We should say:—
Every side of the original square has traced out a new square—that makes 4—and we also have the original square and the "final" square making a total of 6. A cube, therefore, must be bounded by 6 square surfaces.
Similarly we should reflect that the original square and the final square have each 4 linear edges, making 8, and that each of the 4 corner points of the original square would trace out a line, making new lines, and we would therefore conclude that a cube must have 8 + 4 = 12 edges.
Finally, since in a uniform motion no new points will be generated, we should expect the cube to have a total of 8 corner points, i.e., the four corners of the original square and the four corners of the final square.
Now let us apply the same methods to the generation of the tesseract by the movement of a cube.
Observe that just as in the case of the square generating the cube we had the original square to start with and what I called the "final" square to end up with, so, in this case, we shall start and end up with a cube.
In the process of the movement every face of the cube will generate a new cube—that means 6 new cubes, since the cube must have had 6 faces—and there will also be the original cube and the final cube, making a total of 8 cubes all told. A tesseract must therefore be bounded by 8 cubes.
Similarly each line of the original cube will trace out a square. This, since a cube has 12 edges, gives us 12 new squares plus 6 from the original and 6 from the final cube, or a total of 24. A tesseract therefore has 24 plane square faces. Again each point of the original cube will trace out a line, making 8 new lines, and there will also be 12 lines in the original and 12 in the final cube, making a total of 32.
Finally, there will be 8 points in the original cube and 8 in the final cube, but none will have been produced on the way. So a tesseract will therefore have 16 corner points.