We live in the three dimensions of space. We can move forward and backward, left and right, and up and down. Everything that we see has three dimensions. We live in a three dimensional universe. Everything that physically happens to us happens in the third dimension. When we walk forward, we're using the first dimension. When we turn left or right, we're using the second. When we climb stairs, we're using the third dimension.
A quick review of the dimensions just so we're talking in the same terms: Three dimensions is space; two dimensions is a plane; one dimension is a line; and zero dimensions means that there are no directions to point towards, so it would be a point. Each higher dimension adds a direction perpendicular to all the directions of the previous dimension. The second dimension is perpendicular to all the directions of the first dimension. If you take a point and pull it apart from itself, you end up with two points. Connect them and you get a line. If you take a line segment, and pull it out from itself in a direction perpendicular to the line, then you have two parallel lines. Connect the ends and you get a square. A square has two dimensions. Then if you take the square, and pull it out from itself in a direction perpendicular to all the directions of the square, then you get parallel planes. When you connect them, you get a cube. A cube has three dimensions. Since we live in a three-dimensional world, and a cube is three dimensional, that's as far as we can go. We will return to this subject later.
Let us now step down into a two-dimensional universe. We will create this universe for purposes of discussion. I am not saying that this universe actually exists. This entire universe that we will create has only two dimensions. Imagine this universe as if it were on a sheet of paper. You are looking at the paper so that it is vertical, and you see its edge. Looking at it this way, the people in this universe can move up and down, and forward and backward. This universe does not consist of "space", because we already defined space as having three dimensions. This universe is a plane.
Imagine a large circle on the paper. This circle will provide a surface for the people of this universe to walk on. Gravity will pull towards the center of this circle. It is similar to our Earth. Many two-dimensional people live on this circle. We will talk about one in particular named Lisa. Lisa walks along the edge of the circle, and is pulled towards the center of it. For Lisa, down is towards the center of the circle, and up is away from the center. When Lisa sees one of her friends, she sees only one side of them. In order for her to see the other side, her friend would have to turn upside-down. If we draw the people as stick figures, Lisa would not see them as stick figures. She would see them as a line with different colors for different parts of their bodies. To get a better idea of what Lisa sees, take a sheet of paper and turn it so that you are looking straight at the edge of it. You see a line. This is similar to what Lisa sees. But everything she sees is in this line. She doesn't know of anything else besides this line, so to her it wouldn't look like a "line." It would be her entire field of vision. She would see everything this way. She would be able to tell if one object is behind another, because of the shades of color objects have. Since Lisa's paper is vertical, she cannot move left or right. That movement would require another dimension. But she does not feel restricted because she doesn't even know that there is a higher dimension that she could move in.
Everything in Lisa's daily life happens in the second dimension. She eats a two-dimensional breakfast, she sleeps on a two-dimensional bed, she works at a two-dimensional office building. She is perfectly happy moving forward and backward and up and down. That is all she needs to do to live her life. "What is the need for a higher dimension than mine? What good would it do?" Because she doesn't experience the third dimension, she has a hard time imagining it. She would have to imagine a direction perpendicular to those she moves in. If all her memories and experiences are based on her movements in the second dimension, how could she imagine something outside of that? The third dimension is completely incomprehensible to her.
So take the piece of paper that Lisa is on, and bend it. Can she feel this bend? No, because she feels things in the second dimension. Does the surface of the paper change when it is bent? No. The area is still the same as it was when it wasn't bent. You can do anything you want to the paper short of ripping it, and the surface doesn't change. Lisa couldn't tell if the paper was bent any which way. She could still get around in her two dimensions.
Since we can bend the plane Lisa lives on, imagine this plane is the surface of a sphere. A sphere is three-dimensional, but it has a surface, and surfaces are two-dimensional. If it were possible for Lisa to leave her circle and fly, she could leave her planet going straight up and always go straight, and end up flying towards the other side of her planet. This is just the same as when we travel on the earth. If we were to start walking in a straight line, we would end up back where we started, ignoring the fact that there are large bodies of water in the way. The surface of a sphere is not infinite, but it also does not have an end. No matter which way you go on the surface, you will never come to a point where you can't keep going.
We have managed to make Lisa's universe continuous. No boundaries, but not infinite. This is comforting because who would want to live in a universe that goes on forever? A never-ending universe with an infinite number of stars in it is a bit overwhelming. But it also doesn't make sense for the universe to have an end. Once we got to this supposed end, what would keep us from going further? That is why this model is nice. Lisa's universe does not come to an unexplainable end, and it does not go on forever. It has a definite measurable area. We made her universe continuous by curving her plane on a sphere. We curved her plane in the third dimension, not the second. It took the third dimension, a higher dimension, one that she doesn't even know, to curve her plane.
Take a look at Lisa's universe on the paper from the surface, not from the edge. Say she is looking at her friend. She can see the front of her friend only. She can't see the back of her friend. But we looking at her can see her friend's entire outline. We can even see what is inside her friend. We in the third dimension can see inside two-dimensional things. We can see the whole two dimensional universe laid out all at once.
I have kept all the examples so far in three dimensions or less so that you can easily visualize them. Just as Lisa can only visualize things that have two dimensions or less, we can only visualize things that have three dimensions or less. But now it is time to take a peek at the fourth dimension. We will be using Lisa's universe as a model for our universe, plus one dimension.
As we already stated, each higher dimension adds a direction perpendicular to all the directions of the previous dimension. So we have to imagine a direction that doesn't point in any of the directions we know. This is a bit confusing. It is extremely difficult to imagine a direction that is not any of the ones we can see. "What would it look like?" is not an appropriate question. It wouldn't "look" like anything. Lisa could try to imagine the third dimension. She would have her up arrow and her forward arrow. But where else could she point? The third arrow would have to stick out of her paper. If it is sticking out, she can't see it. It is an elementary fact to us that there is a third dimension. We could see this arrow because we are three-dimensional, but Lisa couldn't.
So now we can take this model and apply it to our universe. If you imagine three arrows, one for each of our dimensions, pointing up, forwards and right, then the fourth arrow would not be able to point anywhere. Anywhere it pointed would be pointing some direction of the third dimension. In order for it to point towards the fourth dimension, it would be invisible to us.
We can try to continue our pattern of squares that we explored earlier into the fourth dimension. First we pulled a point away from itself and got a line. We pulled the line away from itself and made a square. Then we pulled the square away from itself in a direction perpendicular to those of the square and made a cube. Imagine if we take the cube and pull it out from itself in a direction perpendicular to all three dimensions of the cube. This sounds impossible. In a way it is. It's impossible in three-dimensional space. But in four-dimensional space, it is possible, just like it was possible to pull a two-dimensional square out of itself in three-dimensional space. Now we have a four-dimensional cube, which, for lack of a word in the English language, we will call a hypercube.
To get a better idea of what a hypercube looks like, we can follow some patterns: A line (1d) has two points (0d) at the ends. A square (2d) has four lines (1d) on its edges. A cube (3d) has six squares (2d) on its surface. So following this sequence, (2,4,6), a hypercube (4d) has eight cubes (3d) on its surface. Another pattern to follow is this: a line has two points, a square has four points, a cube has eight points, so, a hypercube must have sixteen points.
Notice when I drew the cube on this paper, I wasn't really drawing it in three dimensions. I was using only the two that were available on the paper. But in your mind, you saw a cube, not two squares and some diagonal lines. Just as it was possible to draw a three-dimensional object in two dimensions, it is possible to draw a four-dimensional object in two dimensions. We start out with the representation of a cube, and pull it down and right to form a sort of octagon figure. But unfortunately for us, we can comprehend only the two or three-dimensionality of it. If a four-dimensional person took a look at it, they would recognize it as one of their solids.
The reason this looks like an octagon with some pretty designs in the middle and not a four-dimensional hypercube, is because your brain cannot comprehend it that way. Look at the cube. You see it as a cube. But look at it again, this time ignoring any clues that it represents a three-dimensional object. Look at only the lines. Notice the patterns they create. There is a little square in the middle. There are diagonal lines coming off opposite corners, and the other opposite corners are extended. When you look at it this way, and then look at the hypercube, you see similar patterns. Notice, the little square in the cube does not actually exist in a three-dimensional cube. It exists on the paper because of the angle we are looking at it. Similarly, the pattern in the middle of the hypercube does not actually exist, we see it because we are looking at it from a certain angle.
Now that we understand a little more about the fourth dimension, we can look at the model of Lisa's universe to get an idea about ours. We will take the concepts we explored in Lisa's two-dimensional universe, and apply them to the three-dimensional universe that we live in. We said her flat universe was curved around a sphere. This is a good explanation for a universe that isn't infinite but also has no boundaries. There is no way to curve a two-dimensional surface in the second dimension, so we curved it in the third. We can apply this same idea to our three-dimensional space.
Our universe, the one we live in, is curved around a four-dimensional sphere, a hypersphere. We can't feel the curve because we feel only three dimensions. Just as Lisa could go straight up and always go up and she would come back around to her planet, we can do the same. If we were to take a spaceship, with enough fuel, and fly straight up, always going up, we would come back to where we came from. Our space is essentially the surface of a hypersphere. No matter where you go in space, you will never find an end, and yet it is not infinitely huge. Curved space is a difficult concept to grasp. It is impossible to see, but not impossible to understand.

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