the apparent need for an edge, a problem that has only recently been grappled
with. Aristotle's argument, that the universe is finite, and that a boundary was
Line necessary to fix an absolute reference frame, held only until scientists wondered
(5) what happened at the far side of the edge. In other words, why do we not
redefine the "universe" to include that other side？
Riemann ingeniously replied by proposing the hypersphere, the three-
dimensional surface of a four-dimensional ball. Previously it was supposed that
the ultimate physical reality must be a Euclidean space of some dimension, and
(10) thus if space were a hypersphere, it would need to sit in a four-dimensional
Euclidean space that allows us to view it from the outside. But according to
Riemann, it would be perfectly acceptable for the universe to be a hypersphere
and not embedded in any higher-dimensional space; nature need not therefore
cling to the ancient notion. According to Einstein's powerful but limited theory
(15) of relativity, space is a dynamic medium that can curve in one of three ways,
depending on the distribution of matter and energy within it, but because we are
embedded in space, we cannot see the flexure directly but rather perceive it as
gravitational attraction and geometric distortion of images. Thus, to determine
which of the three geometries our universe has, astronomers are forced to
(20) measure the density of matter and energy in the cosmos, whose amounts appear
at present to be insufficient to force space to arch back on itself in "spherical"
geometry. Space may also have the familiar Euclidean geometry, like that of a
plane, or a "hyperbolic" geometry, like that of a saddle. Furthermore, the
universe could be spherical, yet so large that the observable part seems
(25) Euclidean, just as a small patch of the earth's surface looks flat.
We must recall that relativity is a purely local theory: it predicts the
curvature of each small volume of space-its geometry-based on the matter
and energy it contains, and the three plausible cosmic geometries are consistent
with many different topologies: relativity would describe both a torus and a
(30) plane with the same equations, even though the torus is finite and the plane is
infinite. Determining the topology therefore requires some physical
understanding beyond relativity, in order to answer the question, for instance,
of whether the universe is, like a plane, "simply connected", meaning there is
only one direct path for light to travel from a source to an observer. A simply
(35) connected Euclidean or hyperbolic universe would indeed be infinite-and seems
self-evident to the layman-but unfortunately the universe might instead be
"multiply-connected", like a torus, in which case there are many different such
paths. An observer could see multiple images of each galaxy and easily interpret
them as distinct galaxies in an endless space, much as a visitor to a mirrored
(40) room has the illusion of seeing a huge crowd, and for this reason physicists have
yet to conclusively determine the shape of the un
A．searching for an accurate method of determining whether the universe is finite or infinite
B．discussing problems and possibilities involved in providing a definite picture of the shape of the universe
C．declaring opposition to the notion that spherical geometry is a possible model for the shape of the universe
D．criticizing discredited theories about the possible topologies of the universe
E．refuting the idea that there is no way to tell whether the universe is finite and if so what shape it has
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