Four-Dimensional Vistas

by Claude Fayette Bragdon

III PHYSICAL PHENOMENA

LOOKING FOR THE GREATER IN THE LESS

After the assured way in which the author has conducted the reader repeatedly up and down the dimensional ladder, it may be a surprise to learn that physical phenomena offer no irrefragable evidences of hyper-dimensionality. We could not think in higher space if consciousness were limited to three dimensions. The mathematical reality of higher space is never in question: the higher dimensions are as valid as the lower, but the hyper-dimensionality of matter is still unproven. Man's ant-like efforts to establish this as a truth have thus far been vain.

Lest this statement discourage the reader at the very outset, he should understand the reason for such failure. We are embedded in our own space, and if that space be embedded in higher space, how are we going to discover it? If space is curved, how are we going to measure its curvature? Our efforts to do so may be compared to measuring the distance between the tips of a bent bow by measuring along the bow instead of along the string.

Imagine a scientifically-minded threadworm to inhabit a page of Euclid's solid geometry: the evidences of three-dimensionality are there, in the very diagrams underneath his eyes; but you could not show him a solid—the flat page could not contain it, any more than our space can contain a form of four dimensions. You could only say to him, "These lines represent a solid." He would have to depend on his faith for belief and not on that "knowledge gained by exact observation and correct thinking" in which alone the scientist finds a sure ground for understanding.

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. The point is there, on his page: it is what he calls a fact. "Why raise" (he says) "these puzzling and merely academic questions? Why attempt to turn the universe completely upside down?"

But though no proofs of hyper-dimensionality have been found in nature, there are equally no contradictions of it, and by using a method not inductive, but deductive, the Higher Space Hypothesis is plausibly confirmed. Nature affords a sufficient number of representations of four-dimensional forms and movements to justify their consideration.

SYMMETRY

Let us first flash the light of our hypothesis upon an all but universal characteristic of living forms, yet one of the most inexplicable—symmetry.

Animal life exhibits the phenomenon of the right-and left-handed symmetry of solids. This is exemplified in the human body, wherein the parts are symmetrical with relation to the axial plane. Another more elementary type of symmetry is characteristic of the vegetable kingdom. A leaf in its general contour is symmetrical: here the symmetry is about a line—the midrib. This type of symmetry is readily comprehensible, for it involves simply a revolution through 180 degrees. Write a word on a piece of paper and quickly fold it along the line of writing so that the wet ink repeats the pattern, and you have achieved the kind of symmetry represented in a leaf.

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. To make the (two-dimensional) ink figure symmetrical, it had to be folded over in the third dimension. The revolution took place about the figure's line of symmetry, and in a higher dimension. In three-dimensional symmetry (the symmetry of solids) revolution must occur about the figure's plane of symmetry, and in a higher—i.e., the fourth dimension. Such a movement we can reason about with mathematical definiteness: we see the result in the right- and left-handed symmetry of solids, but we cannot picture the movement ourselves because it involves a space of which our senses fail to give any account.

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.

OTHER ALLIED PHENOMENA

In the mirror image of a solid we have a representation of what would result from a four-dimensional revolution, the surface of the mirror being the plane about which the movement takes place. If such a change of position were effected in the constituent parts of a body as a mirror image of it represents, the body would have undergone a revolution in the fourth dimension. Now two varieties of tartaric acid crystallize in forms bearing the relation to one another of object to mirror image. It would seem more reasonable to explain the existence of these two identical, but reversed, varieties of crystal, by assuming the revolution of a single variety in the fourth dimension, than by any other method.

There are two forms of sugar found in honey, dextrose and levulose. They are similar in chemical constitution, but the one is the reverse of the other when examined by polarized light—that is, they rotate the plane of polarization of a ray of light in opposite ways. If their atoms are conceived to have the power of motion in the fourth dimension, it would be easy to understand why they differ. Certain snails present the same characteristics as these two forms of sugar. Some are coiled to the right and others to the left; and it is remarkable that, like dextrose and levulose, their juices are optically the reverse of each other when studied by polarized light.

Revolution in the fourth dimension would also explain the change in a body from producing a right-handed, to producing a left-handed, polarization of light.

ISOMERISM

In chemistry the molecules of a compound are assumed to consist of the atoms of the elements contained in the compound. These atoms are supposed to be at certain distances from one another. It sometimes happens that two compound substances differ in their chemical or physical properties, or both, even though they have like chemical elements in the same proportion. This phenomenon is called isomerism, and the generally accepted explanation is that the atoms in isomeric molecules are differently arranged, or grouped, in space. It is difficult to imagine how atoms, alike in number, nature, and relative proportion, can be so grouped as somehow to produce compounds with different properties, particularly as in three-dimensional space four is the greatest number of points whose mutual distances, six in number, are all independent of each other. In four-dimensional space, however, the ten equal distances between any two of five points are geometrically independent, thus greatly augmenting the number and variety of possible arrangements of atoms.

This just escapes being the kind of proof demanded by science. If the independence of all the possible distances between the atoms of a molecule is absolutely required by theoretical chemical research, then science is really compelled, in dealing with molecules of more than four atoms, to make use of the idea of a space of more than three dimensions.

THE ORBITAL MOTION OF SPHERES: CELL SUB-DIVISION

There is in nature another representation of hyper-dimensionality which, though difficult to demonstrate, is too interesting and significant to be omitted here.

Imagine a helix, intersected, in its vertical dimension, by a moving plane. If necessary to assist the mind, suspend a spiral spring above a pail of water, then raise the pail until the coils, one after another, become immersed. The spring would represent the helix, and the surface of the water the moving plane. Concentrating attention upon this surface, you would see a point—the elliptical cross-section of the wire where it intersected the plane—moving round and round in a circle. Next conceive of the wire itself as a lesser helix of many convolutions, and repeat the experiment. The point of intersection would then continually return upon its own track in a series of minute loops forming those lesser loops, which, moving circle-wise, registered the involvement of the helix in the plane.

It is easy to go on imagining complicated structures of the nature of the spiral, and to suppose also that these structures are distinguishable from each other at every section. If we think of the intersection of these with the rising surface, as the atoms, or physical units, of a plane universe, we shall have a world of apparent motion, with bodies moving harmoniously amongst one another, each a cross-section of some part of an unchanging and unmoving three-dimensional entity.

Now augment the whole by an additional dimension—raise everything one space. The helix of many helices would become four-dimensional, and superficial space would change to solid space: each tiny circle of intersection would become a sphere of the same diameter, describing, instead of loops, helices. Here we would be among familiar forms, describing familiar motions: the forms, for example, of the earth and the moon and of their motion about the sun; of the atom, as we imagine it, the molecule and the cell. 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.

The above exercise of the imagination is a good preparation for the next demand upon it. Conceive a dichotomous tree—one that always divides into two branches—to pass through a plane. We should have, as a plane section, a circle of changing size, which would elongate and divide into two circles, each of which would do the same. This reminds us of the segmentation of cell life observed under the microscope, as though a four-dimensional figure were registering its passage through our space.

THE ELECTRIC CURRENT

Hinton conceived of an electric current as a four-dimensional vortex. He declared that on the Higher Space Hypothesis the revolution of the ether would yield the phenomenon of the electric current. The reader is referred to Hinton's book, The Fourth Dimension, for an extended development of this idea. What follows is a brief summary of his argument. First, he examines the characteristics of a vortex in a three-dimensional fluid. Then he conceives of what such a vortex would be in a four-dimensional medium of analogous properties. The whirl would be about a plane, and the contour of this plane would correspond to the ends of the axis line in the former vortex; and as before, the vortex would extend to the boundary. Every electric current forms a closed circuit: this is equivalent to the hyper-vortex having its ends in the boundary of the hyper-fluid. The vortex with a surface as its axis, therefore, affords a geometric image of a closed circuit.

Hinton supposes a conductor to be a body which has the property of serving as a terminal abutment to such a hyper-vortex as has been described. The conception that he forms of a closed current, therefore, is of a vortex sheet having its edge along the circuit of the conducting wire. The whole wire would then be like the centers on which a spindle turns in three-dimensional space, and any interruption of the continuity of the wire would produce a tension in place of a continuous revolution. The phenomena of electricity—polarity, induction, and the like—are of the nature of the stress and strain of a medium, but one possessing properties unlike those of ordinary matter. The phenomena can be explained in terms of higher space. If Hinton's hypothesis be the true explanation, the universality of electro-magnetic action would again point to the conclusion that our three-dimensional world is superficial—the surface, that is, of a four-dimensional universe.

THE GREATER UNIVERSE

This practically exhausts the list of accepted and accredited indications of hyper-dimensionality in our physical environment. But if the collective human consciousness is moving into the fourth dimension, such indications are bound to multiply out of all measure. It should be remembered that in Franklin's day electricity was manifest only in the friction of surfaces and in the thunderbolt. To-day all physical phenomena, in their last analysis, are considered to be electrical. The world is not different, but perception has evolved, and is evolving.

There is another field, in which some of our ablest minds are searching for evidences of the curvature of space, the field of astronomy and astro-physics. But into this the layman hesitates to enter because the experts themselves have found no common ground of understanding. The ether of space is a battlefield strewn with dead and dying hypotheses; gravitation, like multiplication, is vexation; the very nature of time, form and movement is under vivid discussion, in connection with what is known as the Theory of Relativity.

Notwithstanding these counter-currents of speculation, which should make the wise man speak smilingly of his wisdom, this summary remains incomplete without a reference to the pressure of higher space upon those adventurous minds that essay to deal with the profound problems of the greater universe, and a statement of the reasons for their feeling this pressure. These reasons are well suggested by Professor B.G. Harrison, in his Popular Astronomy. He says: "With the idea of a universe of finite dimensions there is the obvious difficulty of the beyond. The truth is that a universe of finite proportions is equally difficult to realize as one of infinite extent. Perhaps the nearest analogy to infinity that we can understand lies in our conception of a closed curve. It seems easier to imagine the endless movement of a sphere in a circular path than the case of one travelling in a straight line. Possibly this analogy may apply in some way to fourth-dimensional space, but the manner of its application is certainly not easy to understand. If we would imagine that all co-ordinates of time and space were curved, and eventually return to the same point, it might bring the ultimate comprehension one degree nearer."

A HINT FROM ASTRONOMY

The physical evidence that our space is thus curved in higher space, some have considered astronomy to furnish in what is called the "negative parallax" of certain distant stars. This cannot be passed by, though it is too deeply involved with the probable error of the observers themselves to be considered more than an interesting fact in this connection. Every one knows that the difference of angle under which an object is seen from two standpoints is called its parallax. The parallax of the stars—and the consequent knowledge of their distance—is obtained by observing them from opposite points of the earth's orbit around the sun. When a star is within measurable distance, these angles are acute, and the lines from the star to the earth at opposite sides of its orbit converge, therefore. But when these lines, as sometimes happens, appear to be divergent, the result is called a negative parallax, and is explainable by higher space relationships. Obviously, the divergence of the lines would indicate that the object lies behind the observer instead of in front of him. This anomaly can be explained by the curvature of space in the fourth dimension. If space is so curved, the path of light itself is curved also, and a man—were his vision immeasurably keen, not to say telescopic—could see the back of his own head! It is not worth while to give this question of negative parallax too much importance, by reason of the probability of error, but in this connection it should be stated that there appears to be an undue number of negative parallaxes recorded.

GRAVITATION

Gravitation remains a puzzle to science. The tendency of modern physics is to explain all material phenomena in terms of electrons and the ether, but the attempt to account for gravitation in this way is attended with difficulties. In order to cope with these, it seems necessary to assume that our universe is only a portion of a greater universe. This assumption readily lends itself to the conception of our universe as a three-dimensional meeting place of two portions of a universe of four dimensions—that is, its conception as a "higher" surface. This is a fundamental postulate of higher space speculation.

One hypothesis advanced to explain gravitation assumes the existence of a constant hydrostatic pressure transmitted through the ether. A steady flow of ether into every electron in a gravitating system of bodies would give rise to forces of attraction between them, varying inversely as the square of the distance, according to Newton's law. But in order to avoid the conception of the continual destruction and creation of ether, it is necessary to assume a steady flow through every electron between our universe and the greater universe of which it is assumed to form a part Now because the electrons, in order to receive this flow, must lie on the boundary of this greater universe, the latter must be four-dimensional. Every electron, in other words, must be the starting point of a pathway into—and a terminal point out of—four-dimensional space. Here we have another familiar higher space concept.

THE ETHER OF SPACE

The ether of space, because it has at last found entrance, must be given a grudging hospitality in these pages, even though the mysterious stranger prove but a ghost. The Relativists would have it that with the acceptance of their point of view the ether may be eliminated; but if they take away the ether, they must give us something in its stead. In whatever way the science of the future disposes of this problem, it must take into account the fact of light transmission. On the theory that the ether is an elastic solid of amazing properties, in which the light waves vibrate transversely to their direction, it assists the mind to think of the ether as four-dimensional, because then a light wave would be a superficial disturbance of the medium—superficial, but three-dimensional, as must needs be the case with the surface of a four-dimensional solid.

* * * * *

This search for evidences of hyper-dimensionality in the universe accessible to our senses is like looking, not for a needle in a haystack, but for a haystack in a needle—for the greater in the less. From the purely physical evidences, all that can with certainty be said is that the hypothesis is not inconsistent with the facts of science or its laws; that it is being verified and rendered more probable by the investigations of science; that it is applicable to the description or explanation of all the observed phenomena, and assigns a cause fully adequate to have produced them.

Now there is an order of phenomena that we call psychic. Because they are phenomenal they cannot occur outside of time and space altogether; because they are psychic they defy explanation in terms of the space and time of every-day life. Let us next examine these in the light of our hypothesis.