DOC.
42
SPECIAL AND GENERAL
RELATIVITY 367
The
Possibility
of
a
"Finite" and
Yet "Unbounded"
Universe
125
because the
"piece
of
universe"
to
which
they
have
access
is
in
both
cases
practically plane,
or
Euclidean.
It
follows
di-
rectly
from
this
discussion,
that
for
our
sphere-beings
the
cir-
cumference of
a
circle
first
increases with the radius until the
"circumference of the universe"
is reached,
and that
it
thence-
forward
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
[77]
all
points
on
it
are
equivalent.
I
admit that the ratio
of
the
circumference
c
of 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 two-dimensional
sphere-universe
there
is
a
three-
dimensional
analogy,
namely,
the three-dimensional
spherical
space
which
was
discovered
by
Riemann. Its
points
are
like-
wise
all
equivalent.
It
possesses
a
finite
volume,
which
is
de-
termined
by
its
"radius"
(2n2R3).
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"
bod-
ies.
In this
sense we can
imagine
a
spherical space.
Suppose
we
draw lines
or
stretch
strings
in all
directions