Four-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? ”

“ 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. ”

“ The Higher Space Hypothesis is of assistance here. It is only necessary to remember that from the fourth dimension the interior of a solid is as much exposed as the interior of a plane figure is exposed from the region of the third dimension. A four-dimensional being would experience no difficulty, under suitable conditions, in possessing itself of any part of the bodily mechanism of another. ”

“ 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. ”

“ And in the first place, a being in four dimensions would have to us exactly the appearance of a being in space. A being in a plane would only know solid objects as two-dimensional figures—the shapes namely in which they intersected his plane. So if there were four-dimensional objects, we should only know them as solids—the solids, namely, in which they intersect our space. Why, then, should not the four-dimensional beings be ourselves, and our successive states the passing of them through the three-dimensional space to which our consciousness is confined? ”

“ How is it possible to conceive that in a four-dimensional space any creatures should be confined to a three-dimensional existence?
In reply I would say that we know as a matter of fact that life is essentially a phenomenon of surface. The amplitude of the movements which we can make is much greater along the surface of the earth than it is up or down. ”

“ 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. ”

“ Obviously we must start with a four-dimensional Euclidean space in which a three-dimensional spherical surface is embedded. But as a four-dimensional Euclidean space transcends our immediate experience and is in the nature of a mathematical fiction, we should be led to the erroneous conclusion that three-dimensional non-Euclidean geometry must likewise be a purely conceptual construction having nothing in common with measurements which might be conducted with material rods. ”

“ 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? ”

“ I prefer, therefore, in proposing to consider Time as a fourth dimension of our existence, to speak of it as a fourth dimension rather than the fourth dimension. Since this fourth dimension cannot be introduced into space, as commonly understood, we require a new kind of space for its existence, which we may call time-space. There is then no difficulty in conceiving the analogues in this new kind of space, of the things in ordinary space which are known as lines, areas, and solids. A straight line, by moving in any direction not in its own length, generates an area ”

“ 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. ”

“ It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space.1 In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate t by an imaginary magnitude proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) ”

“ It matters little that he replies, "Your four-dimensional world is only an analytic explanation of my phenomena," for the fact remains a fact, that in the mathematician's four-dimensional space there is a space not derived in any sense of the term as a residue of experience, however powerful a distillation of sensations or perceptions be resorted to, for it is not contained at all in the fluid that experience furnishes. ”

“ Hence, if the central sphere rotates, it will not necessarily drive those around it so that their motions will be antagonistic to each other, but the velocities will not arrange themselves in a systematic manner.
In four-dimensional space the figure which forms the next term of the series hexagon, tetrakaidecagon, is a thirty-sided figure. It has for its faces ten solid tetrakaidecagons and twenty hexagonal prisms. Such figures will exactly fill four-dimensional space, five of them meeting at every point. ”

“ We must suppose a sense which we have not, a sense [Pg 158] of direction wanting in us, something which a being in a four-dimensional world has, and which we have not. It is a sense corresponding to a new space direction, a direction which extends positively and negatively from every point of our space, and which goes right away from any space direction we know of. The perpendicular to a plane is perpendicular, not only to two lines in it, but to every line, and so we must conceive this fourth dimension as running perpendicularly to each and every line we can draw in our space. ”

“ 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. ”

“ 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. ”

“ Discarding experience and reasoning wholly from analogy, we introduce some properties of the fourth dimension as follows.
Four-dimensional measure depends upon length, breadth, height and a fourth dimension all multiplied together. In the graphical representation of -space, points are referred to three mutually perpendicular planes formed by three lines mutually at right angles. In a similar way, to represent -space we must assume another axis at right angles to each of the other three. ”

“ 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. ”

“ If the angles of a celestial triangle do not together equal two right angles, there would be an evidence for the physical reality of a fourth dimension.
This conclusion deserves a word of explanation. If space is really four-dimensional, certain conclusions follow which must be brought clearly into evidence if we are to frame the questions definitely which we put to Nature. To account for our limitation let us assume a solid material sheet against which we move. This sheet must stretch alongside every object in every direction in which it visibly moves. ”

“ Such a being would be able to make but a part of himself visible to us, for a cube would be apprehended by a two-dimensional being as the square in which it stood. Thus a four-dimensional being would suddenly appear as a complete and finite body, and as suddenly disappear, leaving no trace of himself, in space, in the same way that anything lying on a flat surface, would, on being lifted, suddenly vanish out of the cognizance of beings, whose consciousness was confined to the plane. ”

“ 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. ”

“ For four-dimensional existence carries with it an occupation of space which is quite miraculous to us—as, for instance, the power of appearing in two places at the same time; while a number of ultra-microscopic atoms, by their electrostatic attractions and repulsions, may maintain definite relations of distance from 178 each other, and may altogether constitute a cloud of considerable size and complex organization—quite imperceptible as a rule, yet occupying a definite area and fully capable of affecting material things. ”

“ We are not empty space so there is no limit to the kind of curvature entering into our composition; in fact our unusual sort of curvature is what distinguishes us from empty space. We are, so to speak, ridges in the four-dimensional world where it is gathered into a pucker. The pure mathematician in his unflattering language would describe us as “singularities”. These two non-empty ridges are joined by empty space, which must be free from those kinds of curvature described by the ten principal coefficients. ”

“ Each section in space being a sphere.
Again, just as solid figures can be represented on paper by perspective, four-dimensional figures can be represented perspectively by solids. If there are two squares, one lying over the other, and the underneath one be pushed away, its sides remaining parallel with the one that was over it, then if each point of the one be joined to the corresponding point of the other, we have a fair representation on paper of a cube. ”

“ The absolute curvature at any point is measured by a single quantity called the radius of spherical curvature. But space-time is a four-dimensional manifold embedded in—well, as many dimensions as it can find new ways to twist about in. Actually a four-dimensional manifold is amazingly ingenious in discovering new kinds of contortion, and its invention is not exhausted until it has been provided with six extra dimensions, making ten dimensions in all. ”

“ In this simple case the two points correspond to the bounding surface of the cube. In the case of a two-dimensional figure an infinite plane represents the whole of space. The square is separated off by four straight lines, and it is impossible for an entry to be made into the interior of the square, except by passing through the straight lines. Now, in these cases, it is evident that the boundaries of the figure are of one dimension less than the figure itself. Points bound lines, lines bound plane figures, planes bound solid figures. Solids then must bound four dimensional figures. ”

“ If we turn the cube which is its base in our space about a line, if, e.g., in fig. 36 we turn the cube about the line AB, not only it but each of the parallel cubes moves about a line. The cube we see moves about the line AB, the cube beyond it about a line parallel to AB and so on. Hence the whole four-dimensional body moves about a plane, for the assemblage of these lines is our way of thinking about the plane which, starting from the line AB in our space, runs off in the unknown direction. ”