AB = A'B', AC = A'C', BC = B'C', AD = A'D', BD = B'D', CD = CD'. It is evidently impossible in 3-space to put "b" in the position "a" or vice versa. It would be possible to make "h" coincide with the image of "a" in a mirror. In fact it is obvious that "b" is the image of "a" as seen in a mirror.
Readers of that classic nonsense book by Lewis Carroll (Rev. C. L. Dodgson), "Alice Behind the Looking-Glass," will be interested in the fact that Mr. Dodgson, himself a mathematician of no mean note, is poking fun at the fourth-dimension students.
Now, while it is impossible for a tridim to make "b" take the position "a," there would be no difficulty in a fourth-dimensional animal interchanging "a" and "b." In other words, to make "a" and "b" coincide, one must be taken up into 4-space, turned over and put down on the other.
It is' easy now to see that, while there is no proof of the material existence of 4-space or space of any dimension higher than three, and while we can not even say that there is any likelihood that such exists, yet the conception of hyperspace is a perfectly real and logical conception; moreover, it is by no means an idle question or a useless idea. Assuming hyperspace, mathematicians have built up a perfectly consistent geometry which throws much light upon problems of 3-space.
We have seen that by many analogies it is a simple matter to conceive of hyperspace. Let us next observe how algebra invites us to consider the possible existence of higher space.
The solution of two simultaneous equations in two variables x, y, gives us a point in a plane. The solution of three simultaneous equations in three variables x, y, z, gives us a point in 3-space. The solution of four simultaneous equations in four variables, x, y, z, w, which is easily performed, gives what? Is there a geometrical equivalent here? Can the values of x, y, z, w be represented graphically? The answer to both questions is No, at least not in our space. Four-space is necessary if we are to give a geometrical representation to the solution of four simultaneous equations, such as:
a1x + b1y + c1z + d1w = e1,
a2x + b2y + c2z + d2w = e2,
a3x + b3y + c3z + d3w = e3,
a4x + b4y + c4z + d4w = e4,
x = a
represent a point in 1-space. [Incidentally it would also denote a line in 2-space and a plane in 3-space.]