DOC. 42 SPECIAL
AND GENERAL RELATIVITY
365
The
Possibility
of
a
"Finite" and
Yet "Unbounded" Universe
123
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
ex-
actly
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
[76]
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" desig-
nate
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
com-
pared
with the
area
of
a
square
constructed with
rods.
The
great
charm
resulting
from
this consideration lies in
the
rec-
ognition
of
the fact
that
the
universe
of
these
beings
is
finite
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
too
small
a
piece
of
it.
Starting
from
a
point, they
draw
"straight
lines"