The fourth dimension

160 THE FOURTH DIMENSION

Each of the six surrounding cubes carried on in the same motion will make a tesseract also, and these will be grouped around the tesseract formed by a. But will they enclose it completely ?

All the cubes an, Af, etc., lie in our space. But there is nothing between the cube a and that solid sheet in contact with which every particle of matter is. When the cube 4 moyes in the fourth direction it starts from its position, say Ak, and ends in a final position an (using the words “ana” and “ kata” for up and down in the fourth dimension). Now the movement in this fourth dimension is not bounded by any of the cubes an, af, nor by what they form when thus moved. The tesseract which A becomes is bounded in the positive and negative ways in this new direction by the first position of a and the last position of a. Or, if we ask how many tesseracts lie around the tesseract which a forms, there are eight, of which one meets it by the cube A, and another meets it by a cube like a at the end of its motion.

We come here to a very curious thing. The whole solid cube A is to be looked on merely as a boundary of the tesseract.

Yet this is exactly analogous to what the plane being would come to in his study of the solid world. The square A (fig. 96), which the plane being looks on as a solid existence in his plane world, is merely the boundary of the cube which he supposes generated by its motion.

The fact is that we have to recognise that, if there is another dimension of space, our present idea of a solid body, as one which has three dimensions only, does not correspond to anything real, but is the abstract idea of a three-dimensional boundary limiting a four-dimensional solid, which a four-dimensional being would form. The plane being’s thought of a square is not the thought of what we should call a possibly existing real square,

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