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66
Relativity: The Special and General Theory
Albert Einstein: Relativity
Part III: Considerations on the Universe as a Whole
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 Poincare, 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
Section 24. 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 from this
consideration lies in the recognition of the fact that the universe of these beings is finile and yet has
no limits.
But the spherical?surface beings do not need to go on a world?tour in order to perceive 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 to small a piece of it. Starting from a point, they draw " straight
lines " (arcs of circles as judged in three dimensional 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 ?, which is independent of
the diameter of the circle. On their spherical surface our flat beings would find for this ratio the
value