A +------------------------------+ B | | | | | | | | | | | | C +------------------------------+ D
It will be noted that in figures 8, 9 and 10, the element of perpendicularity enters as a necessary determination. In figure 8, the lines _ab_ and _bd_ are perpendicular to each other. Similarly, in Fig. 10, lines _ab_, _bc_, _bb'_ and _h'b_ are perpendicular to one another. That is, at their intersections, they make right angles. Similarly, figures representing any number of dimensions may be constructed.
The line _ab_ represents a one-space. An entity living in a one space is called a "unodim." The plane, _abcd_, represents a two-space, and entities living in such a space are called _duodims_. The cube, _abcdefgh_, represents a three-space and entities inhabiting such a space are called _tridims_. Figure 10 represents a four-space, and its inhabitants are called _quartodims_. Each of the above-mentioned spaces is said to have certain limitations peculiar to itself.
The fourth dimension is said to lie in a direction at right angles to each of our three-space directions. This, of course, gives rise to the possibility of generating a new kind of volume, the hypervolume. The hypercube or tesseract is described by moving the generating cube in the direction in which the fourth dimension extends. For instance, if the cube, Fig. 9, were moved in a direction at right angles to each of its sides a distance equal to one of its sides, a figure of four dimensions, the tesseract, would result.
The initial cube, _abcc'e'fhh'_, when moved in a direction at right angles to each of its faces, generates the hypercube, Fig. 10. The lines, _aa'_, _bb'_, _cc'_, _dd'_, _ee'_, _ff'_, _gg'_, _hh'_, are assumed to be perpendicular to the lines meeting at the points, _a_, _b_, _c_, _d_, _e_, _f_, _g_, _h_. Hence _a'b'_, _b'd_, _dd'_, _d'a'_, _ef_, _fg_, _gg'_, _g'e_, represent the final cube resulting from the hyperspace movement. Counting the number of cubes that compose the hypercube we find that there are eight. The generating cube, _abcc'e'f'hh'_, and the final cube, _a'b'_, _b'd_, _dd'_, _d'a'_, _ef_, _fg_, _gg'_, _g'e_, make two cubes; and each face generates a cube making eight in all. A tesseract, therefore, is a figure bounded by eight cubes.