Dual Planes Research Articles

Geometries with dual affine planes and symplectic quadrics

Farmer and Hale [3] prove that every copolar space fully embedded in a finite projective space PG( n, q), with q>, is the copolar space arising from a symplectic polarity. We show that this result is still valid in arbitrary projective spaces; this provides a different and shorter proof of [3] in the finite case.

Finite geometries which contain dual affine planes

Finite geometries in which each plane is projective or dual affine over the field of two elements, or affine over the field of three elements, are studied. It is shown that no connected geometry can mix all three species of planes, and the geometries in which projective and dual affine planes occur are classified.

A Classification of Semi-Translation Planes

The classification of certain types of projective planes has recently been of considerable interest to both geometers and group theorists. Due in part to the current general interest in finite mathematics and the developments connecting group theory and finite geometry, the Lenz-Barlotti classification of finite projective planes (2; 10), in particular, has generated a tremendous amount of research. A great deal of this research has been related to the construction of non-Desarguesian planes.Fryxell (6), Hughes (7), Luneburg (11), and Ostrom (13; 15; 18) have found examples of projective planes, all of which are of a general type that we call semi-translation planes. Many of these planes are of the same Lenz-Barlotti class I-1. (The Lùneburg planes are translation planes. However, the planes dual to the Luneburg planes are semi-translation planes as well as dual translation planes.)

Billy Budd on the Stage

Producer:The main problem seems to me to decide how far we are going to be realistic and how far not. For me the most realistic scene is the Battle, Act III, i., where the crew have very definite things to do, loading guns and firing them, stowing hammocks and so on. Then the mist descends again and they are cut off from the enemy. But the mist is as much a mist of doubt and fear in the mind of Vere when he is about to close with Claggart at the beginning and finally at the end of the scene. And these dual planes of action seem to me to run throughout the opera, as they do in Melville.