Page:Popular Science Monthly Volume 83.djvu/394

Again, to repeat in words slightly different from the foregoing (Fig. 11) considering the $a$ units as $a$ points (an indefinite number), the square $ABCD$ is derived from the line $AB$, which for convenience suppose to be one foot in length, by letting $AB$ with its a points move through a distance of one foot in a direction perpendicular to itself, that is, perpendicular to the one dimension of $AB$, every point of $AB$ describes a line, and $ABCD$ contains therefore $a$ lines and $a^{2}$ points.

Fig. 11.

The cube $ABCD-G$ is derived from the square $ABCD$ which moves one foot in a direction perpendicular to its two dimensions, its $a$ lines and $a^{2}$ points describing $a$ squares and $a^{2}$ lines respectively. The cube $ABCD-G$ therefore contains $a$ squares, $a^{2}$ lines and $a^{3}$ points.

Similarly, the four-dimensional unit is derived from the cube, $ABCD-G$, by letting that cube move one foot in a direction perpendicular to each of its three dimensions, that is, in the direction of the fourth dimension; its $a$ squares, $a^{2}$ lines, and $a^{3}$ points describing respectively $a$ cubes, $a^{2}$ squares, $a^{3}$ lines. The hypercube, therefore, contains $a$ cubes, $a^{2}$ squares, $a^{3}$ lines and $a^{4}$ points.

 

Boundaries

Now, as to the boundaries of the units, $AB$ has two bounding points, $ABCD$ has four, two each from the initial and the final position of the moving line, $ABCD-G$ has eight,—four each from the initial and the final position of the moving square,—and the hypercube^[1] has sixteen,—eight each from the initial and the final position of the moving cube.

Bounding Lines.—Of bounding lines, $AB$ has one (or is itself one), $ABCD$ has 4, one each from the initial and the final position of the moving line and 2 generated by the 2 bounding points of that line; $ABCD-G$ has 12,—4 each from the initial and the final position of the moving square and 4 generated by the 4 bounding points of that square; and the hypercube has 32,—12 each from the initial and the final position of the moving cube and 8 generated by the 8 bounding points of that cube.

Bounding Squares.—Of bounding squares, $ABCD$ has one (itself); $ABCD-G$ has 6,—one each from the initial and the final position of

1. This four-dimension unit is often called the "tesseract."