no disc overlaps another, the surface of the globe will finally
become so full that there is no room for another disc. This simply
means that the spherical surface of the globe is finite in relation
to the paper discs. Further, the spherical surface is a non-Euclidean
continuum of two dimensions, that is to say, the laws of disposition
for the rigid figures lying in it do not agree with those of the
Euclidean plane. This can be shown in the following way. Place
a paper disc on the spherical surface, and around it in a circle
place six more discs, each of which is to be surrounded in turn
by six discs, and so on. If this construction is made on a plane
surface, we have an uninterrupted disposition in which there are
six discs touching every disc except those which lie on the outside.

[Figure 1: Discs maximally packed on a plane]

On the spherical surface the construction also seems to promise
success at the outset, and the smaller the radius of the disc
in proportion to that of the sphere, the more promising it seems.
But as the construction progresses it becomes more and more patent
that the disposition of the discs in the manner indicated, without
interruption, is not possible, as it should be possible by Euclidean
geometry of the the plane surface. In this way creatures which
cannot leave the spherical surface, and cannot even peep out from
the spherical surface into three-dimensional space, might discover,
merely by experimenting with discs, that their two-dimensional
"space" is not Euclidean, but spherical space.

From the latest results of the theory of relativity it is probable
that our three-dimensional space is also approximately spherical,
that is, that the laws of disposition of rigid bodies in it are