finite plane

A finite plane (synonym linear space (http://planetmath.org/LinearSpace2)) is the finite (discrete) analogue of planes in more familiar geometries. It is an incidence structure where any two points are incident with exactly one line (the line is said to “pass through” those points, the points “lie on” the line), and any two lines are incident with at most one point — just like in ordinary planes, lines can be parallel i.e. not intersect in any point.

A finite plane without parallel lines is known as a projective plane. Another kind of finite plane is an affine plane, which can be obtained from a projective plane by removing one line (and all the points on it).

Example

An example of a projective plane, that of order $2$, known as the Fano plane (for projective planes, order $q$ means $q+1$ points on each line, $q+1$ lines through each point):

An edge here is represented by a straight line, and the inscribed circle is also an edge. In other words, for a vertex set $\{1,2,3,4,5,6,7\}$, the edges of the Fano plane are

$$\{1,2,4\},\{2,3,5\},\{3,4,6\},\{4,5,7\},\{5,6,1\},\{6,7,2\},\{7,1,3\}$$

Notice that the Fano plane is generated by the triple $\{1,2,4\}$ by repeatedly adding $1$ to each entry, modulo $7$. The generating triple has the property that the differences of any two elements, in either order, are all pairwise different modulo $7$. In general, if we can find a set of $q+1$ of the integers (mod $q^2+q+1$) with all pairwise differences distinct, then this gives a cyclic representation of the finite plane of order $q$.

Title	finite plane
Canonical name	FinitePlane
Date of creation	2013-03-22 13:05:34
Last modified on	2013-03-22 13:05:34
Owner	marijke (8873)
Last modified by	marijke (8873)
Numerical id	18
Author	marijke (8873)
Entry type	Definition
Classification	msc 51E20
Classification	msc 05C65
Classification	msc 51E15
Classification	msc 05B25
Related topic	LinearSpace2
Defines	Fano plane