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Volume 34, 1901
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Art. L.—Some Observations on the Fourth Dimension.

[Read before the Hawke's Bay Philosophical Institute, 9th September 1901.]

Helmholtz was the earliest writer to attempt to present the conception of transcendental space in a form inviting popular investigation, and his efforts have been ably seconded in recent times by the author of “Flatland,”* in the first place, and by Mr. C. H. Hinton, in the second place. The former has produced a work which has attractions beyond the mere consideration of the fairyland of mathematics; while the latter, beginning with pamphlets of a distinctly popular, nature, has in his latest work laid down, still without- ab-struse mathematics, a scheme of mental training the avowed object of which is to enable the student to form a perfect mental image of a figure in four dimensions.

[Footnote] * “Flatland, A Romance in Two Dimensions, by a Square.” Seeley and Co.

[Footnote] † Author of “Scientific Romances” and “A New Era of Thought.” Swarm, Sonnenscheim, and Co.

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The method of treatment does not admit of much originality. A straigh line bounded by two points will, if moved in a direction perpendicular to itself, trace out a square, bounded by four lines and four points. By moving this square in an independent direction at right angles to the two original directions we shall obtain a cube, bounded by six squares, twelve lines, and eight points. If the cube be now moved in an in dependent direction compounded of none of the three original directions, but at right angles to them all, it will trace out a four - dimensional figure (called by Mr. Hinton a “tessaract.”) which will be bounded by eight cubes, twenty-four squares, thirty-two lines, and sixteen points.

A very small amount of consideration will show how these latter figures are arrived at. The bounding cubes consist of the cube in its original position, the cube in its final position, and the six cubes traced out by the motion of the six squares which bounded the cube. of the squares we had six in the initial and six in the final position, while each of the twelve lines of the cube traced a square, making twenty-four in all. So too with the lines: twelve in the initial and twelve in the final position, with eight traced by the eight points, bring up the total to thirty-two. We may tabulate these results as follows:—

[The section below cannot be correctly rendered as it contains complex formatting. See the image of the page for a more accurate rendering.]

Dimensious. Points. Lines. Squares Cubes.
Line 1 2
Square 2 4 4
Cube 3 8 12 6
Tessaract 4 16 32 24 8

We might, of course, carry on the enumeration for figures in five, six, or “n” dimensions.

Mr. Hinton remarks that, if we take two equal cubes and place them with their sides parallel and connect the corresponding corners by lines, we shall form the figure of a tessaract. But it seems to the present writer that this suggestion ignores the limitation of our three-dimensional space. It is just these limitations which prevent our placing the cubes in a satisfactory position. The suggestion contains the assumption that, as we may project a cube on to a plane, so we may project a tessaract on to a three-dimensional system. In point of fact, such a projection might be made by a being in four dimensions, but we three-dimensional beings must be content with projecting our tessaract upon a plane.

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Use has accustomed us to the fact that we may represent by projection the figure traced by the square ABCD when moved in a direction (parallel to AE) perpendicular to all the lines contained in itself. And in a projection there is nothing to hinder us from moving the cube AG, thus obtained, in a direction represented by AK, which shall be at right angles to AE, AB, and AD. In fact, we might, had we been so disposed, have moved our square AC, in that direction, and traced, instead of AG, the cube AM,

without doing any violence to our notions of the propriety of our dealings with the projection. By moving, then, either the cube AG in a direction AK perpendicular to each of its sides, or the cube AM in a direction AE perpendicular to each of its sides, we shall obtain a projection of the tessaract AQ which will be found to have all the defining elements which are contained in the table above. There are the eight cubes, AM, EQ, AR, BQ, KF, NG, AG, KQ: the twenty-four squares, AC, EG, KM, OQ; AL, EP, DM, HQ; AN, ER, BM, FQ; AH,

KB, BG, LQ; AO, BP, DR, CQ; AF, KP, DG, NQ: the thirty-two lines, AE, KO, DH, NR, BF, LP, CG, MQ, AB, KL, EF, OP, HG, RQ, DC, NM, AK, EO, DN, HR, BL, FP, CM, GQ, AD, KN, EH, OR, BC, LM, FG, PQ: and the sixteen points, A, B, C, D, E, F, G, H, K, L, M, N, O, P, Q, R.

As has been indicated above, any one of the eight cubes here enumerated

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rated might have been considered the generating-cube, which in turn gives us the option of starting from any one of the twenty-four squares or the thirty-two lines.

In two dimensions revolution takes place about a point, while figures are bounded by lines. In three-dimensional space revolution takes place about a line, and figures are bounded by surfaces. In four dimensions revolution takes place about a plane, and figures are bounded by solids.

To a two-dimensional being the figures ABC, DEF are essentially different, no amount of revolution about a point effecting coincidence. To us, however, it is obvious that one can be made to coincide with the other by performing half a revolution about a line. Similarly, to us the cubes figured below are essentially different, and no revolution

about a line can make them coincide. To effect this one must be taken into four-dimensional space and revolved about a plane. Another aspect of the same fact is that, as we appreciate the identity of the two triangles by concentrating our attention upon one face or the other of either triangle, so a four-dimensional being appreciates the identity of the two cubes by virtue of the fact that either side is equally accessible to him. As the faces of the cube are no greater hindrance to him than are the edges of a triangle or square to us, he can apprehend the cube with the angle (6) forward or with the angle (3) forward at will; the latter being the cube B, the former being the cube A, above.

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The realisation of the possibility of the existence of fourth-dimensional space leads naturally to two questions, which, of course, may suggest others: (1.) Seeing that we may conceive of our space system as being made up of innumerable two-dimensional systems, each possibly inhabited by beings quite without cognisance of the companion systems, may not four-dimensional space be compounded of innumerable three-dimensional systems, similar to our own, but lying completely outside our cognisance? (2.) Seeing that figures in a two-dimensional system might be regarded as sections of solids, might not our so-called solids be in reality sections in three dimensions of four-dimensional figures?

With regard to the first question, it does not appear that any reason can be adduced why it should not be answered in the affirmative. This leads to the curious consideration that, in spite of preconceived notions, two bodies may, apparently, occupy the same space. If one plane be superimposed upon another, a figure moved out of the latter an infinitely small distance passes into the other. Two beings might in this way be separated by the smallest possible distance, and yet for all practical purposes be at an infinite distance from one another. In the same way, if we conceive of a cube, say of 1 ft. side, moved one millionth part of an inch in the direction of the fourth axis, it will pass immediately out of our system, and presumably its place may be occupied by another cube similar to itself. The centres of gravity of the two would be separated by an infinitesimal distance, and yet each in its own space system might be a solid, the two cubes to all intents and purposes occupying the same space. In this connection it may be mentioned that it has been suggested that, as we may imagine a plane to be bent over so as to re-enter itself, with or without a twist in the process, so we may suppose it possible for our space system to have been similarly treated. This would make it possible to arrive at one's starting-point by travelling along an apparently straight line for a considerable distance. But though the notion of limited space thus introduced had attractions for so great a thinker as W. K. Clifford, it seems that such a process would involve an extension of our present three-dimensional limitations.

With regard to the second question, while solids may mathematically be such sections, the answer must, when we come to the case of animate beings, assuredly be a negative one, for it is scarcely conceivable that a section could contain the consciousness of the whole. If what we imagine to be independent figures proper to our own space system are but sections of four-dimensional figures, it would seem to be necessary that the innumerable sections of these solids are also playing their parts in an infinite number of two-dimen-

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sional space systems. The author of “Flatland” makes a sphere pass in and out of two-dimensional space, and thereby conveys the suggestion that higher space beings might similarly visit our space and similarly disappear. In fact, the idea has been seized upon as explaining many of the socalled phenomena of Spiritism. But writers on this point have not reckoned with the difficulty of insuring that the higher space being should always offer the same three-dimensional section on entering our space system. Even so simple a figure as a cube might appear in a two-dimensional universe as a point, a line, a triangle, quadrilateral, or five-, or six-sided figures. In fact, under each of the four last headings an infinite variety of forms might be offered. And the possible sections of a four-dimensional figure in space of three axes offers, of course, a far greater variety of forms. It is not conceivable that a being moving freely in space of four dimensions could present itself repeatedly to us in sections even suggesting identity of form.

There is one further objection which must be dealt with in reference to both the above questions. The assumption is usually made by writers on this subject, and has been tacitly accepted in this paper, that a figure might be removed from a plane and afterwards replaced in that plane; and, by analogy, that one of our solids might conceivably be lifted into four-dimensional space and afterwards replaced in our system. Now, either of these processes endues the body dealt with for the time being with an existence in a system higher by one dimension than that in which it was assumed to exist. Two-dimensional beings, if such there be, are by the nature of their limitations placed absolutely without the scope of our cognisance, and we, of course, without the scope of theirs. So too we, as long as we continue to be three-dimensional beings, are absolutely cut off from such four-dimensional beings as there may be. The method of treatment adopted by writers in endeavouring to place the conception of the fourth dimension within reach of their readers consists in developing figures from one space system to another. But it is a fallacy to suppose that the matter occupying a figure can be similarly dealt with. In fact, we have no knowledge or conception of fourth-dimensional matter, any more than of two-dimensional matter. It is this fallacy which vitiates the application of the fourth dimension to reported spiritist wonders, and the recognition of it restores confidence in the old theory that two bodies cannot occupy the same space. J. B. Stallo remarks* that “the analytical argument in favour of the existence or possibility of transcendental

[Footnote] * “The Concepts and Theories of Modern Physics,” p. 269

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space is another flagrant instance of the reification of concepts.” It would appear, however, that his strictures apply not to the arguments for the possibility of transcendental space, but to the arguments that we can have, under our present limitations, any practical acquaintance with such space.