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 dimensions 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 themselves.
The rule will be made more evident by a simple application.
Let us consider one dimension and one position. I will call the axis i, and the position o.
———————————————-i
o
Here the figure is the position o on the line i. Take now two dimensions and two positions on each.
Fig. 63.