Dimension (1967)

We live in a three-dimensional world, space. We can move in and out, up and down, back and forth. Almost all common objects, a boat, a child running, a rose, all these are solid, three-dimensional. There are, however, examples of two-dimensional objects. The world a child creates on a piece of paper is two-dimensional. Up and down is a direction that simply does not exist. Her bridge has height and breadth, but no depth. Her child is also trapped in the two-dimensional world of the paper. There are other examples of the second dimension. The silhouette of a bridge, a boat, or a bird, a reflection, trees silhouetted against a flat sky. Look carefully and you can even find examples of a one-dimensional world. Lines. But is there a fourth dimension? If so, what is it like? Are there common examples of objects from this dimension beyond our own world? No. But this model of a tesseract, a fourth-dimensional cube dreamed up by mathematicians, helps us imagine how strange a world the fourth dimension would lead us into. To help understand the fourth dimension better, start with a zero-dimensional object, a point. Let it trace itself out into the first dimension and it becomes a line. As the line travels into the second dimension, it becomes a square. Suppose this square moves into the third dimension. It then traces out a three-dimensional cube. And now suppose finally that a cube were to travel into the fourth dimension. In so doing, it would trace out a fourth-dimensional cube, a tesseract. In our one-dimensional world, there can be left- and right-handed segments. Limited to motion along a line, the red and yellow ends can never be made to coincide. But if the right-handed segment travels into the second dimension, it can turn itself around and return to its own world as a left-handed segment. Left-handed and right-handed objects can be found in a two-dimensional world. Try as it will, the white triangle can never turn so as to coincide with the red. But if it travels into the third dimension, it can turn over, becoming a left-handed triangle, return to its own world, and match its counterpart. Here are two three-dimensional blocks that may seem alike. Each has the same shape and both are the same size. But no amount of turning or flipping No effort can make one line up with the other. They can never be made to coincide. One is left-handed, the other right. But if the right-handed one were to travel into the fourth dimension it could turn itself inside out returning to its own world as a left-handed block. It would be very easy to build a prison in a one-dimensional world. As long as the point is limited to traveling only in the first dimension it is trapped. Magic? No, let's go back. If the point travels into the second dimension it can easily return to its own world free without ever having broken through the prison walls. Prisons are easy to build in a two-dimensional world, too. As long as the triangle knows only of travel in its own two-dimensional world, there is no escape. Magic, this time? Again, no. A look backward shows that the triangle passed through its prison wall by jumping into the third dimension. In a three-dimensional world, a prison must completely enclose the prisoner. But if the prisoner were to travel into the fourth dimension, it could return to its own world free without ever having cut through the prison walls. If you lived in a one-dimensional world, a segment would look like this. Its end point would hide its interior from you. But as you move into the second dimension, you can see the inside of the inside of the inside. If you lived in a two-dimensional world, a square would be almost unrecognizable. You would see only its edge, and thus the square would seem to be merely a segment. But by moving up into the third dimension, you can look into the square and see its inside of its inside, of its inside, and so on. In our own three-dimensional world, the surface of an object like a cube hides its interior. We cannot look through to see what is inside, but suppose we travel into the fourth dimension. As with the square, we could see the inside of the inside of the inside all while remaining on the outside of the cube. When we look at a triangle or a square from above, from the third dimension, we can see all sides at once. It makes no difference whether they are turning or not. To help distinguish one side from another, we could mark them with colored chalk. All this may seem natural, but look how strange this paper triangle looks if we leave the third dimension and descend into its two-dimensional world. Now, we can see only one side. And if we take a second look at the triangle and square marked with chalk, this time looking at them from their two-dimensional world, all sides cannot be seen simultaneously. First, one side and then another comes into view. If we return to our three-dimensional world and view a cube, the same thing is true. As it turns, sides that were hidden come into view and others disappear. But, let's travel into the fourth dimension and take another look at the cube. All sides are visible simultaneously. This model of a tesseract brought us into the mathematicians' world of triangles, squares, and cubes. Return to the real world and see how much we would gain by knowing where the fourth dimension is. We saw how a right-handed cube can be turned into a left-handed one in the fourth dimension. Watch this shoe salesman. He may lose a customer because his stock is in poor order. Two left feet. What is to done? Send it into the fourth dimension where it can turn around into a right-footed sandal and let it return as a mate to the left. Puzzling? Yes, but possible. We saw how easy it would be to escape from prison if we could travel into the fourth dimension. This commuter is about to solve a problem by traveling there. >> [music] [music] [music] >> The door is jammed. No problem. By traveling into the fourth dimension, escape is easy. Travel into the fourth dimension and you can see the inside of the inside of the inside all while still outside the house. Or travel into the fourth dimension so that you can see all sides of the bridge. of a girl of rows. All these newly gained powers presuppose our knowing where the fourth dimension is. Where is it? No one knows. Does it really exist? Is it only an idea? What do you think? >> [music]