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Title    : Is Space Finite?
Author :
Date    :


Conventional wisdom says the universe is infinite. But
it could be finite, merely giving the illusion of infinity.
Upcoming measurements may finally answer this ancient question

by Jean-Pierre Luminet, Glenn D. Starkman and Jeffrey R. Weeks



Comfort in the Finite

Cosmic Crystals

Circular Reasoning


Local Geometry

Doughnut Space

Finite Hyperbolic Space

Distances Between Galaxy Clusters

Wrapped Around

Three Possible Universes



Image: Bryan Christie

Looking up at the sky on a clear night, we feel we can see forever. There
seems to be no end to the stars and galaxies; even the darkness in between
them is filled with light if only we stare through a sensitive enough
telescope. In truth, of course, the volume of space we can observe is limited
by the age of the universe and the speed of light. But given enough time,
could we not peer ever farther, always encountering new galaxies and

Maybe not. Like a hall of mirrors, the apparently endless universe might be
deluding us. The cosmos could, in fact, be finite. The illusion of infinity
would come about as light wrapped all the way around space, perhaps more than
once--creating multiple images of each galaxy. Our own Milky Way galaxy would
be no exception; bizarrely, the skies might even contain facsimiles of the
earth at some earlier era. As time marched on, astronomers could watch the
galaxies develop and look for new mirages. But eventually no new space would
enter into their view. They would have seen it all.

The question of a finite or infinite universe is one of the oldest in
philosophy. A common misconception is that it has already been settled in
favor of the latter. The reasoning, often repeated in textbooks, draws an
unwarranted conclusion from Einstein's general theory of relativity.
According to relativity, space is a dynamic medium that can curve in one of
three ways, depending on the distribution of matter and energy within it.
Because we are embedded in space, we cannot see the flexure directly but
rather perceive it as gravitational attraction and geometric distortion of
images. To determine which of the three geometries our universe has,
astronomers have been measuring the density of matter and energy in the
cosmos. It now appears to be too low to force space to arch back on itself--a
"spherical" geometry. Therefore, space must have either the familiar
Euclidean geometry, like that of a plane, or a "hyperbolic" geometry, like
that of a saddle [see illustration]. At first glance, such a universe
stretches on forever.

One problem with this conclusion is that the universe could be spherical yet
so large that the observable part seems Euclidean, just as a small patch of
the earth's surface looks flat. A broader issue, however, is 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. Neither
relativity nor standard cosmological observations say anything about how
those volumes fit together to give the universe its overall shape--its
topology. The three plausible cosmic geometries are consistent with many
different topologies. For example, relativity would describe both a torus (a
doughnutlike shape) and a plane with the same equations, even though the
torus is finite and the plane is infinite. Determining the topology requires
some physical understanding beyond relativity.

The usual assumption is that the universe is, like a plane, "simply
connected," which means there is only one direct path for light to travel
from a source to an observer. A simply connected Euclidean or hyperbolic
universe would indeed be infinite. But the universe might instead be
"multiply connected," like a torus, in which case there are many different
such paths. An observer would see multiple images of each galaxy and could
easily misinterpret them as distinct galaxies in an endless space, much as a
visitor to a mirrored room has the illusion of seeing a huge crowd.

A multiply connected space is no mere mathematical whimsy; it is even
preferred by some schemes for unifying the fundamental forces of nature, and
it does not contradict any available evidence. Over the past few years,
research into cosmic topology has blossomed. New observations may soon reach
a definitive answer.

Comfort in the Finite

Many cosmologists expect the universe to be finite. Part of the reason may be
simple comfort: the human mind encompasses the finite more readily than the
infinite. But there are also two scientific lines of argument that favor
finitude. The first involves a thought experiment devised by Isaac Newton and
revisited by George Berkeley and Ernst Mach. Grappling with the causes of
inertia, Newton imagined two buckets partially filled with water. The first
bucket is left still, and the surface of the water is flat. The second bucket
is spun rapidly, and the surface of the water is concave. Why?

The naive answer is centrifugal force. But how does the second bucket know it
is spinning? In particular, what defines the inertial reference frame
relative to which the second bucket spins and the first does not? Berkeley
and Mach's answer was that all the matter in the universe collectively
provides the reference frame. The first bucket is at rest relative to distant
galaxies, so its surface remains flat. The second bucket spins relative to
those galaxies, so its surface is concave. If there were no distant galaxies,
there would be no reason to prefer one reference frame over the other. The
surface in both buckets would have to remain flat, and therefore the water
would require no centripetal force to keep it rotating. In short, it would
have no inertia. Mach inferred that the amount of inertia a body experiences
is proportional to the total amount of matter in the universe. An infinite
universe would cause infinite inertia. Nothing could ever move.

In addition to Mach's argument, there is preliminary work in quantum
cosmology, which attempts to describe how the universe emerged spontaneously
from the void. Some such theories predict that a low-volume universe is more
probable than a high-volume one. An infinite universe would have zero
probability of coming into existence [see "Quantum Cosmology and the Creation
of the Universe," by Jonathan J. Halliwell; Scientific American, December
1991]. Loosely speaking, its energy would be infinite, and no quantum
fluctuation could muster such a sum.

Historically, the idea of a finite universe ran into its own obstacle: the
apparent need for an edge. Aristotle argued that the universe is finite on
the grounds that a boundary was necessary to fix an absolute reference frame,
which was important to his worldview. But his critics wondered what happened
at the edge. Every edge has another side. So why not redefine the "universe"
to include that other side? German mathematician Georg F. B. Riemann solved
the riddle in the mid-19th century. As a model for the cosmos, he proposed
the hypersphere--the three-dimensional surface of a four-dimensional ball,
just as an ordinary sphere is the two-dimensional surface of a
three-dimensional ball. It was the first example of a space that is finite
yet has no problematic boundary.

One might still ask what is outside the universe. But this question supposes
that the ultimate physical reality must be a Euclidean space of some
dimension. That is, it presumes that if space is a hypersphere, then that
hypersphere must sit in a four-dimensional Euclidean space, allowing us to
view it from the outside. Nature, however, need not cling to this notion. It
would be perfectly acceptable for the universe to be a hypersphere and not be
embedded in any higher-dimensional space. Such an object may be difficult to
visualize, because we are used to viewing shapes from the outside. But there
need not be an "outside."

By the end of the 19th century, mathematicians had discovered a variety of
finite spaces without boundaries. German astronomer Karl Schwarzschild
brought this work to the attention of his colleagues in 1900. In a postscript
to an article in Vierteljahrschrift der Astronomischen Gesellschaft, he
challenged his readers:

"Imagine that as a result of enormously extended astronomical experience, the
entire universe consists of countless identical copies of our Milky Way, that
the infinite space can be partitioned into cubes each containing an exactly
identical copy of our Milky Way. Would we really cling on to the assumption
of infinitely many identical repetitions of the same world?. . . We would be
much happier with the view that these repetitions are illusory, that in
reality space has peculiar connection properties so that if we leave any one
cube through a side, then we immediately reenter it through the opposite

Schwarzschild's example illustrates how one can mentally construct a torus
from Euclidean space. In two dimensions, begin with a square and identify
opposite sides as the same--as is done in many video games, such as the
venerable Asteroids, in which a spaceship going off the right side of the
screen reappears on the left side. Apart from the interconnections between
sides, the space is as it was before. Triangles span 180 degrees, parallel
laser beams never meet and so on--all the familiar rules of Euclidean
geometry hold. At first glance, the space looks infinite to those who live
within it, because there is no limit to how far they can see. Without
traveling around the universe and reencountering the same objects, the ship
could not tell that it is in a torus [see illustration]. In three dimensions,
one begins with a cubical block of space and glues together opposite faces to
produce a 3-torus.

The Euclidean 2-torus, apart from some sugar glazing, is topologically
equivalent to the surface of a doughnut. Unfortunately, the Euclidean torus
is food only for the mind. It cannot sit in our three-dimensional Euclidean
space. Doughnuts may do so because they have been bent into a spherical
geometry around the outside and a hyperbolic geometry around the hole.
Without this curvature, doughnuts could not be viewed from the outside.

When Albert Einstein published the first relativistic model of the universe
in 1917, he chose Riemann's hypersphere as the overall shape. At that time,
the topology of space was an active topic of discussion. Russian
mathematician Aleksander Friedmann soon generalized Einstein's model to
permit an expanding universe and a hyperbolic space. His equations are still
routinely used by cosmologists. He emphasized that the equations of his
hyperbolic model applied to finite universes as well as to the standard
infinite one--an observation all the more remarkable because, at the time, no
examples of finite hyperbolic spaces were known.

Image: Bryan Christie


Of all the issues in cosmic topology, perhaps the most difficult to grasp is
how a hyperbolic space can be finite. For simplicity, first consider a
two-dimensional universe. Mimic the construction of a 2-torus but begin with
a hyperbolic surface instead. Cut out a regular octagon and identify opposite
pairs of edges, so that anything leaving the octagon across one edge returns
at the opposite edge. Alternatively, one could devise an octagonal Asteroids
screen [see illustration]. This is a multiply connected universe,
topologically equivalent to a two-holed pretzel. An observer at the center of
the octagon sees the nearest images of himself or herself in eight different
directions. The illusion is that of an infinite hyperbolic space, even though
this universe is really finite. Similar constructions are possible in three
dimensions, although they are harder to visualize. One cuts a solid
polyhedron out of a hyperbolic three-dimensional space and glues pairs of
faces so that any object leaving from one face returns at the corresponding
point on the matching face.

The angles of the octagon merit careful consideration. On a flat surface, a
polygon's angles do not depend on its size. A large regular octagon and a
small regular octagon both have inside angles of 135 degrees. On a curved
surface, however, the angles do vary with size. On a sphere the angles
increase as the polygon grows, whereas on a hyperbolic surface the angles
decrease. The above construction requires an octagon that is just the right
size to have 45-degree angles, so that when the opposite sides are
identified, the eight corners will meet at a single point and the total angle
will be 360 degrees. This subtlety explains why the construction would not
work with a flat octagon; in Euclidean geometry, eight 135-degree corners
cannot meet at a single point. The two-dimensional universe obtained by
identifying opposite sides of an octagon must be hyperbolic. The topology
dictates the geometry.

The size of the polygon or polyhedron is measured relative to the only
geometrically meaningful length scale for a space: the radius of curvature. A
sphere, for example, can have any physical size (in meters, say), but its
surface area will always be exactly 4p times the square of its radius--that
is, 4p square radians. The same principle applies to the size of a hyperbolic
topology, for which a radius of curvature can also be defined. The most
compact hyperbolic topology, discovered by one of us (Weeks) in 1985, may be
constructed by identifying pairs of faces of an 18-sided polyhedron. It has a
volume of approximately 0.94 cubic radian. Other topologies are built from
larger polyhedra.

The universe, too, can be measured in units of radians. Diverse astronomical
observations agree that the density of matter in the cosmos is only a third
of that needed for space to be Euclidean. Either a cosmological constant
makes up the difference [see "Cosmological Antigravity," by Lawrence M.
Krauss; Scientific American, January], or the universe has a hyperbolic
geometry with a radius of curvature of 18 billion light-years. In the latter
case, the observable universe has a volume of 180 cubic radians--enough room
for nearly 200 of the Weeks polyhedra. In other words, if the universe has
the Weeks topology, its volume is only 0.5 percent of what it appears to be.
As space expands uniformly, its proportions do not change, so the topology
remains constant.

In fact, almost all topologies require hyperbolic geometries. In two
dimensions, a finite Euclidean space must have the topology of either a
2-torus or a Klein bottle; in three dimensions, there are only 10 Euclidean
possibilities--namely, the 3-torus and nine simple variations on it, such as
gluing together opposite faces with a quarter turn or with a reflection,
instead of straight across. By comparison, there are infinitely many possible
topologies for a finite hyperbolic three-dimensional universe. Their rich
structure is still the subject of intense research [see "The Mathematics of
Three-Dimensional Manifolds," by William P. Thurston and Jeffrey R. Weeks;
Scientific American, July 1984].

Cosmic Crystals

Despite the plethora of possibilities, the cosmologists of the 1920s had no
way to measure the topology of the universe directly, and so they eventually
lost interest in the issue. The decades from 1930 to 1990 were the dark ages
of the subject. Most astronomy textbooks, quoting one another for support,
stated that the universe must be either a hypersphere, an infinite Euclidean
space or an infinite hyperbolic space. Other topologies were largely
forgotten. But the 1990s have seen the rebirth of the subject. Roughly as
many papers have been published on cosmic topology in the past three years as
in the preceding 80. Most exciting of all, cosmologists are finally poised to
determine the topology observationally.

The simplest test of topology is to look at the arrangement of galaxies. If
they lie in a rectangular lattice, with images of the same galaxy repeating
at equivalent lattice points, the universe is a 3-torus. Other patterns
reveal more complicated topologies. Unfortunately, looking for such patterns
is difficult, because the images of a galaxy would depict different points in
its history. Astronomers would need to recognize the same galaxy despite
changes in appearance or shifts in position relative to neighboring galaxies.
Over the past 25 years researchers such as Dmitri Sokoloff of Moscow State
University, Viktor Shvartsman of the Soviet Academy of Sciences, J. Richard
Gott III of Princeton University and Helio V. Fagundes of the Institute for
Theoretical Physics in S�o Paulo have looked for and found no repeating
images among galaxies within one billion light-years of the earth.

Image: Bryan Christie

Others--such as Boudewijn F. Roukema of the Inter-University Center for
Astronomy and Astrophysics in Pune, India--have sought patterns among
quasars. Because these objects, thought to be powered by black holes at the
cores of galaxies, are bright, any patterns among them can be seen from large
distances. The observers identified all groupings of four or more quasars. By
examining the spatial relations within each group, they checked whether any
pair of groups could in fact be the same group seen from two different
directions. Roukema identified two possibilities, but they may not be
statistically significant.

Roland Lehoucq and Marc Lachi�ze-Rey of the Center for Astrophysical Studies
in Saclay, France, together with one of us (Luminet), have tried to
circumvent the problems of galaxy recognition in another way. We have
developed the method of cosmic crystallography, which in a Euclidean universe
can make out a pattern statistically without needing to recognize specific
galaxies as images of one another. If galaxy images repeat periodically, a
histogram of all galaxy-to-galaxy distances should show peaks at certain
distances, which represent the true size of the universe. So far we have seen
no patterns [see illustration],but this may be because of the paucity of data
on galaxies farther away than two billion light-years. The Sloan Digital Sky
Survey--an ongoing American-Japanese collaboration to prepare a
three-dimensional map of much of the universe--will produce a larger data set
for these studies.

Finally, several other research groups plan to ascertain the topology of the
universe using the cosmic microwave background, the faint glow remaining from
the time when the primordial plasma of the big bang condensed to hydrogen and
helium gas. The radiation is remarkably homogeneous: its temperature and
intensity are the same in all parts of the sky to nearly one part in 100,000.
But there are slight undulations discovered in 1991 by the Cosmic Background
Explorer (COBE) satellite. Roughly speaking, the microwave background depicts
density variations in the early universe, which ultimately seeded the growth
of stars and galaxies [see "The Evolution of the Universe," by P. James E.
Peebles, David N. Schramm, Edwin L. Turner and Richard G. Kron; Scientific
American, October 1994].

Circular Reasoning

These fluctuations are the key to resolving a variety of cosmological issues,
and topology is one of them. Microwave photons arriving at any given moment
began their journeys at approximately the same time and distance from the
earth. So their starting points form a sphere, called the last scattering
surface, with the earth at the center. Just as a sufficiently large paper
disk overlaps itself when wrapped around a broom handle, the last scattering
surface will intersect itself if it is big enough to wrap all the way around
the universe. The intersection of a sphere with itself is simply a circle of
points in space.

Looking at this circle from the earth, astronomers would see two circles in
the sky that share the same pattern of temperature variations. Those two
circles are really the same circle in space seen from two perspectives [see
illustration]. They are analogous to the multiple images of a candle in a
mirrored room, each of which shows the candle from a different angle.

Two of us (Starkman and Weeks), working with David N. Spergel and Neil J.
Cornish of Princeton, hope to detect such circle pairs. The beauty of this
method is that it is unaffected by the uncertainties of contemporary
cosmology--it relies on the observation that space has constant curvature but
makes no assumptions about the density of matter, the geometry of space or
the presence of a cosmological constant. The main problem is to identify the
circles despite the forces that tend to distort their images. For example, as
galaxies coalesce, they exert a varying gravitational pull on the radiation
as it travels toward the earth, shifting its energy.

Unfortunately, COBE was incapable of resolving structures on an angular scale
of less than 10 degrees. Moreover, it did not identify individual hot or cold
spots; all one could say for sure is that statistically some of the
fluctuations were real features rather than instrumental artifacts.
Higher-resolution and lower-noise instruments have since been developed. Some
are already making observations from ground-based or balloon-borne
observatories, but they do not cover the whole sky. The crucial observations
will be made by the National Aeronautics and Space Administration's Microwave
Anisotropy Probe (MAP), due for launch late next year, and the European Space
Agency's Planck satellite, scheduled for 2007.

The relative positions of the matching circles, if any, will reveal the
specific topology of the universe. If the last scattering surface is just
barely big enough to wrap around the universe, it will intersect only its
nearest ghost images. If it is larger, it will reach farther and intersect
the next nearest images. If the last scattering surface is large enough, we
expect hundreds or even thousands of circle pairs. The data will be highly
redundant. The largest circles will completely determine the topology of
space as well as the position and orientation of all smaller circle pairs.
Thus, the internal consistency of the patterns will verify not only the
correctness of the topological findings but also the correctness of the
microwave background data.

Other teams have different plans for the data. John D. Barrow and Janna J.
Levin of the University of Sussex, Emory F. Bunn of Bates College, and Evan
Scannapieco and Joseph I. Silk of the University of California at Berkeley
intend to examine the pattern of hot and cold spots directly. The group has
already constructed sample maps simulating the microwave background for
particular topologies. They have multiplied the temperature in each direction
by the temperature in every other direction, generating a huge
four-dimensional map of what is usually called the two-point correlation
function. The maps provide a quantitative way of comparing topologies. J.
Richard Bond, Dmitry Pogosyan and Tarun Souradeep of the Canadian Institute
for Theoretical Astrophysics are applying related new techniques to the
existing COBE data, which could prove sufficiently accurate to identify the
smallest hyperbolic spaces.

Beyond the immediate intellectual satisfaction, discovering the topology of
space would have profound implications for physics. Although relativity says
nothing about the universe's topology, newer and more comprehensive theories
that are under development should predict the topology or at least assign
probabilities to the various possibilities. These theories are needed to
explain gravity in the earliest moments of the big bang, when
quantum-mechanical effects were important [see "Quantum Gravity," by Bryce S.
DeWitt; Scientific American, December 1983]. The theories of everything, such
as string theory, are in their infancy and do not yet have testable
consequences. But eventually the candidate theories will make predictions
about the topology of the universe on large scales.

The tentative steps toward the unification of physics have already spawned
the subfield of quantum cosmology. There are three basic hypotheses for the
birth of the universe, which are advocated, respectively, by Andrei Linde of
Stanford University, Alexander Vilenkin of Tufts University and Stephen W.
Hawking of the University of Cambridge. One salient point of difference is
whether the expected volume of a newborn universe is very large (Linde's and
Vilenkin's proposals) or very small (Hawking's). Topological data may be able
to distinguish among these models.

If observations do find the universe to be finite, it might help to resolve a
major puzzle in cosmology: the universe's large-scale homogeneity. The need
to explain this uniformity led to the theory of inflation, but inflation has
run into difficulty of late, because in its standard form it would have made
the cosmic geometry Euclidean--in apparent contradiction with the observed
matter density. This conundrum has driven theorists to postulate hidden forms
of energy and modifications to inflation [see "Inflation in a Low-Density
Universe," by Martin A. Bucher and David N. Spergel; Scientific American,
January]. An alternative is that the universe is smaller than it looks. If
so, inflation could have stopped prematurely--before imparting a Euclidean
geometry--and still have made the universe homogeneous. Igor Y. Sokolov of
the University of Toronto and others have used COBE data to rule out this
explanation if space is a 3-torus. But it remains viable if space is

Since ancient times, cultures around the world have asked how the universe
began and whether it is finite or infinite. Through a combination of
mathematical insight and careful observation, science in this century has
partially answered the first question. It might begin the next century with
an answer to the second as well.


Further Reading:

La Biblioteca De Babel (The Library of Babel). Jorge Luis Borges in
Ficciones. Emec� Editores, 1956. Text available on the World Wide Web at (in Spanish) and at (in English).

Cosmic Topology. Marc Lachi�ze-Rey and Jean-Pierre Luminet in Physics
Reports, Vol. 254, No. 3, pages 135-214; March 1995. Preprint available at on the World Wide Web.

Poetry of the Universe. Robert Osserman. Anchor Books, 1995.

Circles in the Sky: Finding Topology with the Microwave Background Radiation.
Neil J. Cornish, David N. Spergel and Glenn D. Starkman in Classical and
Quantum Gravity, Vol. 15, No. 9, pages 2657-2670; September 1998. Preprint
available at on the World Wide Web.

Reconstructing the Global Topology of the Universe from the Cosmic Microwave
Background. Jeffrey R. Weeks in Classical and Quantum Gravity, Vol. 15, No.
9, pages 2599-2604; September 1998. Preprint available at
astro-ph/9802012 on the World Wide Web. Free software for exploring topology
is available at and
on the World Wide Web.





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