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The Shape of Space Curriculum Materials

What is the shape of space?

What is the shape of space? What does that question even mean? What is the shape of the universe we live in? It is three-dimensional and hard to think about because it is so vast. The shape of the two-dimensional surface of our earth is easier to think about. Centuries ago, it was easy to believe that its surface was flat. It looks flat when we see only a small piece of it at a time. People wondered if the land went on forever, or if it had a boundary. Could they fall off the edge? Today we know. Its surface is like a sphere. It does not go on forever, but still it has no boundaries. Theoretically, we could travel in a straight line on the surface of our planet and we would eventually come back to the place we started from. Could something like this happen in our three-dimensional universe, too? Could we get in a spaceship, head off in a straight line and eventually find ourselves back in the place we started from? And how would we know? Could we be in the same situation looking at the universe as people centuries ago were looking at the earth?

Thinking in different dimensions

To think about the shape of our three-dimensional universe, we should first understand some possible shapes for a two-dimensional universe. The surface of Earth is a nice analogy, but not a perfect one. We are not actually confined to the surface as we sometimes think. When two people pass each other on the street, they have the choice of going to the side of one another or, though usually impractical, going over or under the other person. Like it or not, we are three-dimensional people living in a three-dimensional world. But, that doesn't mean we can't understand or visualize a two-dimensional universe. In fact, we have an advantage over the 2-D insiders, in that as 3-D outsiders, we can use the 3rd dimension to visualize the shape of their space. 2-D insiders, affectionately known as flatlanders, may be able to determine that their universe is a sphere or a torus, but might have difficulty imagining what a sphere or torus looks like. Thinking about how flatlanders visualize the shape of their space is very useful for thinking about how we might visualize ours.

What is a straight line?

When the insiders of any space take a look around their universe, they look in straight lines, or geodesics. Light travels on geodesics. A geodesic is a line that appears straight to an insider, though does not necessarily appear straight to an outsider. The geometry of the space determines what the geodesics look like to an outsider.

Geodesics on spheres are called great circles. They appear straight to 2-D insiders, but look like circles to 3-D outsiders. The equator of our earth is an example of a great circle. It appears straight to us when we are on the surface of the planet, but curved when viewed from space.

Geodesics on a plane appear straight to outsiders, as they do to insiders. A straight line drawn on a piece of paper appears straight no matter how far away or at what angle the paper is held, as long as the paper stays flat.

Inside of any two-dimensional surface, straight lines look the same -- namely, straight. However, depending on how the surface curves or bends, the same lines may not appear straight when viewed from the outside. In a similar way, what appears to be a straight line to us insiders of three-space, may appear curved to a 4-D outsider, depending on the geometry of the space.

Representing a two-dimensional space

There are many ways to visualize a space. Three common techniques are discussed here. One is a tiled picture, which gives an idea of what an insider might see. Objects that can be seen multiple times are shown multiple times. Another is a gluing diagram, which is also from the perspective of the insider, but shows each object only once. A third representation is a rolled up picture, which shows what a space looks like in the next higher dimension. This is an outsider's view.

Gluing diagrams for two-dimensional surfaces

For a two-dimensional surface, a gluing diagram represents the entire surface in one flat region. The edges of the region may be left plain to show a boundary or marked as glued to show where the surface connects to itself. Some simple gluing diagrams made with a square are:

[Disc glued] [Cylinder glued] [Moebius band glued] [Torus glued] [Klein Bottle glued] [Projective Plane (square)]

Each of these represents a different two-dimensional surface. Take a closer look at one of these diagrams:

[Klein Bottle glued]

There are markings on all edges of the square, so the surface has no boundary in any direction. Edges with the same type of marking are glued together. The markings on the top and bottom edges of the square show that a flatlander leaving the bottom of the square would come back in the top. The direction of the markings in a gluing diagram determine the orientation of the flatlander when coming back in the other side. The flipped markings on the left and right show that when a flatlander leaves the left side, it comes back in the right side mirror-reversed from top to bottom. A surface where this is possible is called non-orientable.

There are other ways to glue the edges of a square together besides the ways already shown, and there are other shapes that can be used besides a square:

[442] [huh] [Torus (hexagon)]

Tiling two-space

To make a tiled picture, take many copies of a gluing diagram and connect them at the appropriate marks. Tiling with the gluing diagram from the previous example gives the following tiled picture on the right:

[Klein Bottle glued] -> [Klein bottle tiling]

This type of picture is helpful when trying to visualize what an insider might see. The space's symmetries, like translation and glide reflection are also illustrated in this type of picture. Placing the capital letter R inside each square better demonstrates the symmetries of the space.

[Klein Bottle glued (with R)] -> [Klein bottle tiling]

Here are some examples tilings using different gluing diagrams.

[Torus tiling (square f.d.)] [442 tiling] [Projective plane tiling (square f.d.)] [Torus tiling tiling (hexagonal f.d.)]

Looking at a two-dimensional surface in three dimensions

A third useful way to visualize the shape of a space is to think about what it would look like sitting in the next higher dimension. So, for a two-dimensional space, we visualize it as the surface of a three-dimensional object.

To see what a surface looks like in three dimensions, just take a gluing diagram for the surface and glue it where it's marked! For example, take a simple gluing diagram with only one pair of edges glued with no flip:

[Cylinder glued]

The square can be rolled so that the two marked edges meet. The resulting shape would be...

[Cylinder glued to rolled] ... a cylinder!

If the top and bottom edges were also glued, then the shape in three dimensions would be ...

[Torus glued to rolled] ... a torus!

This visualization technique works for gluing diagrams with flips in them too.

[Moebius glued] [Moebius glued to rolled]

This is the gluing diagram for a Möbius strip.

The 3-D shape that comes from gluing the edges of the following familiar diagram is called a Klein bottle.

[Klein bottle glued] -> [sequence omitted] -> [Klein bottle rolled]

For more information on this type of surface, see http://www.geom.umn.edu/zoo/toptype/klein/

This visualization technique works well for two-dimensional surfaces because we are three-dimensional people and naturally like to think of objects in three dimensions. However, flatlanders themselves would have some trouble imagining this shape.

Summary of representing a surface

Remember that since we are 3-D people, we have an advantage over Flatlanders; our extra dimension allows us to see properties about their space that they can't see directly. We have several ways to visualize their space:
  1. a gluing diagram
  2. a tiled picture
  3. a rolled picture
These are all ways of thinking about the same space. The third way is where we usually get the name for the shape of the space - but that doesn't mean that that is the correct way to think of the shape. Being able to understand what each of these pictures means gives us a very good understanding of the shape of a two-dimensional space.

Representing three-dimensional spaces

Each of the ways presented for visualizing two-dimensional spaces can be extended into three dimensions. It is important to have a good understanding of what each of the three types of pictures means for two dimensions before jumping into three dimensions.

Gluing diagrams for three-dimensional surfaces

As in two-space, a gluing diagram represents a tiling, each tile being a a two-dimensional object such as a square. In three-space the tile is a three-dimensional object, such as a cube. The simplist gluing diagram for three-space is a cube having opposite edges glued with no change in orientation:

[3-Torus glued]

As in 2-D, when a traveler goes out one side, it comes back in the other side with the same marking, and the direction of the markings determines the orientability. Some non-orientable three-spaces are shown here:

[Klein bottle cross a circle, glued] [Quarter turn, glued]

Notice in the second diagram above, the left and right sides are glued with a quarter turn twist. An analogous gluing cannot be made in 2 dimensions. The only type of orientation-changing connection that can be made in 2-D is a mirror-reversing flip. In 3-D, the extra dimension can be used to make rotations connections. And, as in 2-D, there are many possible shapes for a fundamental domain.

[Cube] [Rhombic dodecahedron] [Hexagonal prism]

Representing a space - tiled picture

Tiled pictures can be made for three-space as well. The following tilings are shown in motion in the video The Shape of Space.

[3-torus tiling] [Klein bottle cross a circle tiling]

Being 3-D people ourselves, we can visualize this space as if we were in the space. This is usually the preferred method of visualizing three-space, while the "rolled picture" is often the most meaningful for visualizing two-space.

Representing a space - rolled into a higher dimension

The "rolled picture" would not be the easiest method for a flatlander to understand his space since he is a 2-D being and the rolled picture takes advantage of three dimensions. Similarly, rolling our 3-D space into a fourth dimension would not be the easiest way for us to visualize our space, though it can be done.

So what IS the shape of space?

What does all this have to do with the shape of our three-dimensional universe? Well, it is possible that our universe is not infinitly big but still a person could travel forever in any direction. Our universe could be both finite but boundless.

The following is from The Shape of Space video:

What seems to be a star in a distant galaxy could be our own sun. The light we receive from it could be light which left the sun billions of years ago, travelled around the universe, and is just now completing its trip. If we can someday find a pattern in the arrangement of the galaxies, then we will know the true shape of space.

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