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.

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

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:

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.

Here are some examples tilings using different gluing diagrams.

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:

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

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

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

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.

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.

- a gluing diagram
- a tiled picture
- a rolled picture

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:

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.

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.

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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Created: Tuesday, 01-Apr-97 17:45:20 ---
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