The Shape of the Universe
Stacy Hoehn
Vanderbilt University
stacy.hoehn@vanderbilt.edu
October 13, 2009
Stacy Hoehn The Shape of the Universe
What Does “Shape” Mean to a Topologist?
In topology, two objects have the same shape if one can be
deformed into the other without cutting or gluing. Objects with
the same shape are called homeomorphic.
Stacy Hoehn The Shape of the Universe
What Does “Shape” Mean to a Topologist?
In topology, two objects have the same shape if one can be
deformed into the other without cutting or gluing. Objects with
the same shape are called homeomorphic.
Stretching, shrinking, bending, and twisting are allowed.
Stacy Hoehn The Shape of the Universe
What Does “Shape” Mean to a Topologist?
In topology, two objects have the same shape if one can be
deformed into the other without cutting or gluing. Objects with
the same shape are called homeomorphic.
Stretching, shrinking, bending, and twisting are allowed.
Examples of Homeomorphic Objects:
Stacy Hoehn The Shape of the Universe
Homeomorphic or Not?
A doughnut and a coffee cup are homeomorphic.
Stacy Hoehn The Shape of the Universe
Homeomorphic or Not?
A doughnut and a coffee cup are homeomorphic.
A torus and a sphere are not homeomorphic.
Stacy Hoehn The Shape of the Universe
Surfaces
Even though the torus and sphere are not homeomorphic, they do
have something in common.
Stacy Hoehn The Shape of the Universe
Surfaces
Even though the torus and sphere are not homeomorphic, they do
have something in common.
The area near any point just looks like a solid 2-dimensional disk.
Stacy Hoehn The Shape of the Universe
Surfaces
Even though the torus and sphere are not homeomorphic, they do
have something in common.
The area near any point just looks like a solid 2-dimensional disk.
The torus and the sphere are both called surfaces (or 2-manifolds)
because they share this property.
Stacy Hoehn The Shape of the Universe
Going Up a Dimension
Definition
If the area near any point in a space looks like a solid
3-dimensional ball, the space is called a 3-manifold.
Stacy Hoehn The Shape of the Universe
Going Up a Dimension
Definition
If the area near any point in a space looks like a solid
3-dimensional ball, the space is called a 3-manifold.
Examples:
R3
The Universe
Stacy Hoehn The Shape of the Universe
The Surface of the Earth
The surface of the Earth is a surface (2-manifold).
How can we eliminate the infinite plane and torus as possibilities
for the shape of the surface of the Earth? What other surfaces are
there?
Stacy Hoehn The Shape of the Universe
The Torus
To help us visualize the other surfaces (and eventually
3-manifolds), we will first view the torus a little bit differently. We
will construct a torus by gluing together opposite edges of a square.
Stacy Hoehn The Shape of the Universe
The Torus
To help us visualize the other surfaces (and eventually
3-manifolds), we will first view the torus a little bit differently. We
will construct a torus by gluing together opposite edges of a square.
This square, with its opposite sides identified, helps us depict the
torus in the plane.
Stacy Hoehn The Shape of the Universe
The Torus (continued)
Tic-Tac-Toe on the Torus
Does anyone win?
0X
0 X
X 0
Stacy Hoehn The Shape of the Universe
The Torus (continued)
What would you see if you were a two-dimensional being living in a
torus?
Stacy Hoehn The Shape of the Universe
The Torus (continued)
What would you see if you were a two-dimensional being living in a
torus?
You would see copies of yourself in every direction, as far as your
eye could see!
Stacy Hoehn The Shape of the Universe
The Möbius Band
A Möbius band is constructed from a square by gluing the left side
to the right side of the square after performing a half-twist.
Stacy Hoehn The Shape of the Universe
The Möbius Band (continued)
A Möbius band contains an orientation-reversing curve. Clockwise
becomes counterclockwise along this curve!
Stacy Hoehn The Shape of the Universe
The Klein Bottle
A Klein bottle is constructed from a square by gluing together the
left and right edges the same way as for a torus, but now the top
edge is flipped before being glued to the bottom edge.
Stacy Hoehn The Shape of the Universe
The Klein Bottle
A Klein bottle is constructed from a square by gluing together the
left and right edges the same way as for a torus, but now the top
edge is flipped before being glued to the bottom edge.
The Klein bottle is a surface.
Stacy Hoehn The Shape of the Universe
The Klein Bottle (continued)
Tic-Tac-Toe on the Klein Bottle
Does anyone win?
0X
00 X
X
Stacy Hoehn The Shape of the Universe
The Klein Bottle (continued)
What would you see if you were a two-dimensional being living in a
Klein bottle?
Stacy Hoehn The Shape of the Universe
The Klein Bottle (continued)
What would you see if you were a two-dimensional being living in a
Klein bottle?
You would see copies of yourself in every direction, but sometimes
you would be flipped!
Stacy Hoehn The Shape of the Universe
The Klein Bottle (continued)
The Klein bottle contains an orientation-reversing curve since it
contains a Möbius band.
Surfaces that contain an orientation-reversing curve are called
nonorientable. Surfaces that do not contain an
orientation-reversing curve are called orientable.
Stacy Hoehn The Shape of the Universe
The Shape of the Universe
No matter where we have been in the universe so far, if we choose
a spot and travel out from it a short distance in all directions, we
enclose a space that resembles a solid 3-dimensional ball. Thus,
the universe appears to be some 3-manifold. But which 3-manifold
is it?
Stacy Hoehn The Shape of the Universe
Narrowing Down the Possibilities
Scientists have measured the amount of cosmic microwave
background radiation in the universe, and they have found that it
is distributed surprisingly uniformly.
Stacy Hoehn The Shape of the Universe
Narrowing Down the Possibilities
Scientists have measured the amount of cosmic microwave
background radiation in the universe, and they have found that it
is distributed surprisingly uniformly.
This limits the geometries (notions of distance, angles, and
curvature) that can be placed on the universe’s 3-manifold to the
following:
spherical geometry with positive curvature
Euclidean geometry with zero curvature
hyperbolic geometry with negative curvature.
Stacy Hoehn The Shape of the Universe
Curvature
In Euclidean geometry, the sum of the angles in a triangle is 180
degrees. Meanwhile, in spherical geometry, the sum of the angles
is more than 180 degrees, and in hyperbolic geometry, the sum of
angles is less than 180 degrees.
Stacy Hoehn The Shape of the Universe
Geometry and the Eventual Fate of the Universe
Spherical Geometry ⇒ The universe will eventually recollapse.
Stacy Hoehn The Shape of the Universe
Geometry and the Eventual Fate of the Universe
Spherical Geometry ⇒ The universe will eventually recollapse.
Euclidean Geometry ⇒ The universe will continue to expand
forever, but just barely (i.e. the rate of expansion will
approach 0.)
Stacy Hoehn The Shape of the Universe
Geometry and the Eventual Fate of the Universe
Spherical Geometry ⇒ The universe will eventually recollapse.
Euclidean Geometry ⇒ The universe will continue to expand
forever, but just barely (i.e. the rate of expansion will
approach 0.)
Hyperbolic Geometry ⇒ The universe will continue to expand
forever, gradually approaching a (positive) constant rate of
expansion.
Stacy Hoehn The Shape of the Universe
Which geometry does the universe have?
In the early 1800s, Carl Gauss computed the angles formed by 3
mountain peaks in Germany found that they added up to 180
degrees. However, this does not necessarily imply that the universe
is Euclidean due to possible approximation errors when measuring.
Stacy Hoehn The Shape of the Universe
Which geometry does the universe have?
In the early 1800s, Carl Gauss computed the angles formed by 3
mountain peaks in Germany found that they added up to 180
degrees. However, this does not necessarily imply that the universe
is Euclidean due to possible approximation errors when measuring.
Data from a NASA probe in 2001 suggests that the curvature of
the universe is very close to 0. This either means that we live in a
Euclidean universe or we live in a spherical or hyperbolic universe
with extremely low curvature.
Stacy Hoehn The Shape of the Universe
Euclidean 3-Manifolds
If we assume that the universe is a Euclidean 3-manifold, does this
help us determine which 3-manifold the universe is?
Stacy Hoehn The Shape of the Universe
Euclidean 3-Manifolds
If we assume that the universe is a Euclidean 3-manifold, does this
help us determine which 3-manifold the universe is?
Yes! It narrows the number of possibilities down from infinity to
18!
Stacy Hoehn The Shape of the Universe
Euclidean 3-Manifolds
If we assume that the universe is a Euclidean 3-manifold, does this
help us determine which 3-manifold the universe is?
Yes! It narrows the number of possibilities down from infinity to
18!
Theorem
There are exactly 18 Euclidean 3-manifolds.
6 are compact (finite) and orientable
4 are compact (finite) and nonorientable
4 are noncompact (infinite) and orientable
4 are noncompact (infinite) and nonorientable
Stacy Hoehn The Shape of the Universe
Nonorientable Euclidean 3-Manifolds
The 8 nonorientable Euclidean 3-manifolds all contain an
orientation-reversing loop. If you were to fly from Earth along such
a loop, you would eventually return home with your orientation
reversed. It would appear that you had returned to a mirror image
of Earth.
If the universe was nonorientable, there would be strange physical
consequences that have not yet been observed. While they could
be happening outside of our field of vision, it is unlikely that our
universe is nonorientable. It is more likely that the universe is one
of the 10 orientable Euclidean 3-manifolds.
Stacy Hoehn The Shape of the Universe
The 3-Torus
The simplest orientable, compact, Euclidean 3-manifold is the
3-torus. It is a generalization of the torus in a higher dimension.
Instead of gluing together opposite edges of a square, the opposite
faces of a cube are joined.
Stacy Hoehn The Shape of the Universe
The 3-Torus (continued)
If you were somehow in the 3-torus and looked around, you would
see copies of yourself in each direction, and past these copies,
other copies would be visible as far as the eye could see.
Stacy Hoehn The Shape of the Universe
The 3-Torus (continued)
If you were somehow in the 3-torus and looked around, you would
see copies of yourself in each direction, and past these copies,
other copies would be visible as far as the eye could see.
If the universe is a 3-torus, you could fly from Earth in a particular
direction and, without ever changing course, eventually return
home.
Stacy Hoehn The Shape of the Universe
The Quarter-Twist and Half-Twist 3-Manifolds
In the quarter-twist and half-twist 3-manifolds, four of the faces of
the cube are glued together just as for the 3-torus.
The front and back faces, however, are glued together after a twist
of 90 degrees (quarter-twist) or 180 degrees (half-twist).
Stacy Hoehn The Shape of the Universe
The Quarter-Twist and Half-Twist 3-Manifolds
In the quarter-twist and half-twist 3-manifolds, four of the faces of
the cube are glued together just as for the 3-torus.
The front and back faces, however, are glued together after a twist
of 90 degrees (quarter-twist) or 180 degrees (half-twist).
If you were inside the cube for the quarter-twist manifold and
stared out the front or back face, you would see copy after copy of
yourself, each one a 90-degree rotation of the preceding copy.
Stacy Hoehn The Shape of the Universe
The Sixth-Twist and Third-Twist 3-Manifolds
The sixth-twist and third-twist 3-manifolds are both obtained by
gluing faces on a hexagonal prism instead of a cube. Each
parallelogram face is glued to the face directly opposite it.
The two hexagonal faces are then glued together after a twist of
60 degrees (sixth-twist) or 120 degrees (third-twist).
Stacy Hoehn The Shape of the Universe
The Sixth-Twist and Third-Twist 3-Manifolds
The sixth-twist and third-twist 3-manifolds are both obtained by
gluing faces on a hexagonal prism instead of a cube. Each
parallelogram face is glued to the face directly opposite it.
The two hexagonal faces are then glued together after a twist of
60 degrees (sixth-twist) or 120 degrees (third-twist).
If you looked out of one of the hexagonal faces of the prism for the
sixth-twist manifold, you would see copy after copy of yourself,
each rotated 60 degrees more than the preceding copy.
Stacy Hoehn The Shape of the Universe
The Double Cube 3-Manifold
The last compact, orientable, Euclidean 3-manifold is the Double
Cube manifold.
You would see yourself with a very peculiar perspective in this
3-manifold!
Stacy Hoehn The Shape of the Universe
Non-Compact, Orientable, Euclidean 3-Manifolds
It is likely that the universe has the shape of one of the six
compact, orientable, Euclidean 3-manifolds that we just described.
However, there are also 4 non-compact, orientable, Euclidean
3-manifolds.
Stacy Hoehn The Shape of the Universe
Non-Compact, Orientable, Euclidean 3-Manifolds
It is likely that the universe has the shape of one of the six
compact, orientable, Euclidean 3-manifolds that we just described.
However, there are also 4 non-compact, orientable, Euclidean
3-manifolds.
The simplest one of these is 3-dimensional Euclidean space, R3 .
Stacy Hoehn The Shape of the Universe
Non-Compact, Orientable, Euclidean 3-Manifolds
It is likely that the universe has the shape of one of the six
compact, orientable, Euclidean 3-manifolds that we just described.
However, there are also 4 non-compact, orientable, Euclidean
3-manifolds.
The simplest one of these is 3-dimensional Euclidean space, R3 .
The others are called the Slab Space, the Chimney Space, and the
Twisted Chimney Space.
Stacy Hoehn The Shape of the Universe
Non-Compact, Orientable, Euclidean 3-Manifolds
It is likely that the universe has the shape of one of the six
compact, orientable, Euclidean 3-manifolds that we just described.
However, there are also 4 non-compact, orientable, Euclidean
3-manifolds.
The simplest one of these is 3-dimensional Euclidean space, R3 .
The others are called the Slab Space, the Chimney Space, and the
Twisted Chimney Space.
Many cosmologists believe that the universe is not infinite in
nature, but we still must consider these 4 non-compact options as
possibilities until there is substantial evidence against them.
Stacy Hoehn The Shape of the Universe
Can We Narrow Down the Possibilities Even Further?
The simplest procedure is to look for copies of our galaxy, the
Milky Way, in the night sky. If we find copies, we can look at their
pattern to determine the gluing diagram for the universe.
Stacy Hoehn The Shape of the Universe
Can We Narrow Down the Possibilities Even Further?
The simplest procedure is to look for copies of our galaxy, the
Milky Way, in the night sky. If we find copies, we can look at their
pattern to determine the gluing diagram for the universe.
Possible Problems:
Light travels at a finite speed, so looking out into the
universe, we are looking back in time. Even if we someday
find a copy of our galaxy, we may not recognize it because it
might have looked different in its younger years.
The fundamental domain for the universe is huge (possibly
bigger than our sphere of vision) and is continuing to expand.
Stacy Hoehn The Shape of the Universe
More Information
Take MATH 242: Topology of Surfaces in the spring!
Stacy Hoehn The Shape of the Universe
More Information
Take MATH 242: Topology of Surfaces in the spring!
Adams, Colin, and Robert Franzosa. Introduction to
Topology: Pure and Applied. Upper Saddle River: Prentice
Hall, 2007.
Adams, Colin, and Joey Shapiro. “The Shape of the Universe:
Ten Possibilities.” American Scientist. 89 (2001), no. 5,
443-453.
Weeks, Jeffrey. The Shape of Space: How to Visualize
Surfaces and Three-Dimensional Manifolds. New York:
Marcel Dekker, Inc., 1985.
Stacy Hoehn The Shape of the Universe
