**lew@ihuxr.UUCP (Lew Mammel, Jr.)** (09/28/83)

Silvio Levy made an error in his article on projective spaces. The set of opposite point pairs on a sphere correspond to the projective plane, not the projective 3-d space. This is obvious when you note that each projective point is identified with { X * (a,b,c) for all X != 0 } The euclidian coordinates of the point are given by x=a/c , y=b/c. The points represented by (a,b,0) form the "line at infinity". Linear transformations of the projective coordinates correspond to projective transformations of the plane. These include rotations and translations, as well as a set of transformations which correspond to the projection of a horizontal plane onto a vertical plane, as in perspective drawing. Projective geometry gets short shrift these days, but it is quite fascinating. Some of its interesting ramifications are described in Felix Klein's "Geometry" from his "Elementary mathematics from an advanced viewpoint" series. Lew Mammel, Jr. ihuxr!lew