What is the Fourth Dimension?

You have probably heard people refer to the fourth dimension before but maybe you weren't sure what they mean by it. Well, first lets remember that we live in 3 dimensional space. How do we know? - We have three directions of free movement:

1.) Left/Right

2.) Back/Forth

3.) Up/Down

Because we have three possible directions, it will take three values to describe the position of any point (x,y,z). In two dimensions, we only need two points: (x,y). In three dimensions, I could tell you to go forward 8 feet, right 12 feet, then you reach a rope. Climb up the rope 6 feet. Notice that each of these three directions are perpendicular to each other... That means that any of the directions is its own and not combined of any of the other directions. That means I can go forward or backward all I want but I won't be going left or right, nor up or down. So if we were to consider a fourth dimension (x,y,z,t), it would have to be a new perpendicular direction in addition to the three directions we have free movement in.

Considering the Fourth Dimension:

Often it is helpful to visualize the way dimensions increase when considering the fourth dimension. Examine the chart on the right. Notice that the zero dimension is represented by a point, 1 dimensions is represented by a line, and 2 dimensions is represented by a square. When we try to express three dimensions on a flat surface, like your computer screen, we draw two squares and diagonal lines connecting the vertices. It represents a cube, but it is not actually a cube. So perhaps when we draw a 4 dimensional cube, we can draw two cubes and connect the diagonal vertices. That figure is called a hypercube. As pictured above.

Keep in mind that trying to express a 4 dimensional hypercube on a 2 dimensional surface is not very helpful. But it at least helps you understand the kind of progression going on. Visualizing 4th dimensional objects is not important if you merely want do some simple computations. For instance, in 3 dimensions, the volume of a sphere is 4/3(pi)r^3, however the hypervolume of a 4 dimensional hypersphere is determined by 1/2(pi^2)r^4.

Using Analogy to understand the 4th dimension:

Since it is hard to try to directly picture the fourth dimension in our minds, perhaps using analogy can help us. In 1884, Edwin Abbot wrote a book called "Flatland". The book writes about A. Square and his world, Flatland. You may have already guessed, but Flatland is a 2 dimensional, flat plane and A. Square is a square shaped guy who lives there. He has two dimensions of free movement. He can go left/right and back/forth, however because he is restricted to his 2 dimensional Flatland plane, he cannot go up/down off the plane. By analogy, we humans are restricted to our "plane" of existence... and it would be impossible for us to freely move in the fourth dimension. Let's go back to A. Square again. Note that A. Square can only see what lies in his plane of existence, which means if a 3 dimensional sphere were to pass through Flatland, A. Square would not see the sphere, but just 2 dimensional "slices". Taking this further, imagine if a sphere passed halfway through Flatland but stopped in the middle. the sphere would interesect Flatland as just one circle and A. Square could see it! Furthermore, imagine if as the sphere approaches Flatland, A. Square watches as the sphere slowly moves through his plane. What would A. Square see? Recall that A. Square can only see 2-d slices of the sphere (or circles) so what A. Square would percieve is a circle suddenly appearing, then growing... then reaching a maximum size as the sphere was halfway through and as it exited, the circle would grow smaller until it disappeared. This means that 3d objects could be explained to a 2d being as a bunch of "slices stacked" on top of each other. Try to imagine taking a bunch of circles and stacking them. They would begin to form a skeleton framework of the actual 3d image. Similarly if a 4d hypersphere would intersect our plane of existence, we would see a 3d sphere appear out of no where. It would grow until the hypersphere was halfway through, then it would shrink back to nothing. Theoretically, we could stack these spheres to form a hypersphere, but we can't stack them in the usual sense, but rather it would have to extend in the fourth dimension which takes us back to the original dilemma of trying to visualize it.

Some more curiousities:
If we look down upon a square in a flat, 2d plane, we can see the entire object a single glance. Only one perspective is needed. In fact, we could place our finger inside the object without touching the sides. This would be a profound feat for A. Square, a creature inhabiting Flatland. His house is one big square and he can't just put his finger in the middle of the house without first "entering" through a door on one of the sides. Analogously, fourth dimensional beings have the ability to visualize an entire cube at one glance. Humans can only visualize one half the cube at any given second. Also, four dimensional beings could easily put their finger inside a closed cube without penetrating its sides. Other curiousities involve mirror images. Imagine A. Square again. But now, lets pick him up off Flatland and put him back into the plane upside down. He would now be the mirror image of his old self. It is a bit tricky to imagine a human becoming a mirror image of themself since we are unfamiliar with the fourth dimension rotation needed.

Conclusions:
Ultimately, it is best to think of the fourth dimension analogously. Considering that the 3rd dimension is a "new direction" to A. Square, we realize that the 4th dimension is a kind of new direction to us. To say that we don't have free access to it, it not to say we are in no way effected by it.