The fourth dimension
132 THE FOURTH DIMENSION
Hence these two hexagons fit together, forming one hexagon, and the line Tw is only wanted when we consider a section of the whole figure, we thus obtain the solid represented in the lower part of fig. 74. Equal repetitions of this figure, called a tetrakaidecagon, will fill up three-dimensional space.
To make the corresponding four-dimensional figure we have to take five axes mutually at right angles with five points on each, A catalogue of the positions determined in five-dimensional space can be found thus.
Take a cube with five points on each of its axes, the fifth point is at a distance of four units of length from the first on any one of the axes. And since the fourth dimension also stretches to a distance of four we shall need to
represent the succes-
ia in ie in 1 sive sets of points at SL distances 0, 1, 2, 3, 4,
in the fourth dimenal | TT in rr) Tr) sions,five cubes. Now
all of these extend to
in ) im i im no distance at all in 2L the fifth dimension. To represent what
we in Tr) in ie lies in thefifth dimen
sion we shall have to im a a 1 draw, starting from OL 14 2H 3H 4H each of our cubes, five similar cubes to represent the four steps on in the fifth dimension, By this assemblage we get a catalogue of all the points shown in fig. 75, in which L, represents the fifth dimension. Now, as we saw before, there is nothing to prevent us from putting all the cubes representing the different stages in the fourth dimension in one figure, if we take
Wigs 75.