Part 4

MAN THE GEOMETER

When man essays the rôle of creator he cannot do otherwise than follow similar sequences: it is easy to discern dimensional progression in the products of man's ingenuity and skill. Consider, for example, the evolution of a building from its inception to its completion. It exists first of all in the mind of the architect, and there it is indubitably higher-spatial, for he can interpenetrate and examine every part, and he can consider it all at once, viewing it simultaneously from without and from within, just as one would be able to do in a space of four dimensions. He begins to give his idea physical embodiment by making with a pencil-point, lines on a plane (a piece of paper), the third dimension being represented by means of the other two. Next (if he is careful and wise) he makes a three-dimensional model. From the architect's drawings the engineer establishes his points, lays out his angles, and runs his lines upon the site itself. The mason follows, and with his footing courses makes ponderable and permanent the lines of the engineer. These lines become in due course walls--vertical planes. Floors and roofs--horizontal planes--follow, until some portion of three-dimensional space has been enclosed.

Substantially the same sequence holds, whatever the kind of building or the character of the construction--whether a steel-framed skyscraper or a wooden shanty. A line system, represented by columns and girders in the one case, and by studs and rafters in the other, becomes, by overlay or interposition, a system of planes, so assembled and correlated as to define a solid.

With nearly everything of man's creating--be it a bureau or a battleship--the process is as above described. First, a pattern to scale; next, an actual linear framework; then planes defining a solid. Consider almost any of the industries practiced throughout the ages: they may be conceived of thus in terms of dimensions; for example, those ancient ones of weaving and basket making. Lines (threads in the one case, rushes in the other) are wrought into planes to clothe a body or to contain a burden. Or think, if you choose, of the modern industry of book-making, wherein types are assembled, impressed upon sheets of paper, and these bound into volumes-- points, lines, planes, solids. The book in turn becomes the unit of another dimensional order, in the library whose serried shelves form lines, which, combined into planes, define the lateral limits of the room.

HIGHER--AND HIGHEST--SPACE

These are truisms. What have they to do, it may be asked, with the idea of higher spaces? They have everything to do with it, for in achieving the enclosure of any portion of solid space the limit of known dimensions has been reached without having come to any end. More dimensions--higher spaces--are required to account for higher things. All of the products of man's ingenuity are inanimate except as he himself animates them. They remain as they were made, machines, not organisms. They have no inherent life of their own, no power of growth and renewal. In this they differ from animate creation because the highest achievement of the creative faculty in man in a mechanical way lacks the life principle possessed by the plant. And as the most perfect machine is inferior in this respect to the humblest flower that grows, so is the highest product of the vegetable kingdom inferior to man himself, the maker of the machine; for he can reflect upon his own and the world's becoming, while the plant can only become.

What is the reason for these differences of power and function? According to the Higher Space Hypothesis they are due to varying potencies of movement in the secret causeways and corridors of space. The higher functions of consciousness--volition, emotion, intellection--may be in some way correlated with the higher powers of numbers, and with the corresponding higher developments of space. Thus would the difference between physics and metaphysics become a difference of degree and not of kind. Evolution is to be conceived of as a continuous pushing back of the boundary between representation and reality, or as a conquest of space. We may conceive of space as of an infinite number of dimensions, and of consciousness as a moving--or rather as an expanding--point, embracing this infinity, involving worlds, powers, knowledges, felicities, within itself in everlasting progression.

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.