The fourth dimension

THE SIMPLEST FOUR-DIMENSIONAL SOLID 161

but the thought of an abstract boundary, the face of a cube.

Let us now take our eight coloured cubes, which form a cube in space, and ask what additions we must make to them to represent the simplest collection of four-dimensional bodies—namely, a group of them of the same extent in every direction. In plane space we have four squares. In solid space we have eight cubes. So we should expect in four-dimensional space to have sixteen four-dimensional bodies—bodies which in four-dimensional space correspond to cubes in three-dimensional space, and these bodies we call tesseracts.

Given then the null, white, red, yellow cubes, and those which make up the block, we notice that we represent perfectly well the extension in three directions (fig. 98). From the null point of

White the null cube, travelling one inch, we

Ne come to the white cube; travelling

YK \ N one inch away we come to the yellow

\vetlow\ Ei ht cube ; travelling one inch up we come

ellowY to the red cube. Now, if there is

(Orange hidden) g fourth dimension, then travelling

OIE, from the same null point for one

inch in that direction, we must come to the body lying beyond the null region.

I say null region, not cube; for with the introduction of the fourth dimension each of our cubes must become something different from cubes. If they are to have existence in the fourth dimension, they must be “ filled up from” in this fourth dimension.

Now we will assume that as we get a transference from null to white going in one way, from null to yellow going in another, so going from null in the fourth direction we haye a transference from null to blue, using thus the 11