Charles Hinton (a late 19th century philosopher, Theosophist, and mathematician) coined the word Tesseract, the term that refers to the four-dimensional analog of a cube. In other words, a Tesseract is to a cube what a cube is to a square. I prefer to use his word as opposed to hypercube, as hypercube doesn't have as exact of a definition. A Tesseract is not exactly a hypercube, it is more specifically a four-dimensional hypercube. There are hypercubes of any number of dimensions, real or imagined.
I developed a model loosely inspired by Hinton's work (a warning that some supposedly went insane meditating upon Hinton's model), and decided to actually build it out of wooden cubes, as two-dimensional diagrams are missing two whole dimensions and make it more difficult. It stands around 6 inches tall. Here is a photo of the model, which I've enhanced with white lines:
|A Colored Model of an Unfolded Tesseract|
The shape of the model is an unfolded Tesseract. In a similar sense, if you were to unfold a cube into two dimensions, you would have a cross of six squares. Unfolding a Tesseract into three dimensions produces a sort of cross of eight cubes. This is only one of many ways to unfold it, but the most appealing and simple.
The system is based on color. The model represents an inner mentally-produced world. The colors are set up in pairs. There are four pairs of flashing colors: yellow and violet, red and green, orange and blue, white and black. The black cube is represented by the hidden cube at the center. If you were to travel from the center of any cube in any of the six directions corresponding to the faces of the same cube the distance of two cubes, you will arrive in the cube corresponding to the complimentary color. Travel two more cubes in the same direction (or any for that matter) and you will arrive at the cube where you began. The model is to help keep track of the cube you're in and the other colored cubes relating to your place in space while you are training yourself.
Below is an animation I made in PowerPoint demonstrating the connections between edges in folding up a cross of six squares into a cube. It is followed by lines showing the connecting faces when folding up the Tesseract into the fourth dimension in space. Understanding what faces are connected is crucial to success with this model.
One goal of being able to relate to such geometric figures as the point, line, cube, and tesseract in this context it the expansion of consciousness. As there is no four-dimensional imagery for the eyes to look upon, the mind is not accustomed to perceiving a four-dimensional world and is unlikely to be able to readily imagine such a figure. However, training oneself to observe a four-dimensional figure such as the Tesseract is a readily observable expansion of consciousness: perception of the previously imperceivable.
At first, this is not a four-dimensional experience at all. It is easiest to start out imagining it like a 3D version of PacMan, that somehow you magically arrive at the other end of the cube after traveling through the opposite end. After spending lots of time connecting with and developing the inner model, it can then develop into a four-dimensional experience.
This is but one method of approaching the creation of a mental model of a Tesseract. I recommend utilizing many.