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From masg2.epfl.ch!lafitte Wed May 3 21:25:04 0600 1995
From: Melvyn Lafitte
Date: Wed, 3 May 1995 21:25:04 -0600
To: njas@research.att.com
Subject: 1 0 1 1 1 1 2 0 2 2 2 1 3 1 2 1 2 2 4 1 4 3 3 1 4 2 4 2
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Attn: NJA Sloane
I'm sorry not to have got back to you before.
Like I told you, the sequence is related to a Self-Replicating tile.
(I'm sending another message with an uuencoded image of it)
The self-replicating tile may be obtained by the following iterated functions
system:\\
$
\begin{pmatrix}
\xi' \\
\eta'
\end{pmatrix}
=\frac{1}{2} (
\begin{pmatrix}
\xi \\
\eta
\end{pmatrix}
+
\begin{pmatrix}
0 \\
\frac{\sqrt{3}}{2}
\end{pmatrix} )
$
\\
$
\begin{pmatrix}
\xi' \\
\eta'
\end{pmatrix}
=\frac{1}{2} (
\begin{pmatrix}
\xi \\
\eta
\end{pmatrix}
+
\begin{pmatrix}
0 \\
\frac{\sqrt{3}}{6}
\end{pmatrix} )
$
\\
$
\begin{pmatrix}
\xi' \\
\eta'
\end{pmatrix}
=\frac{1}{2} (
\begin{pmatrix}
\xi \\
\eta
\end{pmatrix}
+
\begin{pmatrix}
\frac{1}{2} \\
0
\end{pmatrix} )
$
\\
$
\begin{pmatrix}
\xi' \\
\eta'
\end{pmatrix}
=\frac{1}{2} (
\begin{pmatrix}
\xi \\
\eta
\end{pmatrix}
+
\begin{pmatrix}
-\frac{1}{2} \\
0
\end{pmatrix} )
$
to the interval $\cal{I}$ \\
It is the attractor of this IFS, a very interesting triangle shaped fractal of
non-empty interior and of Hausdorff dimension equal to 2. (and that satisfies
the
open set condition without allowing a stronger condition)\\
This fractal has been discovered by myself in 1991. I later came to know that
Gosper had encountered it at about the same time, that Mandelbrot also
encountered an object, formed by the union of copies of this fractal, in 1975
and lately I read an article by C. Bandt exposing facts about self-replicating
tiles and giving as example this fractal.\\
At that time, I made a thorough study of this fractal. I've been able to prove
2 very elegant laws of formation (aside from its selfsimilar structure) based
on the way it is "filled", indeed this fractal is composed of filled equal
triangles . (equal in the sense: by similitude)\\
Now congruent filled triangles can be found on lines perpendicular to a side of
the triangle shape of the object (triangle convex hull).\\
As we consider smaller and smaller filled triangles, we get more and more of
them on these lines. \\
Well my sequence 1 0 1 1 1 ... results from counting these filled congruent
triangles on these lines, and so each term corresponds to a line, to the
counting on that line.
(Remark that on these lines, are also empty triangles, triangles congruent to
the triangles which are filled)\\
Understand that this forms a sequence only because there happens (proved) to be
an additional self-similarity given to the object, and that it assures that
counts for
a certain size of triangles will be the same than for a greater size of
triangles, but with more terms (counts) added.\\
I came to prove that this sequence is closely related to the sequence
A2487 M0141 N0056, in fact it is a transformation of this sequence by:
$new(3 n)=\frac{old(3n)+1}{2}$ \\
$new(3 n + 1)=\frac{old(3 n + 1)-1}{2}$ \\
$new(3 n + 2)=\frac{old(3 n + 2)}{2}$ \\
I did not intend to send you this sequence but another one which also results
from this triangle shaped fractal. It is by mistake that I copied this sequence
in my message and not that other one. In fact, this sequence I sent you is
pretty interesting, even if it is the transformation of another known sequence,
because for me it rather is the way those filled triangles are distributed
along these lines on this object before being a transformation of that already
known sequence.
So I can give you 60 terms:
1 0 1 1 1 1 2 0 2 2 2 1 3 1 2 1 2 2 4 1 4 3 3 1 4 2 4 2 3 2 3 0 3 3 4 2 6 3 5 2
5 4 7 2 6 4 4 1 5 3 6 3 6 4 6 1 5 4 5 2 5 2 3 ...\\
Now the intended integer sequence starts by:
1 0 2 1 1 2 5 0 10 6 3 2 18 2 10 ...\\
It is a consideration of the filled and empty triangles on a line as the binary
expansion of an integer:0 for empty and 1 for filled:
\begin{verbatim}
0 1
1 0 0 1
0 1 0 1 0 0 0 0
1 0 1 0 1 1 1 1 1 1 1
1 0 0 1 1 0 1 0 0 0 1 0 1 0 0 ...
\end{verbatim}
$\downarrow$\\
1;0;10;1;001;10;101;0;1010;110;00011;10;10011;010;1010;...
It describes these ``fillings'' on these lines much better than the rather
incomplete description that the counting of the filled (first sequence) only
gives us.\\
I will be sending you about 60 terms in the days coming.\\
There are'nt any references on this sequence. In fact, I plan to finally
publish
some of my results in the near future. For me this rep-tile (and the sequence
related to it), and the study I undertook on it, are only indications of a
general and elegant theory concerning ``critical'' cases of self-similarity,
theory I started to develop precisely after I met this example.
Please consider including these (especially the second one) sequences in your
table and continue this very useful work.\\
Best Regards,\\
MJL.
PS:Please refer me with the email adress:melvyn.lafitte@@dma.epfl.ch
instead of any other non constantly working adress. Thanks.
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______________________________________________________________________________
Melvyn Jeremie Lafitte | Departement de Mathematiques
dit Mandelvyn | Ecole Polytechnique Federale de Lausanne
Email: melvyn.lafitte@dma.epfl.ch | ETH-Lausanne (Sister of ETH-Zurich)
______________________________________________________________________________
Syag LeHochma -- Shtika
( Wisdom is delineated by silence.)
______________________________________________________________________________
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