## 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.
**