Chapter VI (part 1)

p. 59 CHAPTER VI Methods of investigation of the problem of higher dimensions. The analogy between imaginary worlds of different dimensions. The one-dimensional world on a line. "Space" and "time" of a one-dimensional being. The two-dimensional world on a plane. "Space" and "time," "ether," "matter" and "motion" of a two-dimensional being. Reality and illusion on a plane. The impossibility of seeing an "angle." An angle as motion. The incomprehensibility to a two-dimensional being of the functions of things in our world. Phenomena and noumena of a two-dimensional being. How could a plane being comprehend the third dimension? A SERIES of analogies and comparisons are used for the definition of that which can be, and that which cannot be, in the region of the higher dimension. We imagine "worlds" of one, and of two dimensions, and out of the relations of lower-dimensional worlds to higher ones we deduce possible relations of our world to one of four dimensions; just as out of the relations of points to lines, of lines to surfaces, and of surfaces to solids we deduce the relations of our solids to four-dimensional ones. Let us try to investigate everything that this method of analogy can yield. Let us imagine a world of one dimension . It will be a line. Upon this line let us imagine living beings. Upon this line, which represents the universe for them, they will be able to move forward and backward only, and these beings will be as the points, or segments of a line. Nothing will exist for them outside their line—and they will not be aware of the line upon which they are living and moving. For there will exist only two points, ahead and behind, or may be just one point ahead. Noticing the change in states of these points, the one-dimensional being will call these changes phenomena . If we suppose the line upon which the one-dimensional being lives to be passing through the different objects of our world, then of all these objects the one-dimensional p. 60 being will perceive one point only; if different bodies intersect his line, the one-dimensional being will sense them only as the appearance, the more or less prolonged existence, and the disappearance of a point. This appearance, existence, and disappearance of a point will constitute a phenomenon . Phenomena, according to the character and properties of passing objects and the velocity and properties of their motions, for the one-dimensional being will be constant or variable, long or short timed, periodical or unperiodical. But the one-dimensional being will be absolutely unable to understand or explain the constancy or variability, the duration or brevity, the periodicity or unperiodicity of the phenomena of his world, and will regard these simply as properties of such phenomena. The solids intersecting his line may be different, but for the one-dimensional being all phenomena will be absolutely identical —just the appearance or the disappearance of a point—and phenomena will differ only in duration and in greater or less periodicity. Such strange monotony and similarity of the diverse and heterogeneous phenomena of our world will be the characteristic peculiarity of the one-dimensional world. Moreover, if we assume that the one-dimensional being possesses memory, it is clear that recalling all the points seen by him as phenomena, he will refer them to time. The point which was : this is the phenomenon already non-existent, and the point which may appear tomorrow: this is the phenomenon which does not exist as yet . All of our space except one line will be in the category of time, i.e., something wherefrom phenomena come and into which they disappear. And the one-dimensional being will declare that the idea of time arises for him out of the observation of motion, that is to say, out of the appearance and disappearance of points.