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. [Illustration: FIG. 9.] [Illustration: FIG. 10.--The Tesseract.] 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. To find the different elements of a tesseract, the following rules will apply: 1. _To find the number of lines_: Multiply the number of lines in the generating cube by two, and add a line for each point or corner in it. E.g., 2 × 12 = 24 + 8 = 32. 2. _To find the number of planes, faces or squares_: Multiply the number of planes in the generating cube by 2 and add a plane for each line in it. E.g., 2 × 6 + 12 = 24. 3. _To find the number of cubes in a hypercube_: Multiply the number of cubes in the generating cube, one, by two and add a cube for each plane in it. E.g., 2 × 1 + 6 = 8. 4. _To find the number of points or corners_: Multiply the number of corners in the generating cube by 2. E.g., 2 × 8 = 16.