#8431: Substitutions over unit cube faces (Rauzy fractals)
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Reporter: vdelecroix | Owner: tjolivet
Type: task | Status: new
Priority: major | Milestone: sage-4.6
Component: combinatorics | Keywords: word morphism
unit face generalized substitution rauzy fractal
Author: vdelecroix, tjolivet, sstarosta | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Changes (by newvalueoldvalue):
* keywords: word morphism => word morphism unit face generalized
substitution rauzy fractal
* owner: vdelecroix => tjolivet
* author: vdelecroix, sstarosta => vdelecroix, tjolivet, sstarosta
Old description:
> Thinking about it for a long time and motivated
> by [http://trac.sagemath.org/sage_trac/ticket/8423&usg=AFQjCNG8_y-
> UlT3ON3unoD3utIEHQfTYbQ| #8423] this ticket stands for
> * the creation of `WordMorphismExtension` and
> `WordMorphismExtensionDual` (from [http://iml.univ-
> mrs.fr/~arnoux/AISrev.pdf|an article of Sano-Arnoux-Ito])
> * think about easy to use functions for plotting Rauzy fractals using
> those algebraic tools. The broken lines and their projection for
> `WordMorphismExtension` and discrete planes approximation from
> `WordMorphismExtensionDual`.
>
> We could be able to access the extensions as a method of morphisms
> {{{
> sage: s = WordMorphism('a->ab,b->ac,c->a')
> sage: s.extension()
> Word morphism Extension [...]
> sage: s.extension_dual()
> Word morphism dual extension [...]
> }}}
>
> And plot them easily from fractals.
> {{{
> sage: s = WordMorphism('a->ab,b->ac,c->a')
> sage: fractals.RauzyFractal(morphism=s, format='broken_line')
> sage: fractals.RauzyFractal(morphism=s, format='discrete_plane')
> }}}
>
> The algebra requires the creation of few other classes which
> * Face and Patch (which is union of faces)
> * `WordMorphismExtension` and its dual
> * `WordMorphismExtensionIterations` for plotting approximations of the
> limit fractal from the extension or its dual
>
> TODO: think about cython for having fast data structure for Face and
> Patch (which heritates from tuple and set) !
New description:
This patch introduces unit cube faces and substitutions over them, as
defined in the article [http://iml.univ-mrs.fr/%7Earnoux/ArnouxIto.pdf
Pisot substitutions and Rauzy fractals] by Arnoux and Ito.
Three new classes are defined:
* `Face` -- models a unit cube face
* `Patch` -- models a finite collection of faces
* `E1Star` -- models the the ''E_1!^*(sigma)'' substitution (over faces)
defined by a unimodular substitution `sigma`
The plotting features enable us draw approximations of Rauzy fractals, or
to generate patches of discrete planes.
The dimension of the faces can be of any dimension (and the substitutions
work accordingly), but the plotting features work only in dimension three
(with three-letter alphabet substitutions).
--
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8431#comment:4>
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