Thursday, March 22, 2007

Ternary Logic

Earlier this week, Kyle (a fellow physics grad student) got me thinking about the set of single variable functions from a finite set A={a,b,c} to a finite set B={0,1,2}. Such functions involve triples of elements of A x B, and take for example the form f_012={(a,0), (b,1), (c,2)}. I ended up using shorthand for such functions, writing f_012 as 012 for instance, to reveal the nice single variable ternary function structure. Ultimately, I came up with the above diagram to show how multiple copies of the parity cube vertices arise in this set of functions. I guess one can also look at it as a Qutrit function diagram. Kea and Carl may see further applications in particle physics. ;)


CarlBrannen said...

Thanks for putting this up. For some reason, I don't understand even obvious things until I've seen them three times, and now I understand what is going on here.

I've been working recently with the permutations on three objects. This is a subset of the full set of functions that map {0,1,2} to {0,1,2}.

The physical reason for dealing with this particular subset is energetic. One supposes that the set {0,1,2} minimizes energy in some way, so we can ignore the situations when these are sent to {0,1} or whatever.

Another way of describing the same restriction is that the Pauli exclusion principle prevents two preons from being sent to the same state.

And yet a third way of describing this is to say that there is some sort of soliton behavior going on in the measurement process that makes sure that the only possible results when measuring the number of fermion objects with some specific characteristics is either 0 or 1.

Doug said...

Hi Kneemo

Your vertex [node] counterclockwise numbering is consistent with classical MOD[3] or in this specific case perhaps MOD[333].

However, in counterclockwise clockface numbering, one should be able to substitute 333 for 000 or 3 for any 0 within the module or along the edges [arcs].

This method then may mimic the first level of a Cantor set which is also a Borel set and sometimes referred to as a proto-fractal.

The 3 and 0 are essentially equivalent when used as place holders.

In ternary code, 0 might be considered "off" as it is in binary code.

Nature seems to prefer "this" or "that" as contrasted to the anthropic "on" or "off".

For example, matter or antimatter rather than no matter; or mass or energy rather than nothing.

Kea said...

Heh, heh, heh. Your blog should be more popular, kneemo. This should wrap up that damned number pretty soon...

kneemo said...

Yup, the subset of functions involving 0, 1, and 2 can be seen as the elements of the symmetric group S_3. The order is 3!=6, which means we can cycle through all the elements and get back to where we started with 6 swaps (transpositions).

Of all the single variable ternary functions, the S_3 subset appears to consists of all the "non-degenerate" mappings.

Mahndisa S. Rigmaiden said...

03 26 07

Hello there Kneemo:
Yes, nice picture. As I was going through permutations using Levi Civita operator in three d, I constructed a similar diagram. Any three vertex shape could represent three adicity in context and the lines that connect each vertex could be seen as a swapping morphism of sorts.

I played around with this geometry a while ago and you can extend it up to n rational dimensions. A four adic system is represented by a trapezoid, rectangle, rhobus or square, each of the vertices are connected via lines and each of these lines rep swapping. The patterns these lines forms on the interior of shapes are pretty like stars.

Good post.

CarlBrannen said...

Oh, come now, you need to officially announce your new arXiv paper here so we can comment on it. It cleared up several confusions I had had about the trajectories and the Koide saddle and all that. I'm still mulling it over.

The problem I'm working on now is the choice of mapping when one wishes to rewrite a linear wave equation. If d W = 0, then f(W) also satisfies this equation, but might be more convenient for one reason or another.

This is a key part of Bohmian mechanics in that they prefer to write complex \psi as = \rho exp( iS) where \rho and S are real.