Part 4

Has length and breadth but no thickness.

Made by moving a line in a direction perpendicular to its length (that is, into the second dimension).

Represented by two straight lines of indefinite length perpendicular to each other.

The lines are called axes and are labeled x and y.

The point where they meet, the origin, is marked O.

Like this [Illustration]

Three dimensions:

A solid like a cube.

Has length, breadth and thickness.

Made by moving a plane in a direction perpendicular to the other two (that is, into the third dimension).

Cannot be pictured on paper, but is indicated by three axes, x, y, and z, of which x and y are on the plane of the page and z is supposed to be stuck up at right-angles to the other two. Stick a pin into the paper at the point O and you will have the third or z axis.

Like this [Illustration]

Four dimensions:

Has length, breadth, thickness and extension into a fourth dimension, say time.

Made by moving a cube in a direction perpendicular to the other three (that is, into the fourth dimension).

Cannot be pictured on paper, but may be indicated by four axes, x, y, z and t (or u), each at right-angles to the other three.

Like this [Illustration]

More dimensions:

Any desired number of dimensions can be worked out mathematically but with increasing difficulty because of the impracticability of diagrammatical representation. We can generalize the idea by speaking of a “geometry of n dimensions” where n may stand for any number whatever from zero to infinity.

A line of a given length contains an infinite number of points.

A square of a given size contains an infinite number of lines.

A cube of a given size contains an infinite number of plane squares.

A tesseract (four-dimensional cuboid) of a given size contains an infinite number of solid cubes.

And what would there be left of space if you took everything out of it, and what would become of time if nothing ever happened? In other words are not space and time merely forms of thought, the framework of ideas, and if so cannot we fix them over to suit our need of new conceptions? As a matter of fact we do. We have constructed by the aid of Euclid and his successors a geometry of three dimensions that works perfectly for all ordinary requirements and if we need a fourth dimension to accommodate these new astronomical and physical phenomena we will build on the necessary addition to our conception of space. There was no use having a fourth dimension so long as we had nothing to put in it. For ordinary earth measurements (geometry) such as laying out a town lot we only use two dimensions, length and breadth. We speak of “flat ground” and “water-level” regardless of the fact that all our “straight” lines on the earth’s surface are really curves that come back to us after going 25,000 miles or less. It is only when measuring mile lengths that we have to correct for the curvature of the earth in the third dimension. So if, as seems probable, we shall have to make allowance in astronomical measurements for the curvature of the universe in a fourth dimension it will merely mean a little labor to the astronomers and it will relieve their minds of some of their perplexities. There is nothing more mystical or mysterious or “psychical” about a fourth dimension than about the other three. A dimension is simply a measurable direction and we can use five dimensions or n dimensions if we need to.

It does not matter that we cannot “see” a figure in four dimensions even with our mind’s eye. Actually we cannot see any figure of more or less than two dimensions: we have to take the others on faith. Nobody can see the mathematician’s point because it has no dimensions, no size at all. The schoolboy says: “Let that be the point A,” and we let it be although what he is pointing at with his stick is not a point but a vast irregular splotch of white chalk on the blackboard. So, too, we cannot see a mathematical line because it has only one dimension, length and no breadth. But set four lines at right angles to one another and we get a square. This we can really see if the enclosed surface is of a different color such as a shadow or black print. Set six squares together at right angles and we get a cube. This we cannot see in its entirety at one time. All that we see when we look squarely at a cube is a square. If we look at it from an angle we see what looks like a square with a couple of lozenges on the sides. The retina of the eye is practically a plane surface, so all we can get is a two-dimensional projection of a solid.

The best way to get an idea of the construction of a cubical solid in four dimensions is to draw a diagram yourself and trace out in turn each of the eight cubes that inclose it. I am indebted to K. W. Lamson of Barnard College for the following sketch and directions:

Draw the four coördinate axes OX, OY, OZ, OU.

Lay off the unit a{1}a{4} on the X axis, a{1}a{2} on the Y axis, a{1}a{7} on the Z axis and a{1}b{1} on the U axis.