Those people who try to imagine a four-dimensional world by drawing geometrical analogies between the properties of lines, surfaces, and solids, commit some egregious fallacies. Thus we hear of one-dimensional creatures, living in a one-dimensional world, meeting and being unable to pass one another. Why not? What physical obstacle is there to prevent their passing one another? A one-dimensional world is not physical; the physical world, the world of obstacles, is three-dimensional neither more nor less. These reasoners are mixing up physics with pure geometry. Pure geometry treats of abstract magnitudes, which are usually represented by diagrams on paper. These have no physical properties whatever; not even the geometrical solids. A geometrical cube is as abstract as a geometrical square; nobody ever handled a geometrical cube, though he may often have handled solid bodies having a cubical shape. In plane geometry, two-dimensional figures can pass freely through or over each other; there is nothing to prevent; lines can intersect without any splicing or gluing; no amount of superimposed squares will ever attain to any thickness, as would be the case with sheets of paper. The spatial dimensions, or rectangular co-ordinates, usually called length, breadth, and thickness, fill up physical space; no more of such dimensions exist. If one of these three dimensions is reduced to zero, the other two vanish also.
Whatever unit of measurement we may use for the linear dimensions, if one of the numbers denoting these dimensions is zero, the product is zero, and the volume is zero. If there are two-dimensional existences, they do not exist in physical space; their dimensions are not spatial.
We might speak of time as a one-dimensional magnitude (using the word "time" in one of its senses); but we do not regard this as a physical dimension, though in a diagram it may be convenient to represent it by a line. In science the word "dimension" is used in a wider and more general sense; we speak of the dimensions of units, which consist of space, mass, and time.
Any such dimension can be represented on a chart by a line and thus we obtain graphs, like those made by recording barometers, etc. Advanced science now speaks of time as a fourth dimension, but the four dimensions do not characterize physical objects but "events " Now an event may be a sufficiently real thing, but we can hardly talk about what would happen if two events met one another out walking. We cannot make our events have so many faces and so many edges, and so many angles, etc.
We can calculate an abstract mathematical construction with four dimensions, on the analogy of the cube, and say how many faces, edges, etc. it ought to have; but this is not the same thing as imagining an actual physical "tesseract" somewhere in space. We say it is not the same thing; but we do not say that such a thing as a four-dimensional being does not exist; we merely say it is not a physical being.
Some people who mix up pure geometry with physics seem to think that we can build up a line out of points, a surface out of lines, and a solid out of surfaces. This cannot be true in a physical sense; for the point has no magnitude and cannot build up anything. Moreover, unless we begin by presupposing three-dimensional physical space, we shall have nowhere to place our point, and it cannot be regarded as having position or distance in relation to other points. Such reasoners do what generally is done in arguments — they assume their conclusion; they assume physical space and then proceed to construct it. They should begin at the other end and assume as a unit a point having dimension, an atom in fact; and then a line becomes a row of points, and so on.
But if we step beyond the physical plane, and use the word "dimension" in its general sense as "the degree of manifoldness of a magnitude," the case is different. We may evolve our universe out of points, lines, triangles, etc., always remembering that these are not physical distances or pieces of string. Points, lines, triangles, etc., are frequently mentioned in The Secret Doctrine in reference to existences on other planes — non-physical planes; in one passage there is a definite distinction made between "solids" and "solid bodies." The point, as a symbol, stands for a monad, a unitary center, a logos; the line is a motion or force, it has length; a triangle (or, more generally, a superficies) has shape, and shape represents quality or character; and so on, as we prefer to leave such interpretations to the fancy of the individual student, not being sure enough of our own personal dogmas. Analogy is a useful servant, but a bad master.
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