# A brief foray into hypercubes

The discussion on error-correcting codes is about to get a little hypercube heavy (never a good state to be in), and a brief foray into how to construct/visualize them may be in order.  I’ll take the liberty of defining an n-dimensional (unit) hypercube as a shape whose

1. vertices are located at coordinates made of entirely 0’s and 1’s, and
2. has an edge wherever two vertices are distance 1 apart.

This would take two more things to make a complete definition: I should let you move the cube about however you like (no reason to have it fixed is space), and I should tell you about the 2-D faces, 3-D hyperfaces, and so on up to the (n-1)-D hyperfaces.  You can use that first one if you want, but I’ll ignore the second.  I think I did a good job of defining what’s called the 1-skeleton of a very particular n-dimensional hypercube.

Two sample vertices representing (1,1,0,0) and (1,0,1,0). These will not be connected in the hypercube.

Anyways.  Wednesday had pictures of a 2-cube and 3-cube.  What about the 4-cube?  Or 5-cube?  It will help to consider this all from a less analytic, more graph theory (or, if that sounds technical, “pictures and problem solving”) point of view.  Condition 1 for a hypercube says that there are 2n vertices, all the binary sequences of length n. Then condition 2 says that two vertices are connected if you can change one vertex’s binary sequence to the other’s by changing a single bit.   We’ll go one step further, by just coloring particles on a line: white for 0, black for 1 (this is something of a homage to my undergraduate thesis advisor’s work with polyhedra).

The only two things left to do are to draw the vertices and arrange them in nice ways (that is, fine a “nice” projection).

A projection of a 4-cube, with vertices labelled.

Below is the image from the wikipedia 5-, 6-, and 7- cubes.  Note the some of the vertices are laying on top of eachother.  I’ll leave it as an exercise to the reader to label these vertices with the appropriate binary sequences.

5-cube

6-cube

7-cube