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".

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.