Hellenica World

Projective plane

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 geometry and finite geometry. The first definition quickly produces planes that are homogeneous spaces for some of the classical groups, including the real projective plane P². The second is suitable for an exhaustive study of the simple incidence properties of plane geometry.

Visualising the real projective plane

In the projective plane P2, a point x is represented by the homogeneous coordinate (x1, x2, x3). If we think of (x1, x2, x3) as a point in real space R3 with the third value of the homogeneous coordinate as a value in the z direction, then P2 can be visualized as R3.

Points, rays, lines, and planes

A line in P² can be represented by the equation ax + by + c = 0. If we treat a, b, and c as the column vector ℓ and x, y, 1 as the column vector x then the equation above can be written in matrix form as:

x^Tℓ = 0 or ℓ^Tx = 0.

Using vector notation we may instead write

x ⋅ ℓ = 0 or ℓ ⋅ x = 0.

The equation k(x^Tℓ) = 0 (which k is a non-zero scalar) sweeps out a plane that goes through zero in R3 and k(x) sweeps out a ray, again going through zero. The plane and ray are subspaces in R3, which always go through zero.

Ideal points

In P² the equation of a line is ax + by + 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 (from a position further out along the z axis and looking back towards the origin) and there are two parallel lines drawn on the plane. From where we are standing (given our visual capabilities) we can see only so much of the plane, which we represent as the area outlined in red in the diagram. If we walk away from the plane along the z axis, (still looking backwards towards the origin), 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 x^Tℓ = 0 there are two column vectors. You can keep either constant and vary the other. If we keep the point constant x and vary the coefficients 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 x^Tℓ = 0 calculates the inner product of two column vectors. The inner product of two vectors is zero if the vectors are orthogonal. To find the line between the points x1 and x2 you must find the column vector ℓ that satisfies the equations x₁^Tℓ = 0 and x₂^Tℓ = 0, that is we must find a column vector ℓ that is orthogonal to x1 and x2. In the case of P2, the cross product will find such a vector. The line joining two points is given by the equation x1 × x2. To find the intersection of two lines you look to duality. If you plot ℓ in the coefficient space you get rays. To find the point x that is orthogonal to the two rays you find the cross product. That is ℓ1 × ℓ2.

While the cross product works in P2, it is not well-defined in arbitrary dimensions. However, this pair of equations is satisfied by

x₁^Tℓ - λ x₂^Tℓ = 0

Projective transformation
Main article: Projective transformation

A projective transformation in P2 space is an invertible mapping of points in P2 to points in P2 that maps lines to lines. A P2 projectivity has the equation

x′ = Hx

where H is an invertible 3 × 3 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:

1. Given any two distinct points, there is exactly one line incident with both of them.
2. Given any two distinct lines, there is exactly one point incident with both of them.
3. 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. 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.
The Fano plane

The smallest possible projective plane is the Fano plane. It has only seven points and seven lines. (See also finite geometry.) 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. 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.

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,
n² + n + 1 lines,
n + 1 points on each line,
n + 1 lines through each point,

where n is an integer (more specifically a natural number greater than 1) called the order of the projective plane. (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. 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. 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. 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). 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. See for example quaternionic projective space. If we use a finite field with pn elements we get a finite projective plane with order pn. The Fano plane is then the plane over the field with two elements, Z2.

The plane over the octonions is sometimes called the Cayley plane and turns out to be an interesting real manifold, which can be used for geometric constructions and understanding of the exceptional Lie groups. As a homogeneous space, the Cayley plane is F₄/ Spin(9) where F₄ is an exceptional Lie group and Spin(9) is the spin group of nine-dimensional Euclidean space.

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. 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. 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 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.

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. 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.
[edit] 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 }
...
Lm = { 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 }
...
Ln = { 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. 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.

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