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.
In a plane there may be three points each equally distant from one another. These may be joined, forming an equilateral triangle in which there are three vertices or points, three lines or sides and one surface.