On Sat, 21 Dec 1996, Louis A. Mamakos wrote:

> > Hmm... I've looked at your web page and I must say that encoder sure does
> > have a wierd coding sequence, doesn't it.... :-)  Methinks I'll have to
> > brainparse that one a little bit...

Just take a walk on the wild sides of the 8-cube. No need to see all of
the sides, just turn once on every corner. Voila, the turning of the
shaft is in a roundtrip of the cube.

No, I am not on drugs and yes, I have had a good night's sleep. The
hallucinatory stuff is also known as mathematics.

A Gray code can be found by tracing a Hamiltonian cycles on the graph of
the n-cube. For you folks exclamating "now he's doing it again!": an
n-cube is the n-dimensional brother of 2-d square and 3-d cube, a graph
is a network consisting of points and lines (just like a schematic, but
without the parts), and a Hamiltonian cycle is a path over the lines of a
graph, traversing every point exactly once. Not every graph has such a
cycle, but every n-cube graph does.

In practice, you label all the points of an n-cube graph with their
respective "coordinates", like in:

   100           101
    *-------------*
    |\           /|
    | \110   111/ |
    |  *-------*  |
    |  |       |  |
    |  |       |  |
    |  |       |  |
    |  *-------*  |
    | /010   011\ |
    |/           \|
    *-------------*
   000           001

Now, if you go point to point, only one coordinate changes at a time,
making for very unambiguous state transitions, as opposed to a straight
binary scheme.

A Hamiltonian cycle would be:
000 -> 100 -> 110 -> 010 -> 011 -> 111 -> 101 -> 001 (-> 000 )

Note that this method is very easyly extended to higher values of n (if
you can't figure it, notice how this one is already an extension of n=2.)

> There's a simple algorithm using XOR operations which can be used to
> translate between gray code and "normal" binary encoded values.  Actually,
> it's a pretty simple circuit as well using XOR gates.  I'd refer you to
> "The Art of Electronics" by Horowitz and Hill.  ISBN 0-521-23151-5, but
> I know there's a new edition subsequent to this one.

Well, why not post the algorithm, it's an interesting thing to put in a pic.

BTW, I got mine from "Graphs, an introductory approach" by Wilson and
Watkins, ISBN 0-471-51340-7. For all you engineers thinking math is only
integration and differential equations, this is a definite gem. If you
hate math, get a copy of "Alice in wonderland". Also makes a great
Christmas gift to newborns :-)

Joost