2D Universes Readings and Videos

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.

We previously explored Homer's transition. 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. In fact, 2D Marge can only pass others in her space by jumping over them, like a leap frog, since there is no depth to pass in front of or behind them.

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 this figure 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.
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?

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.

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.

The 2D person sees part of 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.

Watch this video: The Shape of Space Video [http://www.youtube.com/watch?v=Uzd484Mvm2k]

This animated video produced by The Geometry Center introduces the two-dimensional space of flatland, looks at possible shapes for flatland from the perspective of three dimensions, and represents those shapes of space in two dimensions. Then the animation uses the same kind of representation to look at possible shapes for three-dimensional space. Viewers are taken on a ride across the boundless three-dimensional surface of a three-torus and a four-dimensional Klein bottle. As viewers see these imaginary universes from inside the spaceship, they experience the illusion of seeing copies of the universes.

Read p. 349 in Heart of Mathematics on the Klein Bottle

Watch surfaces with no edge [https://youtu.be/T0rZ41jV0rI]

This video I created explores tic-tac-toe on surfaces with no edges

Watch Flatland: The Movie - Official Trailer [http://www.youtube.com/watch?v=C8oiwnNlyE4]

Flatland: The Movie is an animated film inspired by Edwin A. Abbott's classic novel, Flatland. Set in a world of only two dimensions inhabited by sentient geometrical shapes, the story follows Arthur Square and his ever-curious granddaughter Hex. When a mysterious visitor arrives from Spaceland, Arthur and Hex must come to terms with the truth of the third dimension, risking dire consequences from the evil Circles that have ruled Flatland for a thousand years.

Mathematics... reason... imagination... will help reveal the truth. -Arthur Square

Think about what a 2D creature like Arthur Square or a 2D Marge Simpsons would see as spherius (a sphere) passes through their 2D plane of existence. They can't see above or below the plane because they are limited to their views of being inside. 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.

However, 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, Arthur Square cannot see an entire circle or 2D curve all at once. As a sphere passes through his plane of existence Arthur Square (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 him) that gets closer to her, then farther from her, then turns into a point that finally disappears. This would seem very strange. In fact, this behavior does not make sense in 2 dimensions - objects don't just appear and disappear. A 3rd dimension would be needed to explain the behavior.

Click on the link below. Think about Arthur Square standing on the right and think about what she would see. 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.

Next, go through the following movie by using the play button. Read 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 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.