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The properties of four-dimensional space will become clearer to us if we compare in detail three-dimensional space with the surface, and discover the differences existing between them.

Hinton, in his book, A New Era of Thought, examines these differences very attentively. He represents to himself, on a plane, two equal rectangular triangles, cut out of paper, the right angles of which are placed in opposite directions. These triangles will be

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equal, but for some reason quite different. The right angle of one is directed to the right, that of the other to the left. If anyone wants to make them quite similar, it is possible to do so only with the help of three-dimensional space. That is, it is necessary to take one triangle, turn it over, and put it back on the plane. Then they will be two equal, and exactly similar triangles. But in order to effect this, it was necessary to take one triangle from the plane into three-dimensional space, and turn it over in that space. If the triangle is left on the plane, then it will never be possible to make it identical with the other, keeping the same relation of angles of the one to those of the other. If the triangle is merely rotated in the plane this similarity will never be established. In our world there are figures quite analogous to these two triangles.

We know certain shapes which are equal the one to the other, which are exactly similar, and yet which we cannot make fit into the same portion of space, either practically or by imagination.

If we look at our two hands we see this clearly, though the two hands represent a complex case of a symmetrical similarity. Now there is one way in which the right hand and the left hand may practically be brought into likeness. If we take the right hand glove and the left hand glove, they will not fit any more than the right hand will coincide with the left hand; but if we turn one glove inside out, then it will fit. Now suppose the same thing done with the solid hand as is done with the glove when it is turned inside out, we must suppose it, so to speak, pulled through itself. . . . If such an operation were possible, the right hand would be turned into an exact model of the left hand. 1

But such an operation would be possible in the higher dimensional space only, just as the overturning of the triangle is possible only in a space relatively higher than the plane. Even granting the existence of four-dimensional space, it is possible that the turning of the hand inside out and the pulling of it through itself is a practical impossibility on account of causes independent of geometrical conditions. But this does not diminish its value as an example. Things like the turning of the hand inside out are possible theoretically in four-dimensional space because in this space different, and even distant points of our space and time touch, or have the possibility of contact. All points of a sheet of paper lying on a table are separated


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one from another, but by taking the sheet from the table it is possible to fold it in such a way as to bring together any given points. If on one corner is written St. Petersburg, and on another Madras, nothing prevents the putting together of these corners. And if on the third corner is written the year 1812, and on the fourth 1912, these corners can touch each other too. If on one corner the year is written in red ink, and the ink has not yet dried, then the figures may imprint themselves on the other corner. And if afterwards the sheet is straightened out and laid on the table, it will be perfectly incomprehensible, to a man who has not followed the operation, how the figure from one corner could transfer itself to another corner. For such a man the. possibility of the contact of remote points of the sheet will be incomprehensible, and it will remain incomprehensible so long as he thinks of the sheet in two-dimensional space only. The moment he imagines the sheet in three-dimensional space this possibility will become real and obvious to him.

In considering the relation of the fourth dimension to the three known to us, we must conclude that our geometry is obviously insufficient for the investigation of higher space.

As before stated, a four-dimensional body is as incommensurable with a three-dimensional one as a year is incommensurable with St. Petersburg.

It is quite clear why this is so. The four-dimensional body consists of on infinitely great number of three-dimensional bodies; accordingly, there cannot be a common measure for them. The three-dimensional body, in comparison with the four-dimensional one is equivalent to the point in comparison with the line.

And just as the point is incommensurable with the line, so is the line incommensurable with the surface; as the surface is incommensurable with the solid body, so is the three-dimensional body incommensurable with the four-dimensional one.

It is clear also why the geometry of three dimensions is insufficient for the definition of the position of the region of the fourth dimension in relation to three-dimensional space.

Just as in the geometry of one dimension, that is, upon the line, it is impossible to define the position of the surface, the side of which constitutes the given line; just as in the geometry of two dimensions,

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i.e., upon the surface, it is impossible to define the position of the solid, the side of which constitutes the given surface, so in the geometry of three dimensions, in three-dimensional space, it is impossible to define a four-dimensional space. Briefly speaking, as planimetry is insufficient for the investigation of the problems of stereometry, so is stereometry insufficient for four-dimensional space.

As a conclusion from all of the above we may repeat that every point of our space is the section of a line in higher space, or as B. Riemann expressed it: the material atom is the entrance of the fourth dimension into three-dimensional space.

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For a nearer approach to the problem of higher dimensions and of higher space it is necessary first of all to understand the constitution and properties of the higher dimensional region in comparison with the region of three dimensions. Then only will appear the possibility of a more exact investigation of this region, and a classification of the laws governing it.

What is it that it is necessary to understand?

It seems to me that first of all it is necessary to understand that we are considering not two regions spatially different, and not two regions of which one (again spatially, "geometrically") constitutes a part of the other, but two methods of receptivity of one and the same unique world of a space which is unique.

Furthermore it is necessary to understand that all objects known to us exist not only in those categories in which they are perceived by us, but in an infinite number of others in which we do not and cannot sense them. And we must learn first to think things in other categories, and then so far as we are able, to imagine them therein. Only after doing this can we possibly develop the faculty to apprehend them in higher space--and to sense "higher" space itself.

Or perhaps the first necessity is the direct perception of everything in the outside world which does not fit into the frame of three dimensions, which exists independently of the categories of time and space--everything that for this reason we are accustomed to consider as non-existent. If variability is an indication of the three-dimensional world, then let us search for the constant and

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thereby approach to an understanding of the four-dimensional world.

We have become accustomed to count as really existing only that which is measurable in terms of length, breadth and height; but as has been shown it is necessary to expand the limits of the really existing. Mensurability is too rough an indication of existence, because mensurability itself is too conditioned a conception. We may say that for any approach to the exact investigation of the higher dimensional region the certainty obtained by the immediate sensation is probably indispensable; that much that is immeasurable exists just as really as, and even more really than, much that is measurable.


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Footnotes
55:1 C. H. Hinton, "A New Era of Thought," p. 44.



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http://www.sacred-texts.com/eso/to/to08.htm

[Crimson Glory]

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CHAPTER VI

Methods of investigation of the problem of higher dimensions. The analogy between imaginary worlds of different dimensions. The one-dimensional world on a line. "Space" and "time" of a one-dimensional being. The two-dimensional world on a plane. "Space" and "time," "ether," "matter" and "motion" of a two-dimensional being. Reality and illusion on a plane. The impossibility of seeing an "angle." An angle as motion. The incomprehensibility to a two-dimensional being of the functions of things in our world. Phenomena and noumena of a two-dimensional being. How could a plane being comprehend the third dimension?

A SERIES of analogies and comparisons are used for the definition of that which can be, and that which cannot be, in the region of the higher dimension.

We imagine "worlds" of one, and of two dimensions, and out of the relations of lower-dimensional worlds to higher ones we deduce possible relations of our world to one of four dimensions; just as out of the relations of points to lines, of lines to surfaces, and of surfaces to solids we deduce the relations of our solids to four-dimensional ones.

Let us try to investigate everything that this method of analogy can yield.

[Crimson Glory]

Let us imagine a world of one dimension.

It will be a line. Upon this line let us imagine living beings. Upon this line, which represents the universe for them, they will be able to move forward and backward only, and these beings will be as the points, or segments of a line. Nothing will exist for them outside their line—and they will not be aware of the line upon which they are living and moving. For there will exist only two points, ahead and behind, or may be just one point ahead. Noticing the change in states of these points, the one-dimensional being will call these changes phenomena. If we suppose the line upon which the one-dimensional being lives to be passing through the different objects of our world, then of all these objects the one-dimensional

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Moreover, if we assume that the one-dimensional being possesses memory, it is clear that recalling all the points seen by him as phenomena, he will refer them to time. The point which was: this is the phenomenon already non-existent, and the point which may appear tomorrow: this is the phenomenon which does not exist as yet. All of our space except one line will be in the category of time, i.e., something wherefrom phenomena come and into which they disappear. And the one-dimensional being will declare that the idea of time arises for him out of the observation of motion, that is to say, out of the appearance and disappearance of points. These will be considered as temporal phenomena, beginning at that moment when they become visible, and ending—ceasing to exist—at that moment when they become invisible. The one-dimensional being will not be in a position to imagine that the phenomenon goes on existing somewhere, though invisibly to him; or he will imagine it as existing somewhere on his line, far ahead of him.

We can imagine this one-dimensional being more vividly. Let us take an atom hovering in space, or simply a particle of dust, carried along by the air, and let us imagine that this atom or particle

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of dust possesses a consciousness, i.e., separates himself from the outside world, and is conscious only of that which lies in the line of his motion, and with which he himself comes in contact. He will then be a one-dimensional being in the full sense of the word. He can fly and move in all directions, but it will always seem to him that he is moving upon a single line; outside of this line will be for him only a great Nothingness—the whole universe will appear to him as one line. He will feel none of the turns and angles of his line, for to feel an angle it is necessary to be conscious of that which lies to right or left, above or below. In all other respects such a being will be absolutely identical with the before-described imaginary being living upon the imaginary line. Everything that he comes in contact with, that is, everything that he is conscious of, will seem to him to be emerging from time, i.e., from nothing, vanishing into time, i.e., into nothing. This nothing will be all our world. All our world except one line will be called time and will be counted as actually non-existent.

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Let us next consider the two-dimensional world, and the being living on a plane. The universe of this being will be one great plane. Let us imagine beings on this plane having the shape of points, lines, and flat geometrical figures. The objects and "solids" of that world will have the shape of flat geometrical figures too.

In what manner will a being living on such a plane universe cognize his world?

First of all we can affirm that he will not feel the plane upon which he lives. He will not do so because he will feel the objects, i.e., figures which are on this plane. He will feel the lines which limit them, and for this reason he will not feel his plane, for in that case he would not be in a position to discern the lines. The lines will differ from the plane in that they produce sensations; therefore they exist. The plane does not produce sensations; therefore it does not exist. Moving on the plane, the two-dimensional being, feeling no sensations, will declare that nothing now exists. After having encountered some figure, having sensed its lines, he will say that something appeared. But gradually, by a process of reasoning, the two-dimensional being will come to the conclusion that the figures

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he encounters exist on something, or in something. Thereupon he may name such a plane (he will not know, indeed, that it is a plane) the "ether." Accordingly he will declare that the "ether" fills all space, but differs in its qualities from "matter." By "matter" he will mean lines. Having come to this conclusion the two-dimensional being will regard all processes as happening in his "ether," i.e., in his space. He will not be in a position to imagine anything outside of this ether, that is, out of his plane. If anything, proceeding out of his plane, comes in contact with his consciousness, then he will either deny it, or regard it as something subjective, the creation of his own imagination; or else he will believe that it is proceeding right on the plane, in the ether, as are all other phenomena.

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Sensing lines only, the plane being will not sense them as we do. First of all, he will see no angle. It is extremely easy for us to verify this by experiment. If we will hold before our eyes two matches, inclined one to the other in a horizontal plane, then we shall see one line. To see the angle we shall have to look from above. The two-dimensional being cannot look from above and therefore cannot see the angle. But measuring the distance between the lines of different "solids" of his world, the two-dimensional being will come continually in contact with the angle, and he will regard it as a strange property of the line, which is sometimes manifest and sometimes is not. That is, he will refer the angle to time; he will regard it as a temporary, evanescent phenomenon, a change in the state of a "solid," or as motion. It is difficult for us to understand this. It is difficult to imagine how the angle can be regarded as motion. But it must be absolutely so, and cannot be otherwise. If we try to represent to ourselves how the plane being studies the square, then certainly we shall find that for the plane being the square will be a moving body. Let us imagine that the plane being is opposite one of the angles of the square. He does not see the angle—before him is a line, but a line possessing very curious properties. Approaching this line, the two-dimensional being observes that a strange thing is happening to the line. One point remains in the same position, and other points are withdrawing back from both sides. We repeat, that the two-dimensional being has no idea of an angle. Apparently the line remains the same as it was, yet something is happening to it, without a doubt. The plane being will say

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that the line is moving, but so rapidly as to be imperceptible to sight. If the plane being goes away from the angle and follows along a side of the square, then the side will become immobile. When he comes to the angle, he will notice the motion again. After going around the square several times, he will establish the fact of regular, periodical motions of the line. Quite probably in the mind of the plane being the square will assume the form of a body possessing the property of periodical motions, invisible to the eye, but producing definite physical effects (molecular motion)—or it will remain there as a perception of periodical moments of rest and motion in one complex line, and still more probably it will seem to be a rotating body.

Quite possibly the plane being will regard the angle as his own subjective perception, and will doubt whether any objective reality corresponds to this subjective perception. Nevertheless he will reflect that if there is action, yielding to measurement, so must there be the cause of it, consisting in the change of the state of the line, i.e., in motion.

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The lines visible to the plane being he may call matter, and the angles—motion. That is, he may call the broken line with an angle, moving matter. And truly to him such a line by reason of its properties will be quite analogous to matter in motion.

If a cube were to rest upon the plane upon which the plane being lives, then this cube will not exist for the two-dimensional being, but only the square face of the cube in contact with the plane will exist for him—as a line, with periodical motions. Correspondingly, all other solids lying outside of his plane., in contact with it, or passing through it, will not exist for the plane being. The planes of contact or cross-sections of these bodies will alone be sensed. But if these planes or sections move or change, then the two-dimensional being will think, indeed, that the cause of the change or motion is in the bodies themselves, i.e., right there on his plane.

As has been said, the two-dimensional being will regard the straight lines only as immobile matter; irregular lines and curves will seem to him as moving. So far as really moving lines are concerned, that is, lines limiting the cross-sections or planes of contact passing through or moving along the plane, these will be for

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the two-dimensional being something inconceivable and incommensurable. It will be as though there were in them the presence of something independent, depending upon itself only, animated. This effect will proceed from two causes: He can measure the immobile angles and curves, the properties of which the two-dimensional being calls motion, for the reason that they are immobile; moving figures, on the contrary, he cannot measure, because the changes in them will be out of his control. These changes will depend upon the properties of the whole body and its motion, and of that whole body the two-dimensional being will know only one side or section. Not perceiving the existence of this body, and contemplating the motion pertaining to the sides and sections he probably will regard them as living beings. He will affirm that there is something in them which differentiates them from other bodies: vital energy, or even soul. That something will be regarded as inconceivable, and really will be inconceivable to the two-dimensional being, because to him it is the result of an incomprehensible motion of inconceivable solids.

If we imagine an immobile circle upon the plane, then for the two-dimensional being it will appear as a moving line with some very strange and to him inconceivable motions.

The two-dimensional being will never see that motion. Perhaps he will call such motion molecular motion, i.e., the movement of minutest invisible particles of "matter."

Moreover, a circle rotating around an axis passing through its center, for the two-dimensional being will differ in some inconceivable way from the immobile circle. Both will appear to be moving, but moving differently.

For the two-dimensional being a circle or a square, rotating around its centre, on, account of its double motion will be an inexplicable and incommensurable phenomenon, like a phenomenon of life for a modern physicist.

[Crimson Glory]

Therefore, for a two-dimensional being, a straight line will be immobile matter; a broken or a curved line—matter in motion; and a moving line—living matter.

The centre of a circle or a square will be inaccessible to the plane being, just as the centre of a sphere or of a cube made of solid matter is inaccessible to us—and for the two-dimensional being even

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the idea of a centre will be incomprehensible, since he possesses no idea of a centre.

Having no idea of phenomena proceeding outside of the plane—that is, out of his "space"—the plane being will think of all phenomena as proceeding on his plane as has been stated. And all phenomena which he regards as proceeding on his plane, he will consider as being in causal interdependence one with another: that is, he will think that one phenomenon is the effect of another which has happened right there, and the cause of a third which will happen right on the same plane.

If a multi-colored cube passes through the plane, the plane being will perceive the entire cube and its motion as a change in color of lines lying in the plane. Thus, if a blue line replaces a red one, then the plane being will regard the red line as a past event. He will not be in a position to realize the idea that the red line is still existing somewhere. He will say that the line is single, but that it becomes blue as a consequence of certain causes of a physical character. If the cube moves backward so that the red line appears again after the blue one, then for the two-dimensional being this will constitute a new phenomenon. He will say that the line became red again.