The Plane is the flat open area of the Periklyne where dialogues were conducted and diagrams drawn in the dust, and surrounded by the vine covered bowers of the shaded Suvin.
To "plane," in the jargon of the avout, is a verb meaning "to thoroughly defeat", and is used most often of formal verbal dialogues. To be planed is thus to be thoroughly defeated, and to plane another is to win the contest soundly. It is possibly derived from the idea of a geometric plane, or flat, two dimensional space; in martial combat, when one is defeated, one is generally left lying on the ground and thus forming a "plane."