A tesseract is a hypercube, the four-dimensional equivalent of a cube, which has been unfolded into three dimensions.

Since four-dimensional space is difficult for the human mind to visualize, the best way to think of this is to remove a dimension. Every school child has made a cube from a two-dimensional piece of paper by cutting out the following shape:

      |     |
     A|     |
|     |     |     |     |
|     |     |     |     |
|___ _|_____|__ __|__ __|
      |     |
      |     |

and folded along the lines to create a three-dimensional cube. This is called the net of the cube. When we fold it up, the edges labelled A, become the same edge in the cube. Similarly, other pairs of edges merge to form one edge in the cube

The three-dimensional 'net' of the tesseract looks like this:

    /\  /\  /\
   /  \/  \/  \
  /   /    \   \
 |\  |\    /|  /|
 | \ | \  / | / |
 |  \|  \/  |/  |
  \  /\  | /\  /
   \/  \ |/  \/
   /    \/    \
  |\    /\    /|
  | \  / |\  / |
  |  \/  | \/  |
   \ |  /|\ |  /
    \| / |/\| /
     \/ \/  \/
      \  | /
       \ |/

/me realises that representing four-dimensional geometry in ASCII art is probably a mistake

In case that isn't clear, it is a stack of 4 cubes, with four more cubes arranged in a cross around the second cube from the top. When we 'fold' this up, the top face of the cube at the top of the stack merges with the bottom face of the bottom cube, the adjacent edges of the cubes in the cross join, and so on to form the hypercube.

As a interesting historical note, in the picture Christus Hypercubus, Salvador Dali depicted Christ being crucified on a tesseract.

The hypercube in R^4 (four-space), also called the 8-cell, is known as a tesseract. It has the Schläfli symbol {4,3,3}, and vertices (+-1,+-1,+-1,+-1). A tesseract has 16 vertices, 32 edges, 24 squares, and 8 cubes.

Also, a concept used in the popular Sci-Fi children's book by Madeleine L'Engle A Wrinkle in Time. The tesseract is a concept used to explain time travel in the sense that a line is the shortest way to travel between two points


However, if you can imagine placing your finger in the middle of that line, and folding the line so that the two endpoints are side by side, then you can see how the line is technically not the shortest way, but rather to fold it. The book used the example of an ant crossing this distance as it would travel between two points on the hem of a skirt.

Also used in the short story '—And He Built a Crooked House' by Robert Heinlein. An architect built an 'unfolded' hypercube as a house, similar to Tristan's drawing above, but upside down. The first cube on the bottom was the garage and utilities, the second floor (TWIAVBP UK: First floor) was the living room and such with a cube on each side, then another two above.

This was built in California, and during an earthquake, the unstable unfolded four-dimensional structure collapsed into its more stable form...leaving the people inside slightly trapped. Each exit had two ways to go through it; the way that would have led to another room of the house and if you were either distracted or had the right mindset when going through, you would go outside of the cube. One of them discovered this by accident, by falling out a window into a rosebush.

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.

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