Projective Plane - Projective Planes in Higher Dimensional Projective Spaces

Projective Planes in Higher Dimensional Projective Spaces

Projective planes may be thought of as projective geometries of "geometric" dimension two. Higher dimensional projective geometries can be defined in terms of incidence relations in a manner analogous to the definition of a projective plane. These turn out to be "tamer" than the projective planes since the extra degrees of freedom permit Desargues' theorem to be proved geometrically in the higher dimensional geometry. This means that the coordinate "ring" associated to the geometry must be a division ring (skewfield) K, and the projective geometry is isomorphic to the one constructed from the vector space Kd+1, i.e. PG(d,K). As in the construction given earlier, the points of the d-dimensional projective space PG(d,K) are the lines through the origin in Kd + 1 and a line in PG(d,K) corresponds to a plane through the origin in Kd + 1. In fact, each i-dimensional object in PG(d,K), with i < d, is an (i+1)-dimensional (algebraic) vector subspace of Kd + 1 ("goes through the origin"). The projective spaces in turn generalize to the Grassmannian spaces.

It can be shown that if Desargues' theorem holds in a projective space of dimension greater than two, then it must also hold in all planes that are contained in that space. Since there are projective planes in which Desargues' theorem fails (non-Desarguesian planes), these planes can not be embedded in a higher dimensional projective space. Only the planes from the vector space construction PG(2,K) can appear in projective spaces of higher dimension. Some disciplines in mathematics restrict the meaning of projective plane to only this type of projective plane since otherwise general statements about projective spaces would always have to mention the exceptions when the geometric dimension is two.

Famous quotes containing the words spaces, dimensional, planes and/or higher:

Though there were numerous vessels at this great distance in the horizon on every side, yet the vast spaces between them, like the spaces between the stars,—far as they were distant from us, so were they from one another,—nay, some were twice as far from each other as from us,—impressed us with a sense of the immensity of the ocean, the “unfruitful ocean,” as it has been called, and we could see what proportion man and his works bear to the globe.
Henry David Thoreau (1817–1862)

I don’t see black people as victims even though we are exploited. Victims are flat, one- dimensional characters, someone rolled over by a steamroller so you have a cardboard person. We are far more resilient and more rounded than that. I will go on showing there’s more to us than our being victimized. Victims are dead.
Kristin Hunter (b. 1931)

After the planes unloaded, we fell down
Buried together, unmarried men and women;
Robert Lowell (1917–1977)

The passion of self-aggrandizement is persistent but plastic; it will never disappear from a vigorous mind, but may become morally higher by attaching itself to a larger conception of what constitutes the self.
Charles Horton Cooley (1864–1929)