The Possibility of a "Finite" and yet "Unbounded" Universe
But speculations on the structure of the universe also move in quite another direction. The development of non-Euclidean geometry led to
the recognition of the fact, that we can cast doubt on the infiniteness of our space without coming into conflict with the laws of thought or with experience (Riemann,
Helmholtz). These questions have already been treated
in detail and with unsurpassable lucidity by Helmholtz and Poincaré, whereas I can only touch on them
briefly here.
In the first place, we imagine an existence in two-dimensional
space. Flat beings with flat implements, and in particular flat rigid measuring-rods, are free to move in a plane. For them nothing exists outside of
this plane: that which they observe to happen to themselves and to their flat "things" is the all-inclusive reality of their plane. In particular, the constructions of plane Euclidean geometry can be carried out by means of the rods, e.g. the lattice construction, considered
in [chapter:XXIV]Section XXIV. In contrast to ours, the universe of these beings is two-dimensional; but, like ours, it 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. If these beings say their universe is "plane," there is sense in the statement,
because they mean that they can perform the constructions of plane Euclidean geometry with their rods.
In this connection the individual rods always represent
the same distance, independently of their position.
Let us consider now a second two-dimensional existence, but this time on a spherical surface instead of on
a plane. The flat beings with their measuring-rods and other objects fit exactly on this surface and they are unable to leave it. Their whole universe of observation extends exclusively over the surface of the sphere. Are these beings able to regard the geometry of their universe as being plane geometry and their rods withal as the realisation of "distance"? They cannot do this. For if they attempt to realise a straight line, they
will obtain a curve, which we "three-dimensional beings" designate as a great circle, i.e. a self-contained line of definite finite length, which can be measured up by means of a measuring-rod. Similarly, this universe has a finite area that can be compared with the area of a square constructed with rods. The great charm resulting