The tesseract is often used as a visual representation for the fourth dimension.

The fourth dimension is generally understood to refer to a hypothetical fourth spatial dimension, added to the standard three dimensions. Not to be confused with the spacetime view, which adds a fourth dimension of time to the universe. The space in which this dimension exists is referred to as four-dimensional Euclidean space.

In the early 19th century, people began to consider the possibilities of a fourth dimension of space. Mobius, for example, understood that, in this dimension, a three-dimensional object could be taken and rotated about its mirror image. The most common form of this, the four-dimensional cube or tesseract, is often used as a visual representation of it. Later in the century, Riemann laid the foundation for true four-dimensional geometry, on which later mathematicians would build.

In the three-dimensional world, people can see all of space as existing on three planes. All things can move along three different axes: altitude, latitude and longitude. Altitude would cover up and down motion, latitude north and south or forward and backward motion, and longitude east and west or left and right motions. Each pair of directions forms a right angle to the others and is therefore called mutually orthogonal.

In the fourth dimension, these same three axes continue to exist. Added to them, however, is another axis entirely. Although the three common axes are often called the x, y, and z axes, the fourth is on the w axis. The directions in which objects move in this dimension are often called ana and kata. These terms were coined by Charles Hinton, a British mathematician and science fiction author, who was particularly interested in the idea. He also coined the term “tesseract” to describe the four-dimensional cube.

Understanding the fourth dimension in practical terms can be quite difficult. After all, if someone were told to take five steps forward, six steps left, and two steps up, he would know how to move and where he would stop in relation to where he started. If, on the other hand, a person were instructed to also move nine ana steps, or five kata steps, he would have no concrete way of understanding this, or visualizing where that would place him.

There is a good tool to understand how to visualize this dimension, however, it is first to look at how the third dimension is drawn. After all, a piece of paper is a two-dimensional object, roughly speaking, and therefore cannot really convey a three-dimensional object such as a cube. However, drawing a cube and representing three-dimensional space in two dimensions turns out to be surprisingly easy. What you do is simply draw two sets of two-dimensional cubes, or squares, and then connect them with diagonal lines connecting the vertices. To draw a tesseract, or hypercube, one can follow a similar procedure, drawing several cubes and connecting their vertices.