Higher-dimensional space

“ It is possible to say a great deal about space of higher dimensions than our own, and to work out analytically many problems which suggest themselves. But can we conceive four-dimensional space in the same way in which we can conceive our own space? Can we think of a body in four dimensions as a unit having properties in the same way as we think of a body having a definite shape in the space with which we are familiar? ”

“ It is an axiom of science never to look outside three-space horizons for an understanding of phenomena when these can logically be accounted for within those horizons. Now because, on the Higher Space Hypothesis, each space is the container of all phenomena of its own order, the futility, for practical purposes, of going outside is at once apparent. The highly intelligent threadworm neither knows nor cares that the point of intersection of two lines in his diagram represents a point in a space to which he is a stranger. ”

“ 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. ”

“ This is only another step along the road which the human mind has always travelled: our conception of the cosmos grows more complete and more just at the same time that it recedes more and more beneath the surface of appearances.
Far from the Higher Space Hypothesis complicating thought, it simplifies by synthesis and co-ordination in a manner analogous to that by which plane geometry is simplified when solid geometry becomes a subject of study. By immersing the mind in the idea of many dimensions, we emancipate it from the idea of dimensionality. ”

“ While a man's space-world is limited by his thought, it is, on the other hand, as boundless as his thought. That the world evolves with our consciousness, is at once the philosophy of 'Creative Evolution' and of the higher space theory. Our present spatial milieu has settled down to a seemingly three dimensional finality because our thought-form has become so habitual as to give rise to certain geometric axioms. All we need in order to come to a fourth-dimensional consciousness, said Henri Poincare, 'the greatest of moderns,' is a new table of distribution ”

“ Space shapes can only be symbolical of four-dimensional shapes; and if we do not deal with space shapes directly, but only treat them by symbols on the plane—as in analytical geometry—we are trying to get a perception of higher space through symbols of symbols, and the task is hopeless. But a direct study of space leads us to the knowledge of higher space. And with the knowledge of higher space there come into our ken boundless possibilities. ”

“ There would be no progress, unless we were to suppose the threads as they pass to interweave themselves in more complex shapes.
Can a representation, such as the preceding, be applied to the case of the existence in space with which we have to do? Is it possible to suppose that the movements and changes of material objects are the intersections with a three-dimensional space of a four-dimensional existence? Can our consciousness be supposed to deal with a spatial profile of some higher actuality? ”

“ With the symmetry of solids, or symmetry with relation to an axial plane, no such simple movement as the foregoing suffices to produce or explain it, because symmetry about a plane implies four-dimensional movement. It is easy to see why this must be so. In order to achieve symmetry in any space—that is, in any given number of dimensions—there must be revolution in the next higher space: one more dimension is necessary. ”

“ 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. ”

“ 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. ”

“ We should be led into still greater difficulties if we wished to represent a three-dimensional non-Euclidean space possessing various degrees of non-Euclideanism from place to place. Reasoning by analogy, we should be tempted to say that just as an irregularly curved surface embedded in three-dimensional Euclidean space yielded a two-dimensional geometry of varying degrees of non-Euclideanism, so now all we should have to do would be to conceive of an irregularly curved three-dimensional surface embedded in a four-dimensional Euclidean space. ”

“ 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. ”

“ 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. ”

“ For is not the sphere, or ovoid, the unit form of nature; and is not the spiral vortex its characteristic motion, from that of the nebula in the sky to the electron in the atom? Thus, on the hypothesis that our space is traversing four-dimensional space, and that the forms of our space are cross-sections of four-dimensional forms, the unity and harmony of nature would be accounted for in a remarkably simple manner. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ Some of you may feel that you could never bring your minds to conceive a curvature of space, let alone of space-time; others may feel that, being familiar with the bending of a two-dimensional surface, there is no insuperable difficulty in imagining something similar for three or even four dimensions. ”

“ According to this, we shall have in the world, not one space, but many spaces,—quite analogous to the case that the three-dimensional space consists of an infinite number of planes. The three-dimensional geometry will be a chapter of four-dimensional physics. Now you perceive, why I said in the beginning that time and space shall reduce to mere shadows and we shall have a world complete in itself. ”

“ Now could it be shown that the two-dimensional symmetry observed in nature is the result of a three-dimensional movement, the right-and left-handed symmetry of solids would by analogy be the result of a four-dimensional movement. Such revolution (about a plane) would be easily achieved, natural and characteristic, in four space, just as the analogous movement (about a line) is easy, natural, and characteristic, in our space of three dimensions. ”

“ They would see shifting surfaces as we walked about, but being sensitive to length and breadth only, and not to height, they could gain no notion whatsoever of what we really look like. It is thus with us when we attempt to picture four-dimensional 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. ”

“ To a flatlander of one world, the other world would appear merely as a thin line – one dimension.
Similarly, he pointed out, one of the common three dimensions might be shifted into the fourth, leaving a vacancy in one of the visible three, whereby objects would be visible and tangible only as the two dimensional proportions left! It was his idea that this hemisphere was the workshop of the Wrongness of Space, that the lines and angles represented two-dimensional bases of walls, instruments, and manipulative devices, one of their dimensions having been shifted to the fourth. ”

“ 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. ”

“ Non-Euclidean geometry, however, showed other possibilities. The surface of a sphere has no boundary, yet it is not infinite. In [Pg 165] traveling round the earth, we never reach “the edge of the world,” and yet the earth is not infinite. The surface of the earth is contained in three-dimensional space, but there is no reason in logic why three-dimensional space should not be constructed on an analogous plan. What we imagine to be straight lines going on for ever will then be like great circles on a sphere: they will ultimately return to their starting point. ”

“ If we were perfectly flat, two-dimensional beings, living on the plane, and if not only all the flat bodies sliding over the plane behaved as did the shadows in our previous example, but also our measuring rods and our own living bodies behaved likewise, we should have no other alternative than to assume that these shadows represented the displacements of bodies that moved while remaining rigid or without change. We should accordingly regard the geometry of our two-dimensional space as Riemannian. ”

“ 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 ”

“ It is this discovery, entailing that of the four-dimensional world of space-time, that constitutes the most important contribution of the theory to our knowledge of the universe. What gives added significance to the four-dimensional world is the fact that those entities with which physicists deal are found to be vectors and tensors in its substance, and not in that of some other world. ”

“ 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. ”

“ Hence, if Higher Space exists and our world borders upon it and lies partially in it, it follows necessarily that we see only portions of all objects. We never see their true and complete shape. We see their three measurements, but not their fourth. The new direction is concealed from us, and when I hold this book and move my hand all round it I have not really made a complete circuit. We only perceive those portions of any object which exist in our three dimensions; the rest escapes us. But, once we learn to see in Higher Space, and objects will appear as they actually are. ”