## Dr. Sarah's 2D Universes

Work with one other person and turn in one per group to be graded.

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 2D creature understand the 3rd dimension? In the process, we will see some ideas that at first appear counterintuitive. We will look at 3D objects in new ways in an attempt to develop visualization skills that we will need in order to understand the 4th physical dimension and the shape of the universe.

In Homer Part 1 we explored Homer's transition, and at the beginning of lab today, we reviewed answers to questions 1 and 2. For now, assume that the Simpson's really were 2D creatures living in an x-y plane of some blackboard, as Dr. Frink suggests, and that Homer and Bart had made the transformation to 3D creatures. While a 2D 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). 2D Marge wouldn't be able to comprehend the concept of depth or an entire 3D Homer, since only 2D pieces would make sense to her.

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 our 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 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 2D 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 2D person's left hand. There's no way within 2-space to move the mitten to fit the other hand. If the 2D 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 2D plane, flip it over in 3-space, and then put it back on the plane around the left hand. The 2D 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.
 How would you explain to the 2D person what happened to the mitten? Figure 12.2.Where did the mitten go?
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 2D person there is no front or back; the entire universe is the 2-dimensional plane.
 Living in a 2D world, the 2D person can easily understand any figures in 2-space, including planes. In order to explain a notion such as "perpendicular," we could ask the 2D person to think about the thumb and fingers on one hand. Figure 12.3. The 2D person sees "perpendicular."
A person living in a 2D world cannot directly experience three dimensions, just as we are unable to directly experience four dimensions. Yet, with some help from you, the 2D 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 2D 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 2D person also has images of things that cannot be seen in their entirety. For example, the 2D person may have an image of a circle. Within a 2-dimensional world, the entire circle cannot be seen all at once; the 2D 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 2D 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 2D person cannot see the entire circle but can imagine in the mind the whole circle including inside and out. Thus, the 2D person can only imagine what we, from three dimensions, can directly see. So, the 2D 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 4D 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 2D 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.

Click on the 2D Universes Link from the class highlights web page and then on the link below. Use the Play Button to watch Davide Cervone's Spheres Sliced in 2D. Notice that you can use the other buttons to play it slide by slide or to rewind or replay the movie.

As a sphere passes through Marge's 2D plane of existence, we would see (in her plane) a point turning into a circle which gets larger and larger then smaller and smaller until it turns into a point and finally disappears.

When we look at a building from the front, we see just one side or face of it. In order to see the entire building, we must walk all the way around it. Just as we cannot see an entire building all at once, 2D Marge cannot see an entire circle or 2D curve all at once. As a sphere passes through Marge's 2D plane of existence 2D Marge (standing to our right of the sphere) might see a point and then a curve (the part of the circle we see that would be visible to her) that gets closer to her, then farther from her, then turns into a point that finally disappears. This would seem very strange to Marge as she would be unable to explain this behavior. In fact, this behavior does not make sense in 2 dimensions - things don't just appear and disappear. A 3rd dimension would be needed to explain the behavior.
Go through the following movies by using the play button and reading the explanations to further help you develop visualization skills that we will need in order to understand the shape of the universe.

Davide Cervone's Perspective and Orthographic Views of a Cube
From Davide Cervone's Orthographic and Stereographic Projections
"This movie shows two projections of a cube: on the left, the view is in perspective, where parts that are farther away are smaller; on the right is an orthographic view, where items are always the same size no matter how far away they are. The orthographic view is what you would see as a shadow cast from a light source that is infinitely far away (so that the light rays are parallel), while the perspective view comes from a light source that are finitely far away, so that the light rays are diverging. We begin with a view that you recognize as a view of a cube, and then rotate so that you are looking directly at a face of the cube. In the perspective view, you see a square within a square (the front face is a large square, and the back face is a smaller square); in the orthographic view, however, the front and back squares are the same size, and are one on top of the other, so you seem to see only one square. The remaining four sides appear as trapezoids in the perspective view; but, in the orthographic view, they are flattened out to simple line segments, since these sides are parallel to our line of sight. As the movie continues, the cube rotates. The view on the left shows the rotation clearly, but we have to think harder about the rotation on the right. If you track the colors carefully, you can see which parts of the two-dimensional shadow correspond to which faces of the cube. This is the "revolving door" illusion, in which we seem to see squares moving past each other. We stop rotating at a view where we are looking directly at an edge of the cube; a picture we might never have considered to be a view of a cube. Before moving to the next rotation, we back up just a bit to make it easier to see the different faces of the cube, and then rotate the cube in a different direction, and end up looking directly at the corner of the cube (along its long diagonal). In the orthographic view, we see a hexagon crossed by six lines meeting at the center. This, too, is an unexpected view of the cube. Note that both the closest and farthest corners appear at the center of the hexagon. Now we shrink the colored edges down to simple, black edges, and follow the rotations again in reverse. Although the orthographic views seem harder to understand, they will help us to interpret the views of the hypercube"

Davide Cervone's Rotating Cube
From Davide Cervone's Orthographic and Stereographic Projections
"This movie shows how projections of a cube can be made more understandable by looking at a sequence of images as the cube rotates above the plane of the projection. The various shadows make it clearer which parts are in front and which are behind. We begin with a view of the 3D cube above the projection plane, with the light source above. The relation between the shadow and the rotating object is quite clear. After seeing the cube rotate, we move to a new viewpoint where all we can see is the shadow; we have to imagine the rotating cube from these 2D images. "

2D Marge would think that the shadow behaves in ways that are impossible since the intersection movements are unlike anything that she would have ever experienced.

Davide Cervone's 2D Shadow of a Rotating Cube
From Davide Cervone's Orthographic and Stereographic Projections
"This movie shows the two-dimensional shadow of a cube as it rotates. Two of the faces are colored to help you follow how the shadow changes as the cube turns in space. At the start of the movie, the red square is closest to the light source (or viewer), and so it appears larger, while the blue face is farther away, and seems smaller. As the cube rotates, the blue square seems to shift to the side and both squares appear to become distorted. On the cube itself, they are still squares, but in the two-dimensional shadow, their images are trapezoids. As the cube rotates further, the blue square "pops" through the side of the red square, and eventually is completely outside it. The red square flattens out and "turns inside out" as we move from seeing one side of this face of the cube to seeing the other side. In a view of the cube where the red and blue faces are equally sized trapezoids, we see that these two faces are now the sides of the cube. As the cube turns further, the blue faces comes to the front, and is the largest square, while the red one moves to the back, and becomes smaller: the red and blue have interchanged positions. The rotation continues through the sequence again until the colored faces are back at the their starting positions. We can easily reconstruct in our minds the three-dimensional cube from these two-dimensional pictures. The hard part is actually thinking of them as flat images! But this is what we have to do in order to make sense of the next movie, which shows the three-dimensional shadows of a hypercube rotating in four dimensions."
Answer the questions with your partner, after discussing them and following the directions carefully. You should both write down the answers, but you will just turn in one per group to be graded. For now, assume that the Simpson's really were 2D creatures living in an x-y plane of some blackboard, as Dr. Frink suggests, and that they really behave like 2D creatures would behave. They cannot experience the z-axis because to them it does not exist.
1. Why would 2D Marge starve to death in her plane of existence if her mouth was located in the middle of her head? Answer with both a picture and with words. (Hint: Think about why she would need the 3rd dimension in order to feed herself.)

2. Where would 2D Marge's eyes have to be located for her to see things in her plane of existence? Answer with both a picture and with words.

3. How could 2D Marge and 2D Lisa pass each other? Answer with both a picture and with words. (Hint: Think of them moving around in their 2D plane of existence that has no depth)

4. Use the middle picture with the caption "square 2D" on Davide Cervone's "Dimensional Connections" to explain how a square can be formed from a line (this link is accessible off of the lab that you have been reading by scrolling down to this question number). Think about the line as a paint roller covered with paint, and think about how you would move it to sweep out a square. Explain with both pictures and words. This is a process we could explain to a 2D Marge, since it could take place entirely within her plane of existence.

5. In order to explain a cube to a 2D Marge, one could draw the following picture in 2D Marge's plane of existence. Yet in her plane the cube appears to intersect itself in places that do not intersect in 3D.
Part a) Label one example of 2 edges that appear to intersect in 2D, but which we know do not intersect in 3D (these are not the vertices since 3 lines really do intersect at a vertex in 3D). We could explain to 2D Marge that in 3D this intersection does not happen since there is enough space in the new dimension for the cube to fit without intersecting.
Part b) Using the Edit/Find... command, search in this text transcript of 3D Homer segment and Did You Notice? by James A. Cherry for Professor Frink's description of how a cube or Frinkahedron can be formed from a square, and copy that down here.
6. There is an alternative way of forming a cube - by gluing edges together.
 Before we try to glue a cube, let's do an easier example together. We can form a cylinder by taking a piece of paper and gluing 2 opposite sides together (try this!). I could explain this to 2D 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.
From Davide Cervone's Folding Cubes and Hypercubes:
"The following movie shows an unfolded cube (red) in space, together with its shadow (pink) on the plane below. The white dot at the top of the image represents the light source. As the red cube folds up, we can follow the results in the shadow below. As the sides of the cube begin to fold up, the shadows of these squares become distorted (one edge is closer to the light than the opposite edge, so one edge has a larger shadow than the other). As the edges of the squares come together in space, so their images come together below. Then the top folds over to complete the cube. In doing so, we see the image get larger (as it moves close to the light), and turn inside out as we go from seeing one side to seeing the other side of the top square. As the top closes up, its shadow forms the well-known "square within a square" view of the cube in perspective. At this point, we rotate the whole arrangement so that we see only the shadow of the cube and must imagine the three-dimensional cube unfolding that causes these shadows."
 After you click on the link, use the Play Button to watch Davide Cervone's Folding Cube Movie and then re-read the explanation above. Notice that you can use the other buttons to play it slide by slide or to rewind or replay the movie. Label gluing instructions on the figure to show which sides you would glue together in order to form a cube. (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...) We could give these gluing instructions to Marge and explain that while the figure can't be glued in 2D, there is enough room to perform the gluing in 3D.

7. The Shape of a 2D Universe A 2D creature could live on many differently shaped universes, such as the surface of a sphere (a spherical universe) or a blackboard (a Euclidean universe). Here are some additional possibilities:

Torus - A Euclidean Universe A torus is a mathematician's name for a donut. The surface of a torus is the topological space found in old-style video games such as Pac Man, where a spaceship goes off the right-hand side of the screen only to reappear on the left, or off the top to reappear on the bottom.
 If you take a sheet and try to glue the left edge to the right edge (the single arrows tell you to do this) and the top edge to the bottom edge (the double arrows tell you to do this), the paper will crumple up and you'll get a big mess.
 But, we can do this identification of a square with a stretchy piece of rubber. First we identify the left and right hand side in order to form a cylinder. We use these arrows and the points that correspond to glue the top and the bottom together and this forms a donut or torus.
Now that you know a bit about a torus universe, experience what it is like to live on one by playing
Torus Tic-Tac-Toe. Play a couple of games with the computer so that you get an idea of life on a torus. Recall for example that the top left square is really next to the top right square in this torus universe. You are allowed to scroll the board (once a square has been labeled X or O, you can click on it, hold down, and move the board around to see the identifications) in order to help develop your intuition.

After you have gotten the hang of the game, you and your partner will then play on one computer by changing the game to "human versus human" and clicking on "new game". Play best out of 7 games where you are allowed to scroll the board during the game and then play best out of 7 games where you are not allowed to scroll the board during the game. Keep track of who is winning and tell me who the winner is!
Scrolling Games Winner - best out of 7_________________________

No Scrolling Allowed Games Winner - best out of 7 ________________________

2-Holed Torus - A Hyperbolic Universe
 Here we glue the side with one line through it with the opposite side that has the same label by using the arrowheads to show us that we should glue the sides straight across. Similarly, we also glue together the 2s (II), the 3s (III) and the IVs. It is an exercise in visualization skills to see that the resulting figure is a 2-holed torus.
Klein Bottle - A Euclidean Universe
 Notice that this identification labeling of a square looks similar to the one that resulted in a torus, but the top and bottom edges are glued with a twist - a reflection in the line between the midpoints of the sides. Just as a 2D Flatlander could not imagine how to construct a cylinder out of a piece of paper, we Spacelanders have problems figuring out how to put together this square, because when we label corresponding points (such as the green squares, which are the same in this space because they are identified via the reflection in the dotted line), there seems to be no way to glue them together. However, an inhabitant of 4-space would have no trouble because he would have enough space to glue the edges together.
 There is a way to for us to put this space together using a slinky. We can match up the corresponding points (in the above picture) by having the slinky pass through itself. Make sure that you try this yourself and can visualize it (as the slinky makes its way around the room). This space is called a Klein Bottle.
The space that we have represented here has a nasty intersection when the slinky passes through itself. Our gluing instructions give no hint of this and in fact, the intersections only arose when we tried to glue the space in 3 dimensions. In order to properly see an actual model of the glued square, we would really need a 4th physical dimension. In the 4th dimension there is enough room to glue the edges together without creating intersections, and it is here where the glued square really exists.

Experience what it is like to live on a Klein Bottle by playing Klein Bottle Tic-Tac-Toe. Change the Torus selection to Klein Bottle (do not click on the link - just change the selection on the tic-tac-toe page on the part that reads "torus" that is just above "human vs. computer"), click on "New Game" and play a couple of games with the computer. The top left square is really just below the bottom right square in this Klein Bottle universe.

After you have played a few games, you and your partner will play on one computer by changing the game to "human versus human" and clicking on "new game". Play best out of 7 games where you are allowed to scroll the board during the game and then play best out of 7 games where you are not allowed to scroll the board during the game. Keep track of who is winning and tell me who the winner is!
Scrolling Games Winner - best out of 7_________________________

No Scrolling Allowed Games Winner - best out of 7 ________________________

Dr. Sarah J. Greenwald, Appalachian State University
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