To illustrate how the analogy of the relation between two and three-dimensional space enables us to determine some of the properties of four-dimensional figures:

(1)

Thus, the lowest sort of space is space of zero dimensions, i.e., a mathematical point. If it moves a distance of one inch, it traces out a Line one inch long—that is to say a one space "figure." If this moves at right angles to itself for a distance of one inch, it traces out a two space figure, viz., a square of side one inch. If this again moves a distance of one inch in a direction at right angles to every direction that can be drawn within it, that is, in a direction perpendicular to itself, it traces out a cube of side one inch, i.e., a three space figure or "solid."

We must, therefore, conclude, from analogy, that if the cube were itself to move, a distance of one inch, in a direction at right angles to every direction that can be drawn in our space—in the unknown direction, that is, of the fourth dimension—it would generate a "higher solid" of side one inch. The higher solid thus generated is called a "Tesseract" and its properties are quite well known.

(2)

Since a point is defined as having "position but no magnitude," it follows that it would require an infinite number of points to make up a line.

Similarly a line has length, but no breadth or thickness, and it would therefore require an infinite number of lines laid side by side to make up a surface.

Again a surface has, theoretically, no thickness, and it would therefore require an infinite number of surfaces superimposed on one another to make up a solid.

We must therefore conclude, by analogy, that it would require an infinite number of solids to make up a "higher solid."

In particular, a Tesseract must be supposed to contain an infinite number of cubes, and, in general, four space must be conceived of as containing an infinite number of three spaces.

(3)

Thus a Line (one space) is bounded by Points (zero space).

A surface (two space) is bounded by Lines (one space).

A solid (three space) is bounded by Surfaces (two space).

We must conclude therefore that "higher solids" (four space) are bounded by Solids (three space).

To take the special case with which we are already familiar. The line AB, is bounded by the points A and B. (Fig. 10). The square, A B C D, is bounded by four lines AB, BC, CD, DA. The cube, A B C D E F G H, is bounded by six surfaces, namely, ABCD, CDEF, EFGH, GHAB, ADEH, BCFG.

Similarly we must conclude that a tesseract is bounded by cubes.

We shall see later that there are eight of them.

(4)

We may put (3) in a slightly different way, by saying that:

The portions AB and BC of the line AC are separated by the point B. (Fig. 11.) The portions ABEF and BCDE of the fig. ACDF are separated by the line EB. The portions ABEFGHIM and BCDEMIKL of the solid ACDFGHKL are separated by the surface BIME.

Similarly we must suppose that any two adjacent portions of four space are separated by a three space figure.

Or, again, to alter it slightly, "any space is no more than a boundary between two adjacent portions of the next higher space." Whence it follows that the whole of our three space is but the boundary between two adjacent portions of four space.

(5)

This may at first sight seem difficult to grasp.

In reality however, it is quite simple.

We have only to remember that the tesseract is generated by the movement of a cube, in a direction at right angles to every direction that can be drawn in the cube, and that whenever a figure of a given dimensionality moves thus it generates a figure of the next higher dimensionality.

Thus every point in the cube will trace out a line, every line a surface, and every surface a solid, and, since the distance moved is equal to the length of the side of the cube, these surfaces will be squares and the solids will be cubes.

But let us first consider the analogous case of the generation of the cube by the movement of a square.

Let A B C D represent the original position of the square. It moves, a distance equal to one of its sides, in a direction at right angles to every direction that can be drawn within itself—at right angles, i.e., to every one of its sides—and finally comes to rest in the position E F G H.

Every side has traced out another square and we have, in addition, the old square ABCD, with which we started and the new square EFGH, with which we end.

Thus even if we had no idea how many sides, edges, and corners a cube had we could deduce them.

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.

There is no reason why this process should not be continued indefinitely. For a tesseract may be supposed to move, in distance equal to the length of one of its edges, in a direction not contained in itself and will generate a five dimensional figure, bounded by ten tesseracts, and having in it 40 cubes, 80 squares, 80 lines, and 32 corner points. Thus a whole series of Higher Space figures may be produced. But these are of little practical interest, and I shall not deal with them here.

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