A Tesseract, as noted above, is a four-dimensional physical object also known as a hypercube.
Our regular old objects are as follows:
- 0: A point. Bounded by zero... things.
- 1: A line. Bounded by two points.
- 2: A square. Bounded by four lines and four points.
- 3: A cube. Bounded by six squares, twelve lines, and eight points.
Thus when we go to define our tesseract, we just have to follow the rules. The rules are kind of weird though.
A tesseract is bounded by eight cubes, that bit is pretty easy to get. Now, how do we understand how many squares it is bound by? Well, when we look at a square, it's bound by four lines, which each are defined by two points. Each point is shared by one other line. On a cube, we have six squares, bound by four lines each. Each line is shared by another square. Therefore, on a tesseract each of the squares of each of the cubes that define the tesseract is shared by one of the other cubes. This ends up defining the second-order boundaries as half of the number of first-orders times the number of second-orders in each first-order. So, for the tesseract that comes to (8*6)/2 = 24. Similar to the last step, each point in the cube is shared by three squares. So the third-order boundaries are one-third of the number of second-orders times the number of third-orders in each second-order. For the tesseract this comes to (24*4)/3 = 32. Extrapolating from there, the formula for fourth-orders should be the one-fourth the number of third-orders times the number of fourth-orders in each third-order: (32*2)/4 = 16.
Now should you ever want to construct your very own working tesseract in your backyard, at least you will know how to check that you fit the pieces together correctly.
Here's the more exciting thing about a tesseract: It's a gate, a jump, a portal, whatever you want to call it. It will lead you to different positions in four-dimensional space. These places will all be somewhat related, because the faces of the cubes match up, but there is nothing to say that they are accessible from each other, through anything besides the tesseract.
At the most basic, a tesseract can allow you to move effectively faster than the speed of light. A journey from the two opposite sides of the tesseract bounds should take roughly twice as long than a journey through the tesseract. If this doesn't make sense, think about going from the center of two opposite faces of a cube going around the outside versus passing through the inside of the cube.
At the most exciting, a tesseract can transmit you to a place you would never be able to access from "normal" three-dimensional space. This could be an alternate timeline, the future, the past, or a completely different universe, depending on various factors.
Kids, don't try this at home. Your mileage WILL vary.