#8431: Substitutions over unit cube faces (Rauzy fractals)
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Reporter: vdelecroix
| Owner: tjolivet
Type: task
| Status: needs_work
Priority: major
| Milestone: sage-4.6
Component: combinatorics
| Keywords: word morphism unit face generalized substitution rauzy fractal
Author: Vincent Delecroix, Timo Jolivet, Franco Saliola, Stepan Starosta
| Upstream: N/A
Reviewer:
| Merged:
Work_issues:
|
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Comment(by tjolivet):
Hi Sébastien and the others,
''[We had a long discussion on skype with Sébastien regarding his last
reply.]''
I just uploaded a new version of the last patch. The face types are now
integers, and only integers (from 1 to d), and the
[http://trac.sagemath.org/sage_trac/search/opensearch?q=wiki%3AWordMorphism
WordMorphism] associated with an E1Star object is also defined on
integers.
Some other things I changed:
* I removed the `coloring` option in `apply_substitution`. Why? Because
we can use `repaint`, and it is better that all the "color business" is
taken care of by `repaint`. The colors of a face in an image by a
substitution is automatically the color of its preimage (which enables to
get the interesting coloring given by the substitution). If we want to
change the color of a patch obtained by iteration, `repaint`!
* I took care of the above remarks concerning `color_from_preimage`, and
used the clever solution suggested by Sébastien: dictionaries
`{types:colors}`.
* Some small fixes.
'''Now, a last and important remark'''. Writing
{{{
sigma = WordMorphism('1->12,2->13,3->1')
}}}
is less painful than writing
{{{
sigma = WordMorphism({1:[1,2], 2:[1,3], 3:[1]})
}}}
Since all the face types are integers and the
[http://trac.sagemath.org/sage_trac/search/opensearch?q=wiki%3AWordMorphism
WordMorphism] defining an `E1Star` must be on integers, we are supposed to
use the latter way to define `sigma`. However, I find it very
inconvenient, so we allow the user to give a `sigma` defined on an
'''arbitrary''' alphabet. When `E1Star` is defined, it is converted and
stored as a substitution on the alphabet `[1, ..., d]` (so that there is a
correspondence between the types of the faces and the alphabet of the
substitutions). It really makes the definitions much more convenient and I
don't think that it harms `sage` so much.
Also, I don't think that it would be useful to allow the user to specify
an arbitrary alphabet for the faces, since they are not used as the
letters of a word, but as three-dimensional objects, which we ''call'' of
type `1`, `2` or `3`. (This is why all the types are represented by
integers.) It corresponds to the mathematical definition of the object.
I strongly insist on this last remark!
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8431#comment:25>
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