The fourth dimension

REMARKS ON THE FIGURES 183

that is represented in the sections 0,, by, bs: in fig. 107 the red and white axes are in our space, the blue out of it; in the other case, the red and blue are in our space, the white out of it. It is evident that the face pink y, opposite the pink face in fig. 107, makes a cube shown in squares in 6,, by, b3, bs, on the opposite side to the J purple squares. Also the light yellow face at the base of the cube }, makes a light green cube, shown as a series of base squares.

The same light green cube can be found in fig. 107, The base square in wh) is a green square, for it is enclosed by blue and yellow axes. From it goes a cube in the white direction, this is then a light green cube and the same as the one just mentioned as existing in the sections Bos bs, bs, Bay by.

The case is, however, a little different with the brown cube. This cube we have altogether in space in the section wh,, fig. 108, while it exists as a series of squares, the left-hand ones, in the sections bp, by, ba, 63, 0, The brown cube exists as a solid in our space, as shown in fig. 108. Inthe mode of representation of the tesseract exhibited in fig. 107, the same brown cube appears as a succession of squares. That is, as the tesseract moves across space, the brown cube would actually be to us a square—it would be merely the lasting boundary of another solid, It would have no thickness at all, only extension in two dimensions, and its duration would show its solidity in three dimensions.

It is obyious that, if there is a four-dimensional space, matter in three dimensions only is a mere abstraction ; all material objects must then have a slight four-dimensional thickness. In this case the above statement will undergo modification. The material cube which is used as the model of the boundary of a tesseract will have a slight thickness in the fourth dimension, and when the cube is

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