Three-dimensional space

“ When an astronomer is found in his Reports "gauging infinitude," even the most intuitional of his class is but too often apt to forget that he is gauging only the superficies of a small area and its visible depths, and to speak of these as though they were merely the cubic contents of some known quantity. This is the direct result of the present conception of a three-dimensional space. The turn of a four-dimensional world is near, but the puzzle of science will ever continue until their concepts reach the natural dimensions of visible and invisible space—in its septenary completeness. ”

“ Three-dimensional space is only a section cut through four-dimensional space-time, and moreover sections cut in different directions form the spaces of different observers. We can scarcely maintain that the study of a section cut in one special direction is the proper subject-matter of geometry and that the study of slightly different sections belongs to an altogether different science. Hence the geometry of the world is now considered to include time as well as space. ”

“ But the three-dimensional space in which a curved two-dimensional space is conceived to exist is quite arbitrary. A two-dimensional observer cannot ascertain by any observation whether he is on a plane or a cylinder, a sphere or any other convex surface of the same total curvature. ”

“ We thus obtain three-dimensional mathematical space. Now mathematical space, in view of its amorphous nature, is essentially relative. The very homogeneity of space, its sameness “here” as “there,” precludes our being able to differentiate one absolute position from another. It will follow that absolute motion, i.e., a variation of absolute position in empty amorphous mathematical space, can have no significance.
Our sole means of giving significance to position and motion will be by selecting some three-dimensional frame of reference as standard. ”

“ The truth seems to be that space—and time also—is much more complicated than it would appear to be from the finished structure of physics, and that the one all-embracing three-dimensional space is a logical construction, obtained by means of correlations from a crude space of six dimensions. The particulars occupying this six-dimensional space, classified in one way, form "things," from which with certain further manipulations we can obtain what physics can regard as matter ”

“ Therefore, despite the oft-repeated statements to the contrary, it is not impossible to imagine four-dimensional space. Only, we must not attempt to represent to ourselves four-dimensional space in three-dimensional space, especially not without a knowledge of its properties. ”

“ It is found that the transformation which is really needed is an inversion in a space of four dimensions, the transition to three-dimensional space being made by replacing the fourth coordinate by ict, where t is the time and c the velocity of light. ”

“ But as the construction progresses it becomes more and more patent that the disposition of the discs in the manner indicated, without interruption, is not possible, as it should be possible by Euclidean geometry of the the plane surface. In this way creatures which cannot leave the spherical surface, and cannot even peep out from the spherical surface into three-dimensional space, might discover, merely by experimenting with discs, that their two-dimensional "space" is not Euclidean, but spherical space. ”

“ Let any man picture to himself the aggregate of his own bodily forms from birth to the present time, and he will have a clear idea of a sur-solid in time-space.
Let us now consider the sur-solid formed by the movement, or rather, the successive existence, of a cube in time-space. We are to conceive of the cube, and the whole of the three-dimensional space in which it is situated, as floating away in time-space for a given time; the cube will then have an initial and a final position, and these will be the end boundaries of the sur-solid. ”

“ We must, therefore, conceive that there is a new three-dimensional space for each successive instant of time; and, by picturing to ourselves the aggregate formed by the successive positions in time-space of a given solid during a given time, we shall get the idea of a four-dimensional solid, which may be called a sur-solid. It will assist us to get a clearer idea, if we consider a solid which is in a constant state of change, both of magnitude and position; and an example of a solid which satisfies this condition sufficiently well, is afforded by the body of each of us. ”

“ The section gives three-dimensional space at some moment, and the perpendicular dimension is (imaginary) time. Clearly the slice may be taken in any direction; there is no question of a true separation and a fictitious separation. There is no conspiracy of the forces of nature to conceal our absolute motion—because, looked at from this broader point of view, there is nothing to conceal. The observer is at liberty to orient his rectangular axes of , , and arbitrarily, just as in three-dimensions he can orient his axes of , , arbitrarily. ”

“ From the latest results of the theory of relativity it is probable that our three-dimensional space is also approximately spherical, that is, that the laws of disposition of rigid bodies in it are not given by Euclidean geometry, but approximately by spherical geometry, if only we consider parts of space which are sufficiently great. ”

“ The distinction between horizontal and vertical is not an illusion; and the man who thinks it can be disregarded is likely to come to an untimely end. Yet we cannot arrive at a comprehensive view of nature unless we combine horizontal and vertical dimensions into a three-dimensional space. ”

“ The reason of this is that we have not supposed it to move in the fourth dimension. If the straight line moves in its own direction, it describes only a straight line; if the area moves in its own plane, it describes only an area; in each case, motion in the dimensions in which the thing exists, gives us only a thing of the same dimensions; and, in order to get a thing of higher dimensions, we must have motion in a new dimension. But, as the idea of motion is only applicable in space of three dimensions, we must replace it by another which is applicable in our fourth dimension of time. ”

“ Our measurements of distance in space are found to be subject to certain laws—the laws of geometry. But it has now become impossible to regard the subject of space-geometry as complete in itself. Consider a triangle formed by three points (or events) in the four-dimensional world; if we happen to have drawn our instantaneous strata so that the three points lie in one stratum, then the triangle is a space-triangle and its properties fall within the scope of our classical geometry. ”

“ If any real existence confined to a superficies could be shown to exist, there would be an argument for one relative to which our three-dimensional existence is superficial. But, both on the one side and the other of the space we are familiar with, spaces either with less or more than three dimensions are merely arbitrary conceptions. ”

“ These three quantities are sometimes called potentials.
We now come to a point of far-reaching importance. The formula for teaches us not only the character of the mesh-system, but the nature of our two-dimensional space, which is independent of any mesh-system. If satisfies any one of the first three formulae, then the space is like a flat surface; if it satisfies the last formula, then the space is a surface curved like a sphere. Try how you will, you cannot draw a mesh-system on a flat (Euclidean) surface which agrees with the fourth formula. ”

“ He concedes that there may be thinking beings, living in our world, whose distribution board has four dimensions, and who do consequently think in hyperspace.
THE NEED OF AN ENLARGED SPACE-CONCEPT It is the contrariety in phenomena already referred to, that is forcing advanced minds to entertain the idea of higher space. Mathematical physicists have found that experimental contradictions disappear if, instead of referring phenomena to a set of three space axes and one time axis of reference, they be referred to a set of four interchangeable axes involving four homogeneous co-ordinates. ”

“ With the introduction, in Chapter X, of the peripheral type of force-field which appertains to levity as the usual central one does to gravity, we are compelled to revise our conception of space. For in a space of a kind we are accustomed to conceive, that is, the three-dimensional, Euclidean space, the existence of such a field with its characteristic of increasing in strength in the outward direction is a paradox, contrary to mathematical logic. ”

“ That Space, as our mathematicians have it, is spoken of as having three dimensions, which one may call Length, Breadth, and Thickness, and is always definable by reference to three planes, each at right angles to the others. But some philosophical people have been asking why three dimensions particularly—why not another direction at right angles to the other three?—and have even tried to construct a Four-Dimensional geometry. ”

“ Analogies have been unsparingly used in efforts to popularize the four-space conception and much of the violence which has been done to the notion is due to this vagary. The mathematical publicist, in trying to give a mental picture of the fourth dimension, examines the appearances of three dimensional beings as they might appear to a two dimensional being or duodim. He imagines a race of beings endowed with all the human faculties except that they live in a land of but two dimensions—length and breadth. ”

“ If enthusiasts would be content not to transport the so-called fourth dimensional space out of the sphere of hyperspace and cease trying to speculate upon the results of its interposal into three space conditions, which is in every way a constructual impossibility, there could not be any possible objection to its due consideration. ”

“ Here, forgetting that space of three dimensions is nothing real that can be experienced, but is a mathematical construction, and on the other hand finding it convenient for mathematical reasons to construct spaces of less or more than three dimensions, or of n dimensions, they end by treating these constructions as conceivable realities, and seriously discuss bi-dimensional beings or four-dimensional worlds. ”

“ As portions of our space are bounded by surfaces, plane or curved, so portions of -space are bounded by hyperspace (three-dimensional) .
In our space, a point moving in an unchanging direction generates a straight line.
This straight line (say of units in length) , moving perpendicular to its initial position through the distance a, generates a square.
This square, moving perpendicular to its initial position through the distance , generates a cube. ”

“ Modern geometry bases itself not on space but on multiply-infinite number-manifolds—amongst which the three-dimensional is simply the undistinguished special case—and within these groups investigates functional formations with reference to their structure ”

“ If we proceed beyond the domain of two-dimensional geometry we merge the ideas of non-Euclidean and hyper-space. The ordinary triply-extended space of our experience is purely Euclidean; and if we approach the conception of curvature in such a space it must be curvature in a fourth dimension, and here the mind refuses to follow, although by pure reasoning we can show what must take place in such a space. ”