*THE FOURTH DIMENSION*

, and 4 described by the bounding lines of the moving square; and the hypercube has 24,—6 each from the initial and the final position of the moving cube, and 12 described by the bounding lines of the moving cube.

*Bounding Cubes.*—Finally, of bounding *cubes*, has one (itself); and the hypercube has 8,—one each from the initial and the final position of the moving cube, and 6 described by the bounding squares of the moving cube.

The results obtained for the boundaries may be conveniently exhibited by the following table:

Boundaries

Points | Lines | Squares | Cubes | |

One-dimensional unit | 2 | 1 | 0 | 0 |

Two-dimensional unit | 4 | 4 | 1 | 0 |

Three-dimensional unit | 8 | 12 | 6 | 1 |

Four-dimensional unit | 16 | 32 | 24 | 8 |

Freedom of movement is greater in hyperspace than in our space. The degrees of freedom of a rigid body in our space are 6, namely, 3 translations along and 3 rotations about 3 axes, while the fixing of 3 of its points, not in a straight line, prevents all movement. In hyperspace, however, with 3 of its points fixed, it could still rotate about the plane of those 3 points. A rigid body has 10 possible different movements in hyperspace, namely, 4 translations along 4 axes, and 6 rotations about 6 planes, while at least 4 of its points must be fixed to prevent all movement.

In hyperspace, a sphere of flexible material could without stretching or tearing be turned inside out. Two links of a chain could be separated without breaking them. Our knots would be useless. In hyperspace, as we have seen, it would be entirely possible to pass in and out of a sphere

(or other enclosed space). A right glove turned over through space of four dimensions becomes a left glove, but notice that when the glove is *turned over,* it is *not turned inside out.*^{[1]} This may be made clear by analogy. Suppose we have in a plane (Fig. 12) a nearly closed polygon

- ↑ A right glove turned inside out in
*our*space becomes a left glove.