In this lesson, we will examine various "spaces", and what it means to have or not have an edge, or to be finite or infinite in volume.

1.) Let us start with one dimension. What would a one-dimensional space look like? Draw it in the space below.

Inhabit your one-dimensional Universe with 5 beings. What do the beings look like? Draw them in your one-dimensional space above. Number them from left to right 1 through 5. Can number 2 ever visit number 5 for dinner? According to each being, how many other beings exist in the Universe?

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Let us think about some of the different one-dimensional Universe we can construct.

Which are finite and which are infinite?

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Which have a boundary and which have no boundary?

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2.) Now let's move on to a two-dimensional Universe. Remember, that you are confined to remaining only on the two-dimensional surface. The fact that we can look and see the surface can sometimes form a 3-dimensional solid figure is cheating on our part!

Which are finite and which are infinite?

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Which have a boundary and which have no boundary?

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3.) Let us investigate movement on a torus and a sphere, as two possible shapes of our 2-dimensional universe. A torus is simply a "rolled up" plane with both ends placed together. You may think that making a torus out of an everyday plane like a piece of paper would be feasible, but it is quicker to use other ordinary materials. For instance, take some modeling clay and roll it out into a long cylinder. Then mold both ends together so that you have a "doughnut"-like object. You may even just simply take a piece of rubber hose and glue the two ends together. To have a model of a sphere, use a styrofoam ball.

Cut a piece of string about 5 ft long. Tightly tie one end of the string to a toothpick and stick it at some "starting point" on the torus or sphere. The string shows a track where you have roamed. Move all over the surface of the sphere and torus. Do you ever come to an edge? Are these Universes finite or infinite? |

4. ) Now let us try to go to a 3-dimensional universe. Imagine that indeed we used a piece of paper to create a torus. Lay out a piece of paper in front of you and make a ray going from the middle of the paper to the top. Keep going.

What happens as you reach the end of the paper? Now remember, that end of paper is not a boundary; it just simply continues on the bottom if it is rolled up and makes a torus!

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Now imagine that the whole universe was contained in one room; your classroom! This room has 3 dimensions, but add in our understanding of a torus. If your classroom became such a 3-dimensional Universe, then the left and right wall would represent the same points in space, the floor and ceiling would be the same points, and the front and back of the classroom would be the same, as well. Imagine yourself moving all over every surface of the classroom.

What would you see if:

You looked up?

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You looked to the left?

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You looked forward?

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Is this shape, called a 3-torus, finite or infinite? Does it have a boundary?

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It's quite possible that our Universe is finite in size, has no edges or boundaries. What seem to be distant galaxies may be images of our own Galaxy. Some of the light that enters our telescopes may be light that left our Galaxy billions of years ago and has made a complete trip around the Universe. If astronomers some day find a pattern to the arrangement of the galaxies, we would be able to deduce the true shape of our space.

Astronomers know what to look for in the Universe to discover if it is a 3-torus, for example. If the Universe is small enough that we can see the same region of space in two different directions in the sky, then the radiation arriving from those two directions will have approximately the same temperature pattern. Such an observation could tell us if the Universe is a 3-torus, or whether it has some other shape.

It would be nice to be able to see the shape of our Universe, however, we cannot do it. We are stuck on the surface. Yet, even though we are stuck on the surface, we can deduce the shape of the Universe with observation. We are all living on the surface of the universe. It was easy for folks to once think that Earth was the center of the universe, because everything is moving away from us and around us...from our perspective. What a sad thing to learn that everyone else, on every other planet around every other star in every other galaxy gets the same view -- and can come to the same conclusion.