Shape of the Universe Worksheet - NAME_________________________________
    From the main web page, click on the class highlights page, and then on the Shape of the Universe link.

    First read the Real-life Applications of Related Material section and be sure to follow directions carefully.

  1. Spherical and hyperbolic geometries have many applications in real life. Describe the advantages of a hyperbolic map of the brain.

  2. Visualizing higher dimensions has real-life applications in the analysis of data. Each variable of measurement represents a physical dimension. Part A: Complete the Excel activity and show this to Dr. Sarah before you go on so that she can check this off here. Part B: How many dimensions is this space? Part C: How many patients were in this study?

    Go through the What is the 4th Physical Dimension? section before working on questions 3-6.

  3. How is a hypercube formed from a cube? Use an analogy similar to Professor Frink's description of how a cube is formed from a square.

  4. How many 3D box boundary pieces (which Davide Cervone calls "faces of a hypercube") does a hypercube have?
  5. Use your answer to the last question to explain why experts think a hypercube is not one of the possible shapes of our universe?

  6. What might one layer of Homer's skin look like in 4D if he were to change from 3D to 4D? (Hint: A layer of skin looks like a 2D piece of paper with holes or pores in it - think about what familiar food this would resemble if it gained a dimension.)

    Read the ENTIRE The Shape of our Universe section before working on the rest of the questions.

  7. Each Euclidean 2-torus or donut below has a familiar shape shaded on it. Draw a tiling view in order to complete the shape and then give the name of each shape (as I've done with the first donut). Remember that the top and bottom sides are identified, as are the left and right sides.

  8. Describe how a Euclidean 3-torus can be formed from a cube or box by explaining how you would glue the faces of a cube together. As we saw in the video, this is one of the possibilities for the shape of our universe.

  9. Each Euclidean 3-torus below has a familiar solid drawn in it once you perform you gluing instructions from the last question. Give the name of each surface.

  10. If our universe were a quarter-turn Euclidean space, how might we be able to tell? We would see repeated patters of stars in different directions which would differ by a what angle?

  11. 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.)

  12. By gluing together opposite pentagonal faces of a dodecahedron, we obtain a universe that satisfies the laws of spherical geometry. The number of spherical possibilities are infinite, but have been classified completely. Dr. Sarah's research relates to spherical universes. Build a dodecahedron in the back of the room. Notice that if you want to glue opposite faces together, we will need to do so with a twist, since the straight across gluing won't work. How many edges does the dodecahedron have? How many faces does it have?

  13. The MAP mission is trying to answer the question What is the shape of space? Use an effective search from to find the date when MAP (Microwave Anisotropy Probe) was actually launched. (It was delayed from its original launch plans in 2000, but was actually launched in 2001). To answer this question, give the exact phrases and words that you used for your successful search, and the date of the launch. Recall our effective searching techniques:
    Simpson*   -Bart  
    will return pages with anything after Simpson (Simpson, Simpson's, ...) but without Bart's name in it. Once you have found promising website links, then you can click on a link to go to that page. It would take a long time if we had to read through entire web pages to find info we want. There is a better way:
    Searching within a web page: Under EDIT, go to FIND..., type in one word you want to find in a web page.
    Searching within an adobe acrobat .pdf file Just above the page itself, look for the BINOCULARS, and click on them. Then type in the word you want to find in the page and then click on FIND.

See the accompanying Shape of the Universe lab
Dr. Sarah J. Greenwald, Appalachian State University
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