ONT Re: Riffs And Rote Arithmetic
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RARA. Note 5
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Subj: Gatomorphisms, Cacti, Riffs, Rotes, Script N
Date: Thu, 08 Jan 2004 15:45:08 -0500
From: Jon Awbrey <jawbrey@att.net>
To: Antti Karttunen <...>
antti,
okay, will try to see what i can see in this again.
if you look at my "how i got into this" confession
on the zoo thread, you will see that i've suffered
a long love-tryst with arboreal primal parentheses:
zoo 3. http://suo.ieee.org/ontology/msg05163.html
but a critical juncture occurred around about 1980,
and it will be too much t'ask right now to say why,
but it forced me to shift to cacti, which are made
on an alphabet of 3 or 4 symbols "{", ",", ")" and
sometimes " ", depending on how you count the coup.
other than painted cacti, riffs and rotes keep the
old faith with trees, but all in all i have to hew
pretty close to the old lambda point trixt algebra,
geometry, and logic in order to keep the campfires
burning in the charmed circle. the burning number-
theoretic question remains, as ever: what portion
of the order of the integers is pure combinatorics?
enough for now,
jon
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>
> antti,
>
> will put this on the middle burner to simmer ...
>
> the perigatetic paper appears to have ambulated to here:
>
> http://www.math.uwaterloo.ca/JIS/VOL3/GUY/catwalks.html
>
> synchronicity strikes again! there is a section of my
> 1001-page dissipation that is headed up "catwalks" ...
> haven't looked at it in so long that i can't remember
> what it was about ...
>
> the paper by awbrey^2 ever-the-more moves, to here:
>
> HTTP://WWW.SAGEPUB.CO.UK/journalIssue.aspx?pid=105723&jiid=507153
> HTTP://WWW.SAGEPUB.CO.UK/JournalIssueAbstract.aspx?pid=105723&jiid=507153&jiaid=17772
>
> it's on "ingenta"(?) somewhere, if you get that, but i don't.
>
> oh, that jogs my mind, since we are using peirce's "ratlot" for our n'orleans efforts,
> if i were starting over with peirce, in a more sensible way this time, i'd start here:
>
> | Charles Sanders Peirce,
> |'Reasoning and the Logic of Things',
> |'The Cambridge Conferences Lectures of 1898',
> | Edited by Kenneth Laine Ketner, Introduction
> | by Kenneth Laine Ketner and Hilary Putnam,
> | Harvard University Press, Cambridge, MA, 1992.
>
> in a fog of a carl sandburg sort ...
>
> later,
>
> jon
>
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> Antti Karttunen wrote:
> >
> > On Wed, 17 Dec 2003, Jon Awbrey wrote:
> >
> > > > http://www.research.att.com/projects/OEIS?Anum=A073200
> > > >
> > > > (this is just a dirty, temporary "scaffold" for experimentation).
> > >
> > > i had thought that gatomorphisms had to do with cats!
> >
> > Yes, the etymology is very dubious, that is why
> > I usually use the word only inside the quotes.
> > (E.g. in CATalan, the cat is "gat", in Spanish "gato",
> > and in modern greek it seems to be "gatos".)
> > I think it was R.K. Guy that started this feline terminology
> > with his "Catwalks, Sandsteps and Pascal Pyramids, J. Integer Seqs.,
> > Vol. 3 (2000), #00.1.6":
> >
> > http://www.research.att.com/~njas/sequences/JIS/VOL3/GUY/catwalks.html
> >
> > > "el gato" (the cat) was a tv show character from my
> > > childhood in texas, kinda like "zorro" (the fox),
> > > but i could not find him on google, kept getting
> > > stuff like this:
> > >
> > > http://www.elgatomusic.com/
> > > http://centros5.pntic.mec.es/ies.victoria.kent/Rincon-C/Curiosid/Rc-31/RC-31.htm
> > > http://www.kids-places.com/El_gato_en_el_sombrero_The_Cat_In_The_Hat_0394816269.html
> > >
> > > > This program gives a list of the most important ones,
> > > > and shows how they can be simply defined in the terms
> > > > of each other (the ones in A089840 forming the most primitive
> > > > set, from which to start the building):
> > > >
> > > > http://www.research.att.com/~njas/sequences/gatomorf.c.txt
> > > >
> > > > And here is some Prolog-code illustrating the
> > > > non-recursive gatomorphisms given in A089840:
> > > >
> > > > http://www.research.att.com/~njas/sequences/A089840p.txt
> > > >
> > > > And the actual C-code to search a lots of them:
> > > >
> > > > http://www.research.att.com/~njas/sequences/gatonore.c.txt
> > >
> > > okay, here i will need some kind of key to
> > > the underlying aesthetic or main question ...
> >
> > Well, start from:
> > http://www.research.att.com/~njas/sequences/a014486.ps.gz
> > and consider the bijections between each of the "manifestations".
> > Now, all the possible symmetry operations that one can subject
> > those manifestations to (e.g. to rotate the non-crossing
> > handshake arrangements or Eulerian polygon diagonalizations)
> > are easily definable as say Lisp- Scheme- or Prolog-functions
> > that act on "symbolless" S-expressions, i.e. ones that contain
> > only () (an empty or NIL pointer) as their terminal nodes,
> > i.e. which are just another incarnation of Catalania.
> >
> > But of course, I haven't limited myself to those "natural symmetry"
> > operations only, but has experimentally constructed many other
> > bijections of binary trees.
> >
> > And furthermore, by those bijective mappings implied in that Postscript
> > illustration, any such bijection IS ALSO a bijection of any other
> > manifestation depicted there.
> >
> > (Actually, the first instance of such bijection (apart from the
> > trivial identity) is not submitted by me, but is Wouter Meeussen's
> > "depth first" <-> "breadth first" transformation of the binary trees.
> > In its first form it occurs as:
> >
> > http://www.research.att.com/projects/OEIS?Anum=A038776
> >
> > but in my "signagture-permutation" standard-form as A057117/A057118.)
> >
> > The tables A073200 & A089840 are my first attempts to systematize
> > the construction of these arbitrary "gatomorphisms". Of course
> > they are just a small subset of all the possible ones (as the
> > latter is a non-enumerable set), and I conjecture that e.g.
> > that Meeussen's transformation will be never encountered
> > from A073200.
> >
> > One of the interesting questions is:
> >
> > How many OEIS-sequences (that is, ones occurring in other contexts
> > than just with gatomorphisms, which I have submitted myself), can
> > be "realized" as the sequences which count say, either the number
> > of fixed points or number of cycles (orbits) obtained when a
> > gatomorphism acts on the binary trees of size n? (OK, here
> > one might add some limiting conditions to the complexity
> > of the defining formula, as to make the questions more
> > meaningful.)
> >
> > I'm just starting to find some general rules regarding this.
> > E.g. If the gatomorphism X has Axxxxxx as its fix-point sequence,
> > then if we have the gatomorphism X' which is defined so that it
> > lets X act on the left subtree of the binary tree, and leaves the
> > right subtree intact, then the corresponding fixpoint sequence is
> > CONV(Axxxxxx, A000108), i.e. the convolution of the original with
> > Catalan numbers. By another obvious construction, we always can
> > form a gatomorphism, whose fixpoint sequence is INVERT(Ayyyyy),
> > if Ayyyyyy is the fixpoint sequence of some gatomorphism we
> > know to construct. (This is the reason why I recently was
> > so incessant to ensure that A000127 indeed is INVERT(A000045).)
> >
> > On deeper level, we can ask: How much potential computing power there
> > is in this framework, which itself is just a subset of lambda-calculus?
> >
> > (Well, especially if one doesn't limit oneself just to bijections.)
> >
> > Also, think that Peano axioms define a linear order, where an integer
> > always has a successor. Now, similar "binary tree axioms" define that
> > each node has two children, the left and the right. We get infinite
> > binary trees, of which instances are Stern-Brocot tree in its various
> > manifestations (one can consider even Kepler's tree:
> >
> > http://www.research.att.com/projects/OEIS?Anum=A086592
> >
> > as one of these), and those gatomorphisms which are well-defined also
> > on infinite binary trees, can then be used to induce various bijections
> > of rationals. (Hmm, actually, many of these might extend to R as well.)
> >
> > Yours,
> >
> > Antti
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http://www.cs.bsu.edu/homepages/mighty/history.html
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