A line drawing illustrating an analogy for dimensionality. At the top, a deck of cards is fanned out in a semi-circular arch, showing various card faces (suits including diamonds, spades, clubs, and hearts) and backs. In the center, beneath the arch, is a single card depicted as a three-dimensional rectangular prism with a cross-hatched pattern on its side. Below the illustration is a two-line caption in a stylized block font.

THE FOURTH DIMENSION

THE INDISPENSABLE ANALOGY

Adventure with me down a precipice of thought, sustained only by the rope of an analogy, slender but strong. This rope, anchored in the firm ground of sensuous perception, extends three paces in the direction of the great abyss, then vanishes at the giddy brink. Let us examine this sustaining simile foot by foot and strand by strand.

Familiar both to the mind and eye are the space systems of one, two, and three dimensions; that is, lines, planes, solids. Lines are bounded by points, and themselves bound planes; line-bound planes in turn bound solids. What, then, do solids bound? Here is where the analogical rope vanishes from sight. If you answer that a solid cannot be a boundary, we part company. No argument of mine can convince you to the contrary. But if you are inter-