Consider a cube in Euclidean 3-space with the opposite faces glued straight across. This forms a 3-torus. Notice that while we can understand the gluing instructions, we cannot actually glue the space because we would need a 4th physical dimension to do so. As an analogy, think back to 2D Marge who could understand the gluing instructions to form a cube but could not visualize it. She could not even understand how there could be enough space for the gluings to take place. We are in a similar situation when trying to understand the shape of the universe. Yet, we can still understand the properties of this space, understand what it is like to live inside of it, and even devise experiments to test and see whether this is the shape of our universe (similar to Greeks who discovered that the earth was round).

In the above figure, I have drawn a closed straight path that starts from A on the bottom right edge and then hits the middle of the front face at B. It continues from the middle of the back face (since the front face is glued to the back face) and finishes at the middle of the top left edge at a point which is glued to A (via the top and bottom face gluings as well as the left and right side face gluings).

In this picture, we glue pieces of the 3-torus together, by identifying opposite edges. Notice that the gluing of the top and bottom faces and the left and right faces reveal a sphere sitting inside of it. The visualization technique is similar to torus tick-tac-toe where the square above the top right corner was the same as the bottom right square, but instead of squares, we visualize identified blocks. Just above the top right corner, we can draw the figure in the bottom right corner, because they are the same via the gluings. Hence, if we think of a tiling view, then we can see that the 4 parts of the sphere glue together to form a regular sphere.

We can also visualize life inside of a torus universe. This requires the same type of imagination that we used to visualize life in a 2D universe.

The Flatlanders can travel about their flat 2-torus universe without falling off an edge. When looking at a fundamental domain, we must imagine that its edges are glued together in higher dimensions.

Here is a picture of life inside of a 3-torus, with a view that is analogous to the above square that gets glued to form a 2-torus. Here, this cube gets glued to form a 3-torus. Even though the 3-torus is finite, we have the illusion of flying in an infinite space because we never reach an edge. The same thing happens on a 2-torus or on the surface of the sphere because we keep going around and around, passing where we have been before. There are only two stars in this universe but we see each one over and over, like a hall of mirrors.

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There are only 10 Euclidean possibilities for the shape of a closed Euclidean universe -- 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. We saw one example like this in the Futurama episode I, Roommate when Fry, Leela, and Bender were looking at an apartment resembling Escher's 1953 Relativity work.

In this quarter-turn space, unmarked walls are glued to one another in the simple, straight-across way while the marked side shows that we should glue that side and its opposite side with a rotation by 90 degrees (a quarter of a turn and hence the quarter-turn universe). We identify corresponding squares because squares that are filled in with the same pattern get glued together. The quarter-turn space is a Euclidean universe.

Here is a picture of life inside of a Klein space. We start with a cube and identify 2 of the 3 sets of opposite faces in the usual straight-across way. We glue the 3rd set of faces with a reflection across a line through the center of each face, just as in the Klein bottle. Even though this Euclidean space is finite, we have the illusion of flying in an infinite space because we never reach an edge. There are only two stars in this universe but we see each one over and over. In the Klein space, we fly one way, and see ships in neighboring rows flying in opposite directions. The mirrored images turn, as we do, to fly along paths that seem to cross ours, but they can never hit us - that's impossible in this space.

The above are just a few of the 10 Euclidean possibilities for the shape of a closed Euclidean universe. The others are similarly obtained by gluing together opposite faces of a cube. Use the reading above to help you answer these questions:

Question 11 Each Euclidean 3-torus below has a familiar solid drawn in it once you perform you gluing instructions from the last question. For example, the fourth picture generates a cone because the top and bottom face gluing yields 2 half cones, and the left and right faces are glued together forming 1 full cone. Give the name of each of the other surfaces.

Question 13 Recall that there are 9 other shapes that we can obtain as possibilities for a Euclidean universe via gluing with twists and turns. The picture below shows the 3-torus and 3 additional Euclidean spaces. Write the name of each space whose fundamental domain appears below. Unmarked walls are glued to one another in the simple, straight-across way while the marked side shows whether to glue straight across, with a reflection or a rotation by identifying corresponding squares (squares that are filled in the same get glued together). (Hint: The answers are 3-torus, quarter-turn space, half-turn space, and Klein space, but not in that order.)