Part 5

Draw the cube a{1}a{2}a{3}a{4}a{5}a{6}a{7}a{8} on the three axes XYZ.

Draw parallel to this the cube b{1}b{2}b{3}b{4}b{5}b{6}b{7}b{8} on the U axis.

Draw the cube a{1}a{2}a{3}a{4}b{1}b{2}b{3}b{4} on the three axes XYU. This is partly drawn already.

Draw parallel to this the cube a{7}a{8}a{5}a{6}b{7}b{8}b{5}b{6} on the three axes XYU.

This completes the figure.

There are four other cubes in the figure besides those described above:

The cube on the XZU axes a{1}a{4}b{1}b{4}a{7}a{8}b{7}b{8} and its opposite a{2}a{3}b{2}b{3}a{5}a{6}b{5}b{6}.

The cube on the YZU axes a{5}a{2}a{7}a{1}b{5}b{2}b{7}b{1} and its opposite a{6}a{3}a{8}a{4}b{6}b{3}b{8}b{4}.

The figure has: 16 corners, 32 edges, 24 bounding squares, 8 bounding cubes.

The heavy line a{1}b{6} might be called the principal diagonal and makes an angle of 60 degrees with each of the four axes. It is foreshortened in the sketch, but its real length is twice that of one edge of the cube. Every line except this is on the outside of the four-dimensional figure.]

A four-dimensional cube-like solid if transparent and looked at with one eye would appear something like this. But it is obviously impossible to depict a four-dimensional figure on a two-dimensional surface like this page.--From “The Fourth Dimension Simply Explained,” Munn and Company, N. Y.]

Since our two eyes present us slightly different pictures of an object we infer from these its size, shape and distance, but this is guesswork.

Still we have a pretty clear idea of a cube although we have never seen it in its solidity. But the attempt to visualize the hypercube, the four-dimensional figure corresponding to the cube, strains our imagination to the breaking point. Some mathematicians endowed with constructive imaginations of high power claim to have got by long hard thinking some sort of a shadowy and fleeting perception of it, but their visions--if they are not imaginary--do not help out us ordinary folks. But if we cannot imagine--that is, image--the hypercube we know all about it, even its name. It is called the “tesseract,” and it is bounded by eight cubes just as the cube is bounded by six squares and the square by four lines. The tesseract has 24 square faces, 32 edges and 16 rightangular corners.

TIME AS THE FOURTH DIMENSION

Although we find it hard to conceive of a fourth dimension in space we have no such difficulty in case the fourth dimension is time. In fact, we use this idea all the while and could not get along without it. To fix the position of any event requires four dimensions. For instance, a man is shot. Where? At the corner of 7th Avenue and 42d Street, New York. This fixes the place by two coördinates crossing at right angles in a plane. But was it above or below this, on the twentieth floor of the Times Building or in the Subway? Knowing this fixes the third dimension, but we have still to fix its position in a fourth dimension, time. Was it today or last week and what hour? If then we find out all four we can distinguish this shooting from any that may have occurred in other places at the same time or at other times in the same place.

Or consider this simple illustration: Cut a strip of motion picture film into its separate scenes and pile them up in order till it is as high as it is broad. You have then a cubical event. Two dimensions of the cube are spatial; the third dimension is essentially temporal, although in a spatial form. If one of the films from the middle of the pack represents the present then the films below represent the past and those above the future. The people on the picture you picked out know only of the scene there depicted though they may have a fading memory of the past and a dim anticipation of the future. But to you who are outside of the film pack all the scenes are equally visible. They are all present to you. This is the way most Christians have conceived of God, as one to whom past and future form one eternal present, so he sees simultaneously all things that have been, are or will be.

If our pile of film were made up of snapshots taken one a day throughout a man’s life we should see at one glance his growth from babyhood to boyhood, to maturity and old age. We could turn the leaves of his life backward or forward as we will. Some day perhaps we shall have stereo-movies, scenes in three dimensions with time as the fourth.

This idea of time as a fourth dimension is not a new one. In 1754 d’Alembert, defining “dimension” in the Encyclopedia, wrote: “A brilliant man of my acquaintance believes that one may regard duration as a fourth dimension.” In 1903 Minkowski worked out the idea in mathematical form. H. G. Wells, always quick to catch up a new scientific theory to use as a plot for a story, wrote in 1895 of “The Time Machine,” a vehicle by which a man could travel back and forth in time as he can travel east and west in a motor car. In this he visits the future and finds mankind split into two species, a subterranean working class living on--literally--a pleasure-loving leisure class.

In “The Plattner Case” Wells tells of a chemical professor who was by an explosion knocked into--not the middle of next week as we commonly say--but into the fourth dimension of space. Ten days later he was knocked back again into our world but the only evidence of the truth of his story was that his heart beat on the right side and he was left handed and otherwise reversed in a way that would be impossible in a space of three dimensions.

We can turn a glove inside out in three dimensions and so make it just like its mate of the other hand, but we cannot turn a solid inside out except in four-dimensional space.

In another of his “Thirty Strange Stories” Wells tells “The Story of Davidson’s Eyes.” While Davidson was working in his London laboratory a lightning shock so affected his eyesight that he could not see the familiar objects about him which he could feel but looked instead at a South Sea island on the opposite side of the globe. This might be possible in a curved space of four dimensions although Wells professes to pooh-pooh such an absurd suggestion while he ingeniously insinuates it. George Macdonald in his fantastic romance “Lilith” also introduces the fourth dimension.

Points that are far apart if measured in three dimensions may be close together in the fourth. We can readily understand this if time is the fourth dimension, for events can happen at the same instant though thousands of miles apart. But it is not impossible to conceive of the fourth dimension as spatial instead of temporal if we approach the problem from a simpler standpoint. Let us think of ourselves as living in a “Flatland” of two dimensions with no thought of a third. There yet survive in enlightened America individuals who believe that “the sun do move” and who deny that the earth is “round like a ball.” That is, they do not recognize the curvature of the earth in the third dimension. But if such an individual were to travel in a “straight” line westward over the “level” land and water he would, much to his surprise, come back to his starting point which he had left 25,000 miles behind him.

A WORM’S-EYE VIEW OF THE WORLD

Suppose yourself a worm--the Bible says you are anyway--and crawling around on a sheet of paper. With your vermicular mind you doubtless would take a superficial view of the universe and find it as impossible to imagine a third dimension as man does a fourth. If in the course of your crawling you came across a triangle you might--if you were a measuring worm--pace it off and find that the distance from A to B was 8 inches, from B to C was 6 inches and from this data, if you knew the law of the hypothenuse, you might calculate that the distance from A to C was 10 inches. On measuring it you would find your prediction verified and so gain perfect confidence in your plane geometry. But unbeknownst to you, poor worm with your eyes fixed on the paper, some man may have picked up the sheet and crumpled it up or rolled it over so that A and C are only one inch apart--in the third dimension. The worm is right when he thinks the distance between these points is 10 inches: so is the man right when he says it is one inch. It depends on the point of view.

Now in Einstein’s view something of this sort happens to our three-dimensional space when matter gets into it. We know for instance that if you divide the circumference of any circle by the diameter the ratio figures out as 3.1415+. It has been calculated to 707 decimal places but we can dispense with the rest of them and call the whole thing Pi for short. Write it in Greek as π and it looks more learned. Now if you place a heavy particle, say a lead bullet, in the center of a circle the ratio of the diameter to the circumference, according to Einstein, becomes a little less than Pi, for the circle has been warped, so to speak, into the fourth dimension by the strain of gravitation. The difference in such a case is too small to be measurable by any known means, but it is supposed to be an actual, not an imaginary, deviation from the geometrical law.

Now the sun being a big heavy body must extend its gravitational strain for a considerable distance around and a ray of light passing through this crumpled up space would not be able to pursue a straight course. And, according to the eclipse observations, it does not. Light like everything else follows “the easiest way” and this is not always the straight and narrow path. A river takes the easiest, not the shortest, way to the sea and this leads it through many meanderings. Most of us, I suppose, have a mental image of Newton’s gravitation as a sort of rope by which the sun pulls the earth into its orbit when it is disposed to fly off on a tangent. But from Einstein’s viewpoint we should rather think of the earth as picking its way as best it can through a space-and-time combination that has been strained and distorted by the power of the sun. I visualize Einstein’s solar system as a spider web with the sun in the middle like the spider and the planets like flies trying to get around through the tangled strands. But it is more complicated than that for each planet has its own lesser web of radiating influence to drag about with it wherever it goes.

Newton’s idea is simpler, but unfortunately light at least seems to follow Einstein’s law, not Newton’s. That is why Einstein is such a troublesome fellow. If he would confine himself to metaphysical speculation nobody need bother about these strange notions of his. But when he points how they can be proved and then British astronomers and American physicists find things according to his deductions he cannot be ignored. The man does not seem to have that decent respect for the opinions of mankind that leads most of us to limit our logic to the sphere of common sense. When he gets an idea in his head he follows it wherever it leads him even though he bumps up against Euclid and Newton and the rest of us. For instance, if you admit the second of his two fundamental postulates, that the speed of light is constant, regardless of the velocity of its source, you are irresistibly led--unless you let go of his hand somewhere on the way--to the conclusion that time is a local affair; that there is no way of telling by light signals whether two clocks at a distance are keeping the same time, or whether two events at different places occur simultaneously. You could not tell this even if you could shoot a watch from one place to the other with the speed of light, for no matter how many seconds--or years--the watch might be on its way it would register the same time. If instead of a watch a man could travel at that speed he would not grow old on the way. According to Einstein no man, watch or any other material thing can travel with the speed of light, for it would require an infinite force to give the smallest particle such a velocity. But let us suppose that a hollow projectile holding a man, such as Jules Verne and Wells used on their voyages to the moon, should be sent off into space with a velocity one twenty-thousandth less than light. If at the end of a year the projectile should be caught like a comet by the gravitation of some star and be swung around and sent back to the earth, the man on stepping out of his shell would be two years older but he would find the world two hundred years older. This would be, as Professor Langevin suggests in Scientia, 1911, an interesting way to study history, but it would be risky, not to say impossible. Still French scientists, like Napoleon, have no place in their dictionaries for so stupid a word as “impossible” and M. Esnault-Pelterie has figured out that a thousand pounds of radium would be sufficient to carry a man to Venus in 35 hours if a hollow projectile could be fitted up like a rocket with the radium in the rear sending out a rapid fire of electrons.

TURNING TIME BACKWARD