The fourth dimension

THE SIMPLEST FOUR-DIMENSIONAL SOLID 169

We get then altogether, as three-dimensional regions, ochre, brown, light purple, light green.

Finally, there is the region which corresponds to a mixture of all the colours; there is only one region such as this. It is the one that springs from ochre by the addition of blue—this colour we call light brown.

Looking at the light brown region we see that it increases in four ways. Hence, the tesseracts of which it is composed increase in number in each of four dimensions, and the shape they form does not remain thin in any of the four dimensions. Consequently this region becomes the solid content of the block of tesseracts, itself; it is the real four-dimensional solid. All the other regions are then boundaries of this light brown region. If we suppose the process of increasing the number of tesseracts and diminishing their size carried on indefinitely, then the light brown coloured tesseracts become the whole interior mass, the three-coloured tesseracts become threedimensional boundaries, thin in one dimension, and form the ochre, the brown, the light purple, the light green. The two-coloured tesseracts become two-dimensional boundaries, thin in two dimensions, e.g., the pink, the green, the purple, the orange, the light blue, the light yellow. The one-coloured tesseracts become bounding lines, thin in three dimensions, and the null points become bounding corners, thin in four dimensions. From these thin real boundaries we can pass in thought to the abstractions—points, lines, faces, solids—bounding the four-dimensional solid, which is this case is light brown coloured, and under this supposition the light brown coloured region is the only real one, is the only one which is not an abstraction.

It should be observed that, in taking a square as the representation of a cube on a plane, we only represent one face, or the section between two faces. The squares,