[seqfan] Re: detective work related to "Floretions"
Creighton Kenneth Dement
creighton.k.dement at mail.uni-oldenburg.de
Sat Nov 21 17:53:09 CET 2009
> First, a big hats-off to Robert Munafo for his "Rosetta stone work", this
> is
> very laudable task.
I think it's wonderful that he's helping me and hope that continues into
the future.
> Now, some of Joerg's criticism makes sense, although unfortunately, he
> represents it in his typical caustic manner. E.g. using scribd.com is not
> necessarily
> a good idea, if you want to distribute your documents. At least it used to
> crash
> my old browser before I could read the embedded documents past the first
> page.
This was Joerg's main point, as I see it.
Due to the trouble I'd run into with my website before, I wanted to find a
place where there was little risk of the site ever being taken down. I was
unaware of the problems with Scribd and, as was pointed out, I immediately
took his advice.
> But, this is clearly over-the-top:
>
> Joerg Arndt <arndt at jjj.de> wrote:
>>
>> Does the python script linked under
>> "Symbolic Calculator in Python" on the page
>> http://fumba.eu/sitelayout/Floretion.html
>> not rate as "code from hell" for you?
>> Compared to this IOCCC appears to be
>> a competition in clarity.
>>
>>
> Note that Creighton didn't originally intend the code for public
> consumption!
Exactly!
> We all write quick-and-dirty code, especially when we don't expect the
> code
> to take part in a public beauty contest.
> But...
> The idea of using integer indexing (instead of letters) for the sixteen
> bases
> of Floretion algebra, is probably a good idea.
I see 4 cases:
1. multiplication as defined internally within a program
2. input/output user interface operations for the above program
3. We wish to explain what the algebra of floretions are to a math expert
4. We wish to show a new-comer to the field how it's done "quick n'
dirty", with no reference to fancy terms.
If case 3 is our main goal, then I see no need to use coefficients at all!
Let Q be the algebra of the quaternions over the reals, then it can be
shown that the tensor product of this algebra with itself is isomorph to
the space of 4x4 real matrices and that the floretion basis vectors 'i,
'j, ... (written as matrices) form a basis for this space.
This was shown by Edwin Clark in his Maple script.
http://fumba.eu/sitelayout/mapleEClarkFloretions1.html
Or... you can take my original path years ago, which was to start with the
quaternion group (8 elements), create a copy, and then define equivalent
classes on the cartesian product. For example 'i = [i,1] = [-i, -1], etc.
Cases 1 and 2: with all of my earlier programming work, I've used the
indices 0-15. For example, FAMP defines
//determines 'kk'
z[8] =( +x[0]*y[12] - x[1]*y[10] + x[2]*y[5] + x[3]*y[14] - x[4]*y[13]
+ x[5]*y[2] + x[6]*y[7] + x[7]*y[6] + x[8]*y[15] - x[9]*y[11] +
x[10]*y[1] - x[11]*y[9] -x[12]*y[0] + x[13]*y[4] - x[14]*y[3] +
x[15]*y[8]);
In this case, however, I wanted to start from scratch creating a symbolic
multiplier where the user can multiply something like this:
((3a-2bc)'i + j')(4a'ij') and get an output like this:
(-12a^2-8bca)j' - 4a'i and the last thing I wanted is for some bug to
arise along the way where there is confusion among indices and
coefficients or exponents. FAMP did not have that functionality so it
didn't matter.
Case 4. Almost every time I've explained this in the past, I've done it
with indices. Yesterday, I happened to use the symbolic multiplier to
print out the list so there weren't any indices (and I certainly wasn't
expecting the furor because of that choice). If we are talking about the
coefficients of higher powers of X, I definitely recommend use the
(linear) projection operators ibase, jbase, ...
Frank is absolutely correct in stating that some theorems are easier to
show if you take another approach (though in my opinion, that is not
always the case). For example, it appears to be easier to why vesseq(X)
always satisfies a fourth order linear recurrence relation or less for any
X using matrices.
Sincerely,
Creighton
> Especially it would probably make the code in the SymbolicMultiplier
> much more concise. One should select the correspondence between indices
> and bases with some care. E.g., probably the neutral element (1) should
> have
> index 0. And do I guess correctly, that i, j, k with < or > on the top at
> the page
> http://fumba.eu/sitelayout/Floretion.html
> mean the non-neutral bases of those two Quaternion-algebras
> embedded in Floretion-algebra, and those with <> on the top are their
> various
> products?
> With proper allocation of indices 1-15 to these (think Gray Code, etc.)
> one
> would probably get some very concise code.
>
>
>
> Yours,
>
> Antti Karttunen
> who has his own mess in OEIS waiting to be cleaned and polished...
> (I hope to do it by myself one day.)
>
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