Dr. Sarah's Homer Part 2 for INTERNET EXPLORER

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

In Homer Part 1 we explored Homer's transition. Yet, in most respects, the Simpsons already represent a 3-d world, while an amoeba confined to a thin layer of water on a slide in biology class, seems to more accurately represent a 2-d world. All of the Simpson's actions indicate that they have use of three full dimensions for movement (for example, drawing on a blackboard requires 3 dimensions, and so does the way they hold out their arms, talk to each other, ...). In addition, they use perspective viewing of objects.

We know that experts think that the universe has many more physical dimensions than we directly experience. In order to try and understand the idea of more physical dimensions, we will step back and try to address an easier, but related question: How could a 2-d creature understand the 3rd dimension? In the process, we will see some ideas that at first appear counterintuitive. We will look at 3-d objects in new ways, in an attempt to develop visualization skills that we will need in order to understand the 4th physical dimension. Next week we will revisit the intuition and ideas we have learned within the context of the 4th dimension, and will look at the practical utility of these abstract notions.

So, for now, assume that the Simpson's really were 2-d creatures living in an x-y plane of some blackboard, as Dr. Frink suggests, and that Homer and Bart had made the transformation to 3-d creatures. While a 2-d Marge can't really understand the 3rd dimension and would feel like there isn't any room for another dimension, she could see weird behavior occurring that suggests that the 3rd dimension exists (for example the "wall" that Homer disappeared into could not be explained using only 2 dimensions). 2-d Marge wouldn't be able to comprehend the concept of depth or an entire 3-d Homer, since only 2-d pieces would make sense to her. In the process, we will continue to explore the relationships between geometry, visualization, art, history, philosophy and physics.

Taken from Hyperspace & A Theory of Everything, by Dr. Michio Kaku When I was a child, I used to visit the Japanese Tea Garden in San Francisco. I would spend hours fascinated by the carp, who lived in a very shallow pond just inches beneath the lily pads, just beneath my fingers, totally oblivious to the universe above them.

I would ask myself a question only a child could ask: what would it be like to be a carp? What a strange world it would be! I imagined that the pond would be an entire universe, one that is two-dimensional in space. The carp would only be able to swim forwards and backwards, and left and right. But I imagined that the concept of "up", beyond the lily pads, would be totally alien to them. Any carp scientist daring to talk about "hyperspace", i.e. the third dimension "above" the pond, would immediately be labeled a crank.

I wondered what would happen if I could reach down and grab a carp scientist and lift it up into hyperspace. I thought what a wondrous story the scientist would tell the others! The carp would babble on about unbelievable new laws of physics: beings who could move without fins. Beings who could breathe without gills. Beings who could emit sounds without bubbles.

I then wondered: how would a carp scientist know about our existence? One day it rained, and I saw the rain drops forming gentle ripples on the surface of the pond. Then I understood. The carp could see rippling shadows on the surface of the pond. The third dimension would be invisible to them, but vibrations in the third dimensions would be clearly visible. These ripples might even be felt by the carp, who would invent a silly concept to describe this, called "force." They might even give these "forces" cute names, such as light and gravity. We would laugh at them, because, of course, we know there is no "force" at all, just the rippling of the water.

Today, many physicists believe that we are the carp swimming in our tiny pond, blissfully unaware of invisible, unseen universes hovering just above us in hyperspace. We spend out life in three spatial dimensions, confident that what we can see with our telescopes is all there is, ignorant of the possibility of 10 dimensional hyperspace. Although these higher dimensions are invisible, their "ripples" can clearly be seen and felt. We call these ripples gravity and light. The theory of hyperspace, however, languished for many decades for lack of any physical proof or application. But the theory, once considered the province of eccentrics and mystics, is being revived for a simple reason: it may hold the key to the greatest theory of all time, the "theory of everything".

Taken from David Henderson's Experiencing Geometry in the Euclidean, Spherical, and Hyperbolic Spaces Try to imagine the possibility of our physical universe being a 3-sphere in 4-space. It is the same kind of imagination a 2-dimensional being would need in order to imagine that it was on a plane or 2-sphere (ordinary surface of a sphere) in 3-space.

How would you explain 3-space to a person living in two dimensions ?

Always try to imagine how things would look from the person's point of view. A good example of how this type of thinking works is to look at an insect called a water strider. The water strider walks on the surface of a pond and has a very 2-dimensional perception of the world around it. To the water strider, there is no up or down; its whole world consists of the 2-dimensional plane of the water. The water strider is very sensitive to motion and vibration on the water's surface, but it can be approached from above or below without its knowledge. Hungry birds and fish take advantage of this fact. For more discussion of water striders and other animals with their own varieties of intrinsic observations, see the delightful book, The View from the Oak, by Judith and Herbert Kohl [Na: Kohl and Kohl, 1977].

Think about the question (How would you explain 3-space to a person living in two dimensions?) in terms of this example: The person depicted in Figure 12.1 lives in a 2-dimensional plane. The person is wearing a mitten on the right hand. Notice that there is no front or back side to the mitten for the 2-D person. The mitten is just a thick line around the hand.

Figure 12.1. 2-dimensional person with mitten.

Suppose that you approach the plane, remove the mitten, and put it on the 2 -D person's left hand. There's no way within 2-space to move the mitten to fit the other hand. If the 2-d person tried to fit the glove onto their left hand, the thumb would point the wrong way. So, you take the mitten off of the 2-D plane, flip it over in 3-space, and then put it back on the plane around the left hand. The 2-D person has no experience of three dimensions but can see the result — the mitten disappears from the right hand, the mitten is gone for a moment, and then it is on the left hand.

Figure 12.2. Where did the mitten go?

How would you explain to the 2-D person what happened to the mitten?

This person's 2-dimensional experience is very much like the experience of a water strider insect. A water strider walks on the surface of a pond and has a very 2-dimensional perception of the universe around it. To the water strider, there is no up or down; its whole universe consists of the surface of the water. Similarly, for the 2-D person there is no front or back; the entire universe is the 2-dimensional plane.

Living in a 2-D world, the 2-D person can easily understand any figures in 2-space, including planes. In order to explain a notion such as "perpendicular," we could ask the 2-D person to think about the thumb and fingers on one hand.

Figure 12.3. The 2-D person sees "perpendicular."

A person living in a 2-D world cannot directly experience three dimensions, just as we are unable to directly experience four dimensions. Yet, with some help from you, the 2-D person can begin to imagine three dimensions just as we can imagine four dimensions. One goal of this problem is to try to gain a better understanding of what our experience of 4-space might be. Think about what four dimensions might be like, and you may have ideas about the kinds of questions the 2-D person will have about three dimensions. You may know some answers, as well. The problem is finding a way to talk about them. Be creative!

One important thing to keep in mind is that it is possible to have images in our minds of things we cannot see. For example, when we look at a sphere, we can see only roughly half of it, but we can and do have an image of the entire sphere in our minds. We even have an image of the inside of the sphere, but it is impossible to actually see the entire inside or outside of the sphere all at once. Another similar example: sit in your room, close your eyes, and try to imagine the entire room. It is likely that you will have an image of the entire room, even though you can never see it all at once. Without such images of the whole room it would be difficult to maneuver around the room. The same goes for your image of the whole of the chair you are sitting on or this book you are reading.

Assume that the 2-D person also has images of things that cannot be seen in their entirety. For example, the 2-D person may have an image of a circle. Within a 2-dimensional world, the entire circle cannot be seen all at once; the 2-D person can only see approximately half of the outside of the circle at a time and can not see the inside at all unless the circle is broken.

Figure 12.4. The 2-D person sees a circle.

However, from our position in 3-space we can see the entire circle including its inside. Carrying the distinction between what we can see and what we can imagine one step further, the 2-D person cannot see the entire circle but can imagine in the mind the whole circle including inside and out. Thus, the 2-D person can only imagine what we, from three dimensions, can directly see. So, the 2-D person's image of the entire circle is as if it were being viewed from the third dimension. It makes sense, then, that the image of the entire sphere that we have in our minds is a 4-D view of it, as if we were viewing it from the fourth dimension.

When we talk about the fourth dimension here, we are not talking about time which is often considered the fourth dimension. Here, we are talking about a fourth spatial dimension. A fuller description of our universe would require the addition of a time dimension onto whatever spatial dimensions one is considering.

Try to come up with ways to help the 2-D person imagine what happens to the mitten when it is taken out of the plane into 3-space. Draw upon the person's experience living in two dimensions, as well as some of your own experiences and attempts to imagine four dimensions.


Work with one other person and turn in one lab per group. For each of the following questions, read through carefully, watch the movies, read again, and drag through the movie in slow motion using your mouse. THEN, ANSWER EACH QUESTION WITH BOTH PICTURES AND WORDS:

In Davide Cervone's "Movie Slices of a Cube Passing Through Flatland" (the blackboard), pretend that 2-d Marge is standing in the middle of the right side of the gray shaded square (far enough to the right that the cube never whacks into her!). Notice that Marge's view of something passing through near her will be very different from your view of what actually passes through her plane since she will not be able to see outside of her plane, and she will only be able to see a part of her plane (think about standing outside a building - you can't see all 4 sides at once). Answer questions 1-4 using these ideas.

  1. What do we see in Marge's plane when the first cube passes through her blackboard?

  2. What would she see when the first cube passes through her blackboard?

  3. What do we see in Marge's plane when the second cube passes through her blackboard?

  4. What would she see when the second cube passes through her blackboard?

  5. Use Davide Cervone's "Dimensional Connections" to explain how a square can be formed from a line.

  6. Search in this text transcript of 3-d Homer segment and Did You Notice? by James A. Cherry for Professor Frink's description of how a cube can be formed from a square, and copy that down here.

  7. Davide Cervone's "Cube and Hypercube Movies". Read the text to the right of the first movie. Then watch the first movie and answer the following question: If 2-d Marge was standing at the bottom of the movie (near to the controls), then what would she see (as a shadow in her plane of existence) as the cube was formed from a square? (I find it best to answer this by repeating the shadow part of the movie.)

  8. There is an alternative way of forming a cube. Recall that we can form a cylinder by taking a piece of paper and gluing the left and right sides together. I could explain this to 2-d Marge by drawing a square and then giving the gluing instructions as follows: . While she wouldn't understand how there could be space to accomplish this feat, (since we need a 3rd dimension to actually glue the edges together to form the cylinder) she could still comprehend the gluing instructions. Take a look at Davide Cervone's "A Cube Unfolded" Explain which sides you would glue together in order to form a cube by labeling the gluing instructions on the figure below. (Hint: You will need to glue 7 sets of edges together, so you may want to give the instructions by labeling a set of 1s to be glued together, a set of 2s, and so on...)

  9. Go through the following movies to further help you develop visualization skills.
    Davide Cervone's "Perspective and Orthographic Views of a Cube"
    Davide Cervone's "Shadows of a Cube Rotating Above a Plane

    Some other questions
  10. Why couldn't Marge eat if her mouth was located in the middle of her head?

  11. Where would 2-d Marge's eyes have to be located for her to see things in her plane of existence? Why?

  12. How could 2-d Marge and 2-d Lisa pass each other? (Hint - think of them moving around in their 2-d plane-without-depth.)