The fourth dimension

CHAPTER X A FOUR-DIMENSIONAL FIGURE

THE method used in the preceding chapter to illustrate the problem of Kant’s critique, gives a singularly easy and direct mode of constructing a series of important figures in any number of dimensions.

We have seen that to represent our space a plane being must give up one of his axes, and similarly to represent the higher shapes we must give up one amongst our three axes.

But there is another kind of giving up which reduces the construction of higher shapes to a matter of the utmost simplicity.

Ordinarily we have on a straight line any number of positions. The wealth of space in position is illimitable, while there are only three dimensions.

I propose to give up this wealth of positions, and to consider the figures obtained by taking just as many positions as dimensions.

In this way I consider ditnensions and positions as two “kinds,” and applying the simple rule of selecting every one of one kind with every other of every other kind, get a series of figures which are noteworthy because they exactly fill space of any number of dimensions (as the hexagon fills a plane) by equal repetitions of

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