The tesseract is often used as a visual representation of the fourth dimension.
The fourth dimension is generally understood to refer to a hypothetical fourth spatial dimension, added to the three standard dimensions. Not to be confused with the space-time view, which adds a fourth dimension of time to the universe. The space in which this dimension exists is called a four-dimensional Euclidean space.
At the beginning of the 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 picked up and rotated around the mirror image of it. 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, upon which later mathematicians would build.
In the three-dimensional world, people can see all of space as existing in three planes. All things can move along three different axes: altitude, latitude, and longitude. Altitude would cover up and down movement, north and south latitude or forward and backward movement, and east and west longitude or left and right movement. Each pair of directions is at right angles to the others and are therefore called mutually orthogonal.
In the fourth dimension, these same three axes continue to exist. Added to them, however, is another axis altogether. 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 is told to take five steps forward, six steps to the left, and two steps up, he will know how to move and where to 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 of 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 sheet of paper is a two-dimensional object, roughly speaking, and therefore cannot convey a three-dimensional object like a cube. However, drawing a cube and representing three-dimensional space in two dimensions is 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, a similar procedure can be followed, drawing several cubes and connecting their vertices.