The fourth dimension

A FOUR-DIMENSIONAL FIGURE 135

These separate figures are the successive stages in which the whole four-dimensional figure in which they cohere can be apprehended.

The first figure and the last are tetrakaidecagons. These are two of the solid boundaries of the figure. The other solid boundaries can be traced easily. Some of them are complete from one face in the figure to the corresponding face in the next, as for instance the solid which extends from the hexagonal base of the first figure to the equal hexagonal base of the second figure. This kind of boundary is a hexagonal prism. The hexagonal prism also occurs in another sectional series, as for instance, in the square at the bottom of the first figure, the oblong at the base of the second and the square at the base of the third figure.

Other solid boundaries ean be traced through four of the five sectional figures. Thus taking the hexagon at the top of the first figure we find in the next a hexagon also, of which some alternate sides are elongated. The top of the third figure is also a hexagon with the other set of alternate rules elongated, and finally we come in the fourth figure to a regular hexagon.

These four sections are the sections of a tetrakaidecagon as can be recognised from the sections of this figure which we have had previously. Hence the boundaries are of two kinds, hexagonal prisms and tetrakaidecagons.

These four-dimensional figures exactly fill four-dimensional space by equal repetitions of themselves,