Dr. Sarah's 3-d into 4-d Homer Lab

WQED Videos - Life by the Numbers

We'll watch excerpts from The Shape of the World Video and Mapping the earth, Global Positioning System, and understanding the shape of the universe. From the Life by the Numbers series. Seeing is Believing video from Life By the Numbers Series on perspective drawings and the 4th dimension.

For each of the following links, read thru carefully, watch the movies, read again, and drag through the movie in slow motion using your mouse:
Devide Cervone's Movie Slices of a Cube Passing Thru Flatland(the blackboard). Pretend that a 2-d creature is standing to the right of the cube (far enough to the right that the cube never whacks into it!) You may answer the following with a sequence of pictures or in words:

  • What would a 2-d creature see when the first cube passes through its blackboard?

  • What would a 2-d creature see when the second cube passes through its blackboard?

  • What would a 2-d creature see when the third cube passes through its blackboard?

    Davide Cervone's Movies of the Formation of the Cube and Hypercube

  • How is the cube formed from a square?

  • How is the hypercube formed from a cube?

    Davide Cervone's Faces of the Hypercube

  • How many faces (2-d sheets) does a cube have?

  • How many "faces" (3-d boxes) does a hypercube have?

    Davide Cervone's Movies of the Rotating Cube and Hypercube

    2-d to 3-d Homer

    While a 2-d Marge or a 2-d creature can't really understand the 3rd dimension, recall that when the 2-d creature sees the orange passing through its plane, it sees a point, then a curve getting bigger and bigger and then smaller and smaller until it becomes a point again and then disappears. The 2-d creature understands that something appears, changes and then disappears completely and in this way sees the existence of the 3rd dimension. Also, while it doesn't feel that there is room for the z-axis, it can understand that there could be some axis perpendicular to both the x and y axis at the same time that it just can't see. Also, the 2-d creature couldn't see an entire 3-d Homer. Instead, it could see a layer of skin, a piece of hair or slice views of his body.

    3-d to 4-d Homer

    In most respects, the Simpsons already represent a 3-d world. For example, when the Simpsons pass one another, they don't jump over each other as 2-d creatures would have to do. Pretend that Homer Simpson and family had been 3-d all along and that he changed into 4-d (length, width, height and some other physical dimension, call it w). Think about how Homer would be different if he changed from 3-d to 4-d.
  • What would happen to one piece of Homer's hair (ie it looks like a 1-d line to the naked eye)?

  • What would happen to one layer of Homer's skin (it looks like a 2-d piece of paper to the naked eye)?

  • What would happen to one tiny birthmark (it looks like a 0-d point to the naked eye)?

    Letter from 3-d into 4-d due Wednesday at 5pm turned in with this completed sheet:
    Write a letter from Homer to his wife Marge discussing his jump from 3-d into 4-d. Think about how you described Homer's jump from 2-d to 3-d in your last letter, and what pieces of this letter you could change to apply to a letter of Homer changing from 3-d into 4-d. In your letter, use ideas from class, movies and the answers to the questions. Be sure to describe the change in Homer's appearance and to use the writing checklist.