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

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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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The Fourth Di mention 1 1' Nl). M KXTAI. I UK AS OK TIIKKK 1)1 M KXSIOXKJ) SI'ACK. IX the ordinary geometry of space of three dimensions. a solid is defined as a portion of space separated from the remainder of space by a surface. This surface. if in parts(e. g. the surface of a prism) has its parts Innind by lines, which, in turn, arc limited by points, the vertices of the solid. Surfaces and lines may l e unlimited or unbounded, and yet finite in extent: as. for example, the surface of a sphere and the circumference of a circle. The point is considered to have no size or dimension ; the straight line one dimension, length, which makes direction | ossi-ble: the Hat surface two dimensions, length and width, and the solid three dimensions. length, width and thickness. If a point moves in a certain way its path is a straight line which has one dimension. iength. due to the motion of the dimensionless point. If a straight line moves in its own direction, its path will be the same line, except that a limited line will increase its length: but if it moves in a direction not its own. say in one perpendicular to its own. the path it passes through is a plane surface, having two dimensions, length, belonging to the generating line, and width, due to the motion of the line in the new direction. I f a flat surface moves in one of its own directions. no change is produced, except that a limited surface will increase its extent; but if a flat surface moves in a direction not its own. say one perpendicular to both its own dimensions, its path is a solid, having three dimensions, the length and the width of the generating surface, and thickness, due to the motion in the new direction. For example, a one-inch cube may l e produced by conceiving first a point to move one inch in any direction, second, the resulting line to move one inch in any direction at right angles to its own. and third, the resulting square to move one inch in the direction at right angles to both its own. Thus it is evident that by the motion of a point, a line, or a surface in a new direction, a new figure can be produced having one more dimension than the generating figure. Xow if a solid could move in a direction not its moil, say one at right angles to all its three dimensions, wc conclude by analogy that a new figure would be formed having four dimensions. This wc cannot picture because wc arc lxnind down in our mental images to the material furnished us by our senses. That is, our images of figures in space arc based wholly on our sense impressions of matter; hence we may lx conditioning space by assuming for it characteristics which belong to matter. A four dimensioned solid may only seem impossible, and not be so in reality. Suppose that triangles A and l (Fig 1) in the same plane have their respective parts equal and arranged in the same orders, i. c. angles 1. 2. 3 and sides a, b. c, are arranged around clockwise in both; that is, they arc the same side up. PAG K N INK.

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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