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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Hamming Error-Correcting Codes

This is part 1 in hopefully a series on the Hamming error-correcting codes, to be continued on Friday.  The problem is this:  suppose you wish to send a message composed entirely of 0’s and 1’s, but suspect part of it might get “corrupted”.  More concretely, I am going to send you the message

01101,

but there’s a chance that you receive instead something like 00101 (so the second bit was flipped), or 01111 (so the fourth bit was flipped).  Is there any way to control for this?  An idea that is both natural and wrong is to send two copies of the message.  In this case, I’d send along

0110101101.

Now if a bit gets flipped (note that there are now more bits that could be flipped), you can see exactly where — perhaps you receive

0010101101,
where the non-matching bits are highlighted.  The problem here is that you cannot tell whether the offending bit should be interpreted as a 0 or a 1 (which might be ok if you are only interested in error detection).  But if we want to be able to correct the error, we’ll have to be a little more clever.  As a very broad outline of our strategy, we are going to take each of the symbols we would like to transmit and encode them into a “big enough” space so that each symbol has a buffer zone around it to account for any errors in transmission.

In some sense, this is a familiar strategy: on touchscreen keyboards you do not have to hit the exact spot for a key to work, only be close enough.  In this case, the “symbols” I’m referring to are letters, and the “buffer zone” is the area of the screen you can touch so the computer recognizes that you are trying to touch this particular letter.

The trick here (and what is sort of confusing about this) is that the symbols we wish to transmit are bits, and the space that we will be embedding our symbols into will also be bits (rather than a shiny retina display!) As a hint of things to come, I’ve displayed below a representation of the space of 2-bit strings (which will not be big enough to detect 1-bit errors), and a representation of the space of 3-bit strings, which is of perfect size to detect 1-bit errors in a 1 bit message.

All 2-bit strings, with a line connecting them if the distance between them is 1, so you can get to one from the other by switching a single bit.

 

 

 

All 3-bit strings, connected whenever the distance between them is 1. Notice that the points (0,0,0) and (1,1,1) have disjoint balls of radius 1 around them. This will be the basis of the error correcting code, by agreeing that anything received in the red “ball” will be interpreted as the symbol (0,0,0), and anything received in the blue “ball” with be interpreted as (1,1,1). Then this can be turned into a 1 bit error correcting code.

 

Bouncing balls

Inspired, as usual, by Leonid’s recent post, I decided to first write a script that would mimic his.  After that, since I had all the numbers worked out, I wrote two more MATLAB programs: one that mimicked elastic collisions in 2-dimensions, and one that mimics them in 3.

In theory, you can specify the number of particles and their radius, as well as the mass, position, and initial velocity for each (I didn’t vectorize radius for some reason, so I cannot model balls of different sizes bouncing around).  However, in practice I just generate random vectors for each of these numbers.  The final aspect is that the domain I put the balls in was a pool table of 9 x 4.5 units, or 9 x 4.5 x 4.5 for the 3D version.  This was just to make calculating the reflecting angle easier when a ball hit the wall.

As with Leonid’s code, mine works by checking whether the next step will cause any collisions, then adjusting the velocity vector so that the collision didn’t happen (using conservation of momentum and kinetic energy).  This algorithm is not “smart” in the sense that by avoiding one collision, it might get pushed into a second collision which it does not detect, and if a particle gets going fast enough, it can reflect off a wall from a large distance (my time step is just 0.01).  You can spot this in some of the figures below.

Anyways, here are some of the outputs.  I did not go through the trouble of turning these into .gifs, but they play fairly smoothly.  What happens is I simulate N particles of varying masses and velocities bouncing around in a 2- or 3- dimensional box for T seconds, then plot the path of one of the particles.  The end position of all the particles, plus this path, is in each picture below (with the “tracked” particle colored in red).

4 particles for 20 seconds. Not many collisions... I can spot 3, I think.

40 particles bouncing for 20 seconds. This particle is involved in a few more collisions than last time (and looks to have been moving faster, too.

Now 40 particles for 20 seconds in 3d. A few more collisions, and it looks like it was going pretty fast in the middle there for a while.

400 particles for 20seconds. Starting to look more like the "random walk" of Robert Brown's pollen, though I would certainly have to mess with how heavy the particles were to more accurately model that.