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.

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).

tesseract

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

5-cube

6-cube

6-cube

 

7-cube

7-cube

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One comment on “A brief foray into hypercubes

  1. […] now we know the general idea behind Hamming error correcting codes, and how one might construct and visualize hypercubes.  Now suppose we want to encode, in an error correcting way, 4 bits.  Recall that this means […]

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