Finite Affine Planes
If the number of points in an affine plane is finite, then if one line of the plane contains n points then:
- all lines contain n points,
- every point is contained in n+1 lines,
- there are n2 points in all, and
- there are a total of n2 + n lines.
The number n is called the order of the affine plane.
All known finite affine planes have orders which are prime or prime power integers. The smallest affine plane (of order 2) is obtained by removing a line (and the three points on that line) from the Fano plane. An affine plane of order n exists if and only if a projective plane of order n exists (the definitions of order in these cases is not the same). Thus, there is no affine plane of order 6 or order 10. The Bruck-Ryser-Chowla theorem provides further limitations on the order of a projective plane, and thus, the order of an affine plane.
Read more about this topic: Affine Plane (incidence Geometry)
Famous quotes containing the words finite and/or planes:
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (1623–1662)
“After the planes unloaded, we fell down
Buried together, unmarried men and women;”
—Robert Lowell (1917–1977)