Dr. Sarah's 3-d into 4-d Homer Lab - Counts as 2 Labs

Homer Changing Dimensions in Treehouse of Horror VI 3F04 (10/30/95)

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 when he changed dimensions, he became 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. Recall that 2-d Marge had problems understanding 3-d Homer, since in her world there didn't seem to be any room for a 3rd dimension. Yet, she could gain some 3-d intuition by envisioning 2-d parts of 3-d Homer, and by noticing strange things happening (like objects appearing and disappearing suddenly into walls) that couldn't be explained by just her 2-d universe. It is hard for us to envision a 4-d Homer because our brains are wired for 3-d understanding. We will try to gain some intuition by understanding 3-d slices and shadows of 4-d, corresponding to the way we helped 2-d Marge understand 3-d. Recall from class that physicists think that the universe is higher dimensional, since things like string theory, the big bang theory, and black holes seem to need them.

• Go to Text transcript of 3-d Homer segment and Did You Notice? by James A. Cherry, and find the equation where numbers are raised to a power of 12. Write this down here.

• Why is the left hand side of the equation equal to an odd number?

• Why is the right hand side of the equation equal to an even number?

• Use the answers to the last two questions to explain why the equation cannot hold.

• Name another reason from Andrew Wiles' mathematics that the equation cannot hold.

Answer each Question with BOTH a Sequence of Pictures and also with Words

• What might one piece of Homer's hair look like in 4-d when he changes dimensions (in 3-d, it looks like a 1-d line to the naked eye)?

• What might one layer of Homer's skin look like in 4-d when he changes dimensions (in 3-d, it looks like a 2-d piece of paper to the naked eye)?

• What might one tiny birthmark look like in 4-d when he changes dimensions (in 3-d, it looks like a 0-d point to the naked eye)?

From INTERNET EXPLORER, go through Davide Cervone's talk on The Cube and the Hypercube: Rotations and Slices. If controls appear below a picture that means that it is a movie. Play the movie by hitting the key (second from the right) that looks like a filled in triangle with a greater than sign on its right. Answer each Question with BOTH a Sequence of Pictures and also with Words

• How is a cube formed from a line?

• How is a hypercube formed from a cube?

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

• How many "faces" (3-d boxes) does a hypercube have? Do not draw a picture.

Two Letters due the day of the final exam turned in with this completed sheet:

• Using my comments, revise your letter from Homer to his wife Marge discussing his jump from 2-d into 3-d from Dr. Sarah's 2-d into 3-d Homer Lab for Internet Explorer (turn in your original letter too).
• Write a letter from Homer to his wife Marge discussing his jump from 3-d into 4-d. Think about what pieces of your first 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 above questions. Be sure to describe the change in Homer's appearance and to use the writing checklist.