Duchamp then devises a wonderful comparison between this almost inconceivable prospect of seeing an entire 3-D object all at once and something we can easily achieve with our analogous sensation of touch. Suppose we hold a small penknife firmly in one hand. We can touch the knife's outer surface at all points simultaneously, and we can reconstruct the form of the object from these sensations even though we cannot see all parts of the knife's surface simultaneously from any single vantage point in a 3-D world. Now, Duchamp suggests, suppose we regard our simultaneous touch in three dimensions as analogous to the possibility of simultaneous sight in four dimensions. Then we will understand how we might “see” the entire surface of a 3-D object all at once.

When the Sphere visits Flatland right at the auspicious moment of transition into the third millennium at the inception of the year 2000, he first tries to teach A Square about the third dimension by verbal argument. But A Square cannot comprehend such an expanded universe of higher dimensionality, so the Sphere tears him from the plane of Flatland and treats him to a view of his entire universe from “above” (a dimension previously inconceivable to A Square). Of course, A Square has learned the shapes of buildings and compatriots in his 2-D world, but he can resolve these forms only by laboriously working his way around their perimeters and measuring their sides and angles. From his new vantage point above his old world, however, A Square can see the entire form of each Flatland object all at once—a wondrously new vision that he can conceive and express only as seeing the “invisible inside” of things in one grand, full, and instantaneous view.