The Possibility of a 'finite' Yet 'unbounded' Universe

What is the structure of the universe?

Non-Euclidean geometry says that outer space is not infinite (Riemann, Helmholtz). These questions were already answered fully by Helmholtz and Poincaré:

In a 2D space, flat beings with flat implements and flat rigid measuring-rods are free to move in a plane. For them, nothing exists outside of this plane. The constructions of plane Euclidean geometry can be done through the rods, e.g. the lattice construction, considered in Section 24.

Like ours, their universe extends to infinity. In their universe, there is room for an infinite number of identical squares made up of rods, i.e. its volume (surface) is infinite.

• Their universe is a “plane” because they can perform the constructions of plane Euclidean geometry with their rods
• Those rods always represent the same distance, independently of their position.

In a spherical 2D existence, flat beings with their measuring-rods and other objects fit exactly on this surface. Their whole universe extends exclusively over the surface of the sphere.

Would these beings regard their universe as a plane geometry and their rods as the realisation of “distance”?

They cannot do this. If they draw a straight line, they will get a curve, which we “3D beings” designate as a great circle. This circle is a self-contained line of definite finite length, which can be measured with a measuring-rod.

Similarly, this universe has a finite area, that can be compared with the area of a square constructed with rods. This spherical geometry shows a space that is finite and yet has no limits.

But the spherical-surface beings do not need to go on a world-tour to realize that they are not living in a Euclidean universe.

They can convince themselves of this on every part of their “world,” provided they do not use too small a piece of it. Starting from a point, they draw “straight lines” (arcs of circles when viewed in 3D space) of equal length in all directions.

They will call the line joining the free ends of these lines a “circle.”

For a plane surface, the ratio of the circumference of a circle to its diameter, both lengths being measured with the same rod, is, according to Euclidean geometry of the plane, equal to a constant value π, which is independent of the diameter of the circle.

On their spherical surface, our flat beings would find for this ratio the value:

π (sin (r/R) / (r/R))

This is a smaller value than π.

The difference is that the more considerable, the greater is the radius of the circle in comparison with the radius R of the “world-sphere.”

Through this relation, the spherical beings can determine the radius of their universe even when only a relatively small part of their world-sphere is available for their measurements.

But if this part is very small, they will no longer be able to demonstrate that they are on a spherical “world” and not on a Euclidean plane, for a small part of a spherical surface differs only slightly from a piece of a plane of the same size.

Thus, if the spherical-surface beings are living on a planet of which the solar system occupies only a negligibly small part of the spherical universe, they cannot determine whether they are living in a finite or in an infinite universe. This is because the “piece of universe” to which they have access is in both cases practically plane, or Euclidean.

It follows that for our sphere-beings, the circumference of a circle first increases with the radius until the “circumference of the universe” is reached. It thenceforward gradually decreases to zero for still further increasing values of the radius.

During this process, the area of the circle continues to increase more and more, until finally it becomes equal to the total area of the whole “world-sphere.”

Why have we placed our “beings” on a sphere rather than on another closed surface?

This is because of all closed surfaces, the sphere has all points on it equivalent.

The ratio of the circumference c of a circle to its radius r depends on r. But for a given value of r, it is the same for all points of the “world-sphere”.

In other words, the “world-sphere” is a “surface of constant curvature.”

To this 2D sphere-universe there is a 3D analogy, namely, the 3D spherical space which was discovered by Riemann.

Its points are likewise all equivalent. It possesses a finite volume, which is determined by its “radius” ( 2 π 2 R 3 ).

Is it possible to imagine a spherical space? To imagine a space means nothing else than that we imagine an epitome of our “space” experience, i.e. of experience that we can have in the movement of “rigid” bodies. In this sense we can imagine a spherical space.

Suppose we draw lines or stretch strings in all directions from a point, and mark off from each of these the distance r with a measuring-rod.

All the free end-points of these lengths lie on a spherical surface. We can specially measure up the area (F) of this surface by means of a square made up of measuring-rods.

• If the universe is Euclidean, then F = 4 π r 2.
• If it is spherical, then F is always less than 4 π r 2.

With increasing values of r, F increases from zero up to a maximum value which is determined by the “world-radius,” but for still further increasing values of r, the area gradually diminishes to zero.

At first, the straight lines which radiate from the starting point diverge farther and farther from one another. But later they approach each other. Finally they run together again at a “counter-point” to the starting point.

Under such conditions they have traversed the whole spherical space.

The 3D spherical space is analogous to the 2D spherical surface. It is finite (i.e. of finite volume), and has no bounds.

“Elliptical space” is another kind of curved space. It has the two “counter-points” are identical (indistinguishable from each other).

An elliptical universe can thus be considered to some extent as a curved universe possessing central symmetry.

Closed spaces without limits are conceivable. From amongst these, the spherical space (and the elliptical) excels in its simplicity, since all points on it are equivalent.

Is our universe infinite or finite (like the spherical universe). Our experience is insufficient to answer this.

GR answers this with a moderate degree of certainty. This solves the problem in Section 30.