Imagery

Think of a plane as a table with three legs. The legs should not be infinite in length, so they may be planted on the walls or ceiling as well as the floor. Of course, the legs are perpendicular to the floor, wall, or ceiling that they are planted on. Name floor, wall, or ceiling as base. Now, relax your imagination, and assume all tables planted on a base have their feet on the same three points of the base. So, the legs overlap for some of their length. In my imagination, more than one thing can occupy the same space. Now imagine flies on a table. Mark out lengths in the corners to find the coordinates of the flies. Construct tables on the floor, with leg lengths given by the fly coordinates. I have claimed, with other words, that if the flies are on a single table, then their constructed tables all intersect in a single point. Consider what happens when one fly walks parallel to a base. One leg of its table remains fixed in length. Similarly (pun intended), if the fly walks in any straight line, a point on its table remains fixed in distance above the floor. In fact, a whole line on its table remains fixed, because its coordinates change at fixed rates wrt each other, and because leg lengths change at rates proportionate to their distances from a fixed line. To get from one point to any other, the fly may walk first parallel to one base, then parallel to another (and so on for hyperflies). Thus, two lines are fixed, intersecting in a point. But no matter where the fly starts, it is the same two lines that are fixed, because its coordinates change in the same ratio when it walks parallel to a base.

Why am I being so colorful about cospaces? Cospaces allow me to extend one space with boundaries from another to find a superspace. And it is because of superspaces that I can find a contradiction to the Simplex Theorem to prove that every linear space is the classification of some collection of hyperplanes.