,2)
3 : all isomorphic with PG(2,3)
4 : all isomorphic with PG(2,4)
5 : all isomorphic with PG(2,5)
6 : impossible as order of a projective plane, proved by Tarry as Euler's thirty-six officers problem
7 : all isomorphic with PG(2,7)
8 : all isomorphic with PG(2,8)
9 : PG(2,9), and three more different (up to isomorphism) non-Desarguesian planes. Gaston Tarry ( September 27, 1843 - June 21, 1913) was a French Mathematician. The thirty-six officers problem is a mathematical puzzle proposed by Leonhard Euler in 1782
10 : impossible as order of a projective plane, proved by heavy computer calculation.
11 : at least PG(2,11), others are not known but possible.
12 : it is conjectured to be impossible as an order of a projective plane, but this is not proven.

References

D. In combinatorial Mathematics, an incidence structure is a triple C=(PLI Projective geometry is a non- metrical form of Geometry, notable for its principle of duality. Construction Consider a Sphere, and let the Great circles of the sphere be "lines" and let pairs of Antipodal points be "points" Hughes and F. Piper (1973). Projective Planes. Springer-Verlag. ISBN 0-387-90044-6.
Eric W. Weisstein, Projective plane at MathWorld. Eric W Weisstein (born March 18, 1969, in Bloomington Indiana) is an Encyclopedist who created and maintains MathWorld MathWorld is an online Mathematics reference work created and largely written by Eric W
Clement W. H. Lam, "The Search for a Finite Projective Plane of Order 10", American Mathematical Monthly 98, (no. 4) 1991, pp. 305 - 318.
Lindner, Charles C. and Christopher A. Rodger (eds. ) Design Theory, CRC-Press; 1 edition (October 31, 1997). ISBN 0-8493-3986-3.
G. Eric Moorhouse, Projective Planes of Small Order, (2003)

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