University of Wisconsin Superior - Gitche Gumee Yearbook (Superior, WI)

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Such a world and such creatures, in order to have material existence must have, according to our ideas of matter, some small thickness in the third dimension. Figures cut out of paper, lying on a smooth surface and moving freely about on it would represent the geometrical conditions well. The intelligent inhabitants of this world would have no consciousness of this thickness, but would be aware only of two dimensions. Any plane figure, as a square, lying in this space, would appear to them as a solid” hounded by lines ; a boundary of lines would be capable of enclosing a portion of their space. Their own outsides would be lines; lines would suffice for house walls, roofs and floors being unnecessary. Their geometricians could have no difficulty in bringing into coincidence two equal triangles the same side up (Fig. 1), but the case of the two equal triangles not the same side up would present a difficulty like our trouble with symmetrical solids (Fig. 3). They might say that if one of such a pair of triangles could be rotated about a line. i. e. through a third dimension, it would come back into their space equal to the other. No idea of such a turning would be possible to them, but we. from our vantage ground of three dimensioned space can sec it clearly. Again, suppose this world of two dimensions instead of being fiat, has, unperceived by the inhabitants, a curvature in the third dimension, c. g. is a spherical surface, under which condition their space, though appearing infinite to them, would be without limits but finite in extent. On measuring the angles of large triangles they would find the angle sum to exceed two right angles, and they might explain the peculiarity bv assuming it to be due to a curvature in the third dimension. Since the excess in the angle sum bears a definite relation to the size of the sphere, and since the larger the sphere the larger the triangle must be to make the excess appreciable, a failure on their part to detect it might lx due merely to the immensity of their space. If a solid, say a cube, were passed slowly through this two dimensioned world, corner first, with the diagonal perpendicular to the two dimensions, it would seem to a two dimensioned observer, at first a triangle, increasing in size; then a changing hexagon, then a triangle again, which decreases and disappears. It would be difficult for the two dimensioned observer to account for the appearance, changes, and disappearance, of an apparent solid, unless he assumed that they were sections of a higher solid containing a third dimension. It would then be possible for him to build up an idea of the character of the cube from its sections in his space, but he would be powerless to imagine its true appearance. If the cube were passed through his space face first, it would seem to him a square which appeared suddenly, remained unchanged for a time, and disappeared as suddenly. THREE DIMENSIONED SPACE VIEWED FROM FOIR DIMENSIONED SPACE. Now let its suppose the existence of an intelligent being living in a world of four dimensions. From the analogy of the two dimensioned world it is plain that our three dimensioned space and our solids in order to be real in his eyes, must have at least a slight thickness in the fourth dimension. The insides of all our solids (including ourselves) arc open to his sight, since he looks into our space from a direction at right angles to all of its dimensions, and our idea that a portion PAGE ELEVEN

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These triangles can be made congruent by merely sliding and turning one of tnem to the position of the other. A point I can be found, about which one of them may be rotated to the position of the other. Therefore superposition can in this case be brought about by the rotation about a point. But if two triangles. C and D (Fig. 2). have their respective parts equal and arranged in opposite orders, i. e. sides a. b. c. and angles 1. 2. 3 of triangle C are arranged clockwise, while sides a. b. c. and angles 1. 2, 3 of triangle I) arc arranged counter-clockwise, that is if they are opposite sides up. they can. by rotating one of them about the proper jx»int be brought to the symmetrical position in Fig. 2. Then by folding over on the line M X as an axis, the triangles become coincident. But folding over brings into use a dimension not belonging to the plane or to the triangles. One of the triangles must Ik- taken out of the plane, in a direction not its own. turned over in the third dimension, and put back into the plane. Any two symmetrical plane figures can be brought into coincidence by rotation about a line. Now if we have two spherical triangles with equal parts in opposite orders and attempt to place them in coincidence by turning one of them over, we find the coincidence im]x ssible. In-cause they curve away from each other. Turning over in the third dimension is not sufficient in this case, because the curvature of the spherical surface involves the third dimension, and we have failed to turn the triangle over in a dimension or direction not its own. Again if we have two solids, say two pyramids A and B (Fig. 3), whose parts arc respectively equal and arranged in opposite orders, i. e. one is the mirror image of the other, we can conceive of no motion which will make them coincide. But if rotation about a point will bring into coincidence two equal figures iu the same plane which arc the same side up; if rotation about a line will bring into coincidence two equal plane figures which are opposite sides up, by turning one of them over in the third dimension; then, by analogy, rotation about a plane should bring into coincidence “equal solids, which have equal parts in opposite orders, by turning one of them over in a fourth dimension not its own. Such a turning wc cannot picture, but we can imagine its results. Animal forms, except the lower ones, exhibit this symmetry about a plane, the right half being the mirror image of the other. In the world of matter this seems to be the result of life only. A SPACE OF TWO DIMENSIONS. Imagine a world of two dimensions, having in it, or on it, intelligent beings. PAGE TEN

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of space can he enclosed by surfaces is an illusion due to our sense limitations. Any closed space, as a room, is “open on the side of the fourth dimension and an entrance or exit may lx made readily if that direction is taken. This higher being could see no difficulty in causing two symmetrical spherical triangles or two symmetrical solids to coincide, the possibility of rotation about a plane, or of turning over in the fourth dimension being as readily perceived by him as rotation alxnit a line is by us. or rotation about a point by a two dimensioned being. .Vow as the inhabitants of a two dimensioned world may justly argue a third dimension from symmetry in plane figures, so may we argue a fourth dimension from the existence of symmetry in solids: and as we can prcccivc the former directly. so would a four dimensioned being prcccivc the latter directly. And further, since in nature only living forms exhibit symmetry about a plane, we may argue a connection between life and a fourth dimension. Again, if our space is not “plane.” but has a curvature, say a positive constant one. then it may not be infinite in extent but finite though unbounded. This curvature might he discovered by measuring the angles of very large triangles, c. g. those whose vertices arc fixed stars, and noting a spherical excess. Indeed an assertion of such an excess has been made, but errors of measurement doubtless account for it. On the other hand, failure to detect an excess may be due to the immensity of our space and the comparatively small size of the triangles used. If this four dimensioned being should pass a solid of his space through our space, we could see the three dimensions which were in our space, but not the fourth. As it passed through our space from the fourth direction it would appear l crhaps like a changing solid, increasing or decreasing in size, coming into view from nowhere and disappearing as mysteriously. Or it might seem a solid appearing suddenly and retaining a permanent form for a time and then disappearing. Perhaps living forms answer this description. GKOMKTRY OF FOt’R DIMENSIONKl Sl'ACK. While we cannot gain a definite concept of space of four dimensions it is possible to work out by analytic methods a great many facts about it. Just as a point in a plane can be determined by two coordinates referred to two axes intersecting at right angles, and as any point in our space can lx? fixed by three coordinates referred to three axes, the third being at right angles to both the other two. so any point in four dimensioned space can be determined by four coordinates, referred to four axes, the fourth being at right angles to each of the other three. Further, an equation in two variables represents a line in a plane; and if the line is a closed curve it bounds a |x rtion of the plane, and thus serves, in two dimensioned space, to determine completely a two dimensioned “solid.” An equation in three variables represents a surface in three dimensioned space; and if the surface is a closed surface (e. g. the surface of a sphere or ellipsoid), it completely determines a three dimensioned solid. Then an equation in four variables represents a surface in four dimensioned space, which, if it is of proper curvature will determine completely a four dimensioned solid. In fact the investigation of space of four dimen- I'AGH TWELVE