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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 ¼, which is independent of
the diameter of the circle. On their spherical surface our flat beings would find for this ratio the
value
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Relativity: The Special and General Theory
i.e. a smaller value than ¼, the difference being the more considerable, the greater is the radius of
the circle in comparison with the radius R of the " world-sphere." By means of this relation the
spherical beings can determine the radius of their universe (" world "), even when only a relatively
small part of their worldsphere is available for their measurements. But if this part is very small
indeed, 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 have no means of determining whether they are
living in a finite or in an infinite universe, because the " piece of universe " to which they have
access is in both cases practically plane, or Euclidean. It follows directly from this discussion, that
for our sphere-beings the circumference of a circle first increases with the radius until the "
circumference of the universe " is reached, and that 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."
Perhaps the reader will wonder why we have placed our " beings " on a sphere rather than on
another closed surface. But this choice has its justification in the fact that, of all closed surfaces, the
sphere is unique in possessing the property that all points on it are equivalent. I admit that 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 " worldsphere "; in other words, the " world-sphere " is a " surface of constant
curvature."
To this two-dimensional sphere-universe there is a three-dimensional analogy, namely, the
three-dimensional spherical space which was discovered by Riemann. its points are likewise all
equivalent. It possesses a finite volume, which is determined by its "radius" (2¼2R3). 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
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