A Primer of Higher Space (The Fourth Dimension)

ested enough to ask, “Well, what do solids bound?” logic compels the answer, “Higher solids: four-dimensional forms (invisible to sight) related to the solids we know as these are related to their bounding planes, as planes to their bounding lines.”

Let us retrace our steps and go over the ground again. A point, moving a given distance in an unchanging direction, traces out a line. This line, moving in a direction at right angles to itself a distance equal to its length, traces out a square. This square, moving in a direction at right angles to its plane a distance equal to the length of one of its sides, traces out a cube. It is easy to picture these processes and the resultant geometrical figures of one, two, and three dimensions, because the line, the square, and the cube have their correlatives in the world of objects; but the imagination fails when the attempt is made to continue this order of form-building. Here again the rope disappears into the void. For the cube to develop in a direction at right angles to its every dimension, a new region of space would be required—a fourth dimension. Should you declare that there is not and cannot be such a region of space, I wave you farewell, as before. But if you hesitate, I cannot forbear to press my advantage. In such a higher space, the cube would trace out a hyper-cube, or tesseract, a four-dimensional figure related to the cube as the cube is related to the square. This figure, invisible to the eye, is known to the mind. The number of its points, lines, planes, and cubic boundaries, and their relation to one another, are as familiar to