Opened 7 years ago

Closed 7 years ago

# Substitutions over unit cube faces (Rauzy fractals)

Reported by: Owned by: vdelecroix tjolivet major sage-4.6.1 combinatorics word morphism unit face generalized substitution rauzy fractal sage-combinat, slabbe, abmasse, tmonteil sage-4.6.1.alpha2 Vincent Delecroix, Timo Jolivet, Franco Saliola, Štěpán Starosta Sébastien Labbé, Alexandre Blondin Massé N/A

This patch introduces unit cube faces and substitutions over them, as defined in the article 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).

### comment:1 Changed 7 years ago by vdelecroix

• Description modified (diff)

### comment:2 Changed 7 years ago by vdelecroix

• Description modified (diff)
• Type changed from defect to task

### comment:3 Changed 7 years ago by slabbe

• Description modified (diff)

I add myself in cc because I am interested.

### comment:4 Changed 7 years ago by tjolivet

• Authors changed from vdelecroix, sstarosta to vdelecroix, tjolivet, sstarosta
• Description modified (diff)
• Keywords unit face generalized substitution rauzy fractal added
• Owner changed from vdelecroix to tjolivet
• Summary changed from Rauzy fractal (discrete planes and broken lines) to Substitutions over unit cube faces (Rauzy fractals)

### comment:5 Changed 7 years ago by tjolivet

• Status changed from new to needs_review

### comment:7 Changed 7 years ago by mhansen

I feel like Patch, Face, E1Star might be too domain specific to import into the global namespace.

### comment:8 Changed 7 years ago by slabbe

• Authors changed from vdelecroix, tjolivet, sstarosta to Vincent Delecroix, Timo Jolivet, Stepan Starosta, Franco Saliola
• Status changed from needs_review to needs_work

Cher Timo,

Thanks a lot for your recent effort in polishing that code. I am actually needing it rigth now!! It will be of great help to me.

I just read through the patch from my web browser and got those comments :

1. As Mike Hanson, I think those three classes should not be in the global namespace. To create a E1Star, I recommand to add a method in the WordMorphism class.
2. Add the file to the documentation. To do this you must edit the file sage-your-branch/doc/en/reference/combinat.rst
3. Each sage block must be preceded by ::.
4. No example using color in Face __init__ function.
5. Examples of all the possible drawings.
6. Rename _image_by_substitution to _apply_substitution in Face class.
7. Face._image_by_substitution should only accept E1Star object. If not, it should raise an TypeError and not try to coerce the input to an E1Star object.
8. Patch.translate method forgets the color of the faces.
9. Patch.apply_substitution should only accept E1Star object. If not, it should raise an TypeError and not try to coerce the input to an E1Star object.
10. _M and _inv_M should become lazy_attributes
11. _inv_M should be computed from _M
12. For the matrix method of E1Star, should _inv_M be the "matrix associated with self" more than _M ?
13. When I used these classes last week, I needed the method __eq__ for Patch. Here is my code :
    def __eq__(self, other):
r"""
EXAMPLES::

TODO!!

"""
return (isinstance(other, Patch) and
set(self) == set(other) )


I am going to download, apply and use the patch real soon...and I guess I will have more comments...

### comment:9 Changed 7 years ago by slabbe

I suggest to add this example for the method __eq__ for Patch class :

    def __eq__(self, other):
r"""
EXAMPLES::

sage: from sage.combinat.e_one_star import E1Star, Face, Patch
sage: x = WordMorphism('0->02,1->012,2->2')
sage: y = WordMorphism('0->012,1->12,2->2')
sage: p = Patch([Face((0,0,0),1), Face((0,0,0),2), Face((0,0,0),3)])
sage: E1Star(x) (p) == E1Star(y) (p)
False
sage: E1Star(x * y) (p) == E1Star(y) (E1Star(x) (p))
True

"""
return (isinstance(other, Patch) and
set(self) == set(other) )



I also suggest to add a __len__ method and change the __repr__ to behave like Graphs (Graph of 45 vertices).

    def __repr__(self):
r"""
String representation of a patch.

EXAMPLES:

sage: x = [Face((0,0,0),t) for t in [1,2,3]]
sage: P = Patch(x)
sage: P
Patch of 3 faces
"""
return "Patch of %s faces"%len(self)

def __len__(self):
r"""
Returns the number of faces.

EXAMPLES::

sage: x = [Face((0,0,0),t) for t in [1,2,3]]
sage: P = Patch(x)
sage: len(P)       #indirect doctest
3
"""
return len(self._faces)


### comment:10 Changed 7 years ago by slabbe

The method __call__ of E1Star should pass the possible kwds to apply_substitution method :

    def __call__(self, patch, **kwds):
r"""
EXAMPLES:

sage: p = Patch([Face((0,0,0),t) for t in [1,2,3]])
sage: sigma = WordMorphism('1->12,2->13,3->1')
sage: E = E1Star(sigma)
sage: E(p)
Patch: [[(1, 0, -1), 1]*, [(0, 1, -1), 2]*, [(0, 0, 0), 3]*, [(0, 0, 0), 1]*, [(0, 0, 0), 2]*]
"""
patch = Patch(patch)
patch.apply_substitution(self, **kwds)
return patch


### comment:11 Changed 7 years ago by slabbe

The last lines of plot_tikz should be :

        s += '\\end{tikzpicture}\n'
return LatexExpr(s)


Make sure to add the following line in the beginning of the file :

from sage.misc.latex import LatexExpr


This is to avoid latex(a_patch.plot_tikz()) be edited wrongly.

### comment:12 Changed 7 years ago by slabbe

The function plot_tikz should be renamed _latex_.

A graph in Sage has methods plot, plot3d and _latex_ and I think we can follow the same behavior.

### comment:13 Changed 7 years ago by tjolivet

• Authors changed from Vincent Delecroix, Timo Jolivet, Stepan Starosta, Franco Saliola to Vincent Delecroix, Timo Jolivet, Franco Saliola, Stepan Starosta
• Description modified (diff)
• Status changed from needs_work to needs_review

### comment:14 in reply to: ↑ description Changed 7 years ago by slabbe

EDIT (2010-09-19): a new version of the patch is attached (trac_8431_e_one_star.patch), taking into account the suggestions given in the comments. Documentation compiles fine, except for the following warning, which I don't think comes from an error in e_one_star.py:

/home/timo/sage-4.5.3/local/lib/python2.6/site-packages/sage/combinat/e_one_star.py:docstring of sage.combinat.e_one_star:56: (WARNING/2) Literal block expected; none found.

Actually, it does. If you don't write a sage block after the line EXAMPLES:, then you must write only one colon:

EXAMPLES:

We start by drawing a simple three-face patch::


EXAMPLES::

We start by drawing a simple three-face patch::


One more detail about ReST :

This module implements the E_1^*(\sigma) substitution
associated with a one-dimensional substitution \sigma,
that acts on unit faces of dimension (d-1) in \mathbb R^d.


The macro \RR, as well as \ZZ and \QQ is defined to get the doc more readable in the terminal, so the following should work : \RR^d.

### comment:15 Changed 7 years ago by tjolivet

Okay, thanks again for the useful comments!

I've updated the patch.

### comment:16 Changed 7 years ago by slabbe

• Status changed from needs_review to needs_work

I just uploaded a patch which fixes some syntax stuff but I did a little bit more than that. More importantly, I :

• moved Face._apply_substitution to the method E1Star._call_on_face because that is where that code belongs.
• reworked Patch.apply_substitution and moved the code that was repainting the faces to a method called Patch.repaint
• moved the code of Patch.apply_substitution to E1Star.__call__ because it *was* creating a copy anyway. Patch.apply_substitution works as before. E1Star.__call__ works twice as fast as before because it doesn't do two copies but only one.
• added some cached method and lazy attributes in E1Star class

I still see one possible problem before giving a positive review, but didn't have time enough to test/fix it. The attribute E1Star._base_iter seems to assume some conditions on the values of faces types. That they are integer for instance. Haven't tested it.


Timo, now that I uploaded a new patch, you will have to create a new separate patch that applies over yours and mine if you want to modify anything.

### Changed 7 years ago by slabbe

Applies over the trac_8431_e_one_star.patch

### comment:17 Changed 7 years ago by slabbe

Okay, I just re-uploaded my review patch. The faces type is now considered as an arbitrary object. It can be any hashable object. Doing this, the code is more readable (e.g. the method _base_iter) and more versatile (no such type + 1 assuming type is an integer).

In order to do so, I changed the way E1Star._base_iter behave. It now returns a dict.

And now, the domain alphabet of the substitution used must correspond to the type of faces. No more '1' -> rank('1') = 0 -> + 1 -> 1 = type of a face. The type of faces iff domain alphabet. If one wants to use integer type of faces, he must define a substitution on integers which is possible using dictionary.

I also moved orig_coords to an attribute of Patch called face_contour. Code gets cleaner that way also. face_contour now gets assigned at the creation of a Patch.

I also added #not tested to many plot test so that the time of testing for that file is decreased from 24s to 7s on my computer. I kept some plot tested at minimum.

My very last concern about the actual state of the ticket is about the methods outputting tikz code. I want to make sure that they work well. I will do more test later (I'm done for the day). The necessity of \newcommand needs to be studied.

Sorry Timo, I am very serious on the review. But I think it is better and easier to do changes now than after inclusion in Sage.

Cheers,

### comment:18 follow-up: ↓ 19 Changed 7 years ago by tjolivet

Okay I'll continue working on it! Thanks again for all you advices and improvements!

I'll do the remaining things we talked about and upload a new patch.

As for the tikz code, I am sure that it compiles fine (with a normal texlive installation). The \newcommand is nice because then one face is reprensented by a small line instead of three bulky ones. It could be two lines, but in tikz there is no other way to specify an RGB color than using \definecolor, which takes one more line. (Something similar to black!30!green for RGB would be the great...)

### comment:19 in reply to: ↑ 18 ; follow-up: ↓ 20 Changed 7 years ago by abmasse

Hello, Timo and Sébastien !

As for the tikz code, I am sure that it compiles fine (with a normal texlive installation). The \newcommand is nice because then one face is reprensented by a small line instead of three bulky ones. It could be two lines, but in tikz there is no other way to specify an RGB color than using \definecolor, which takes one more line. (Something similar to black!30!green for RGB would be the great...)


I might have a solution that would avoid using them, but which is not as neat. You could use

\foreach \x / \y / \z / \c1 / \c2 / \c3 in {...}


and replace the ... by the appropriate very long list of parameters for each face and then put in the body of the \foreach command the 3-lines Latex statement that draws the face with these parameters.

What do you say? Do you understand what I'm talking about (I don't feel my explanation is very clear)?

### comment:20 in reply to: ↑ 19 ; follow-up: ↓ 21 Changed 7 years ago by tjolivet


Yes, you made yourself clear, and it seems like a good idea. I'll try it on big instances, and if it works, I'll make it that way in the new patch. Otherwise, moving the \newcommand outside the tikzpicture is not a problem at all (it should have been that way, I probably made a typo''...).

### comment:21 in reply to: ↑ 20 ; follow-up: ↓ 22 Changed 7 years ago by abmasse

Yes, you made yourself clear, and it seems like a good idea. I'll try it on big instances, and if it works, I'll make it that way in the new patch. Otherwise, moving the \newcommand outside the tikzpicture is not a problem at all (it should have been that way, I probably made a typo''...).

The problem is that if you use the tikz_plot twice in the document with sagetex there will be a compilation problem (since you can't use \newcommand on an already define command (you need local variables!).

There might be another solution, using \def\loza#1#2#3#4#5#6 (which is local when included in a tikzpicture environment), but I haven't tried it out.

### comment:22 in reply to: ↑ 21 Changed 7 years ago by tjolivet

The problem is that if you use the tikz_plot twice in the document with sagetex there will be a compilation problem (since you can't use \newcommand on an already define command (you need local variables!). There might be another solution, using \def\loza#1#2#3#4#5#6 (which is local when included in a tikzpicture environment), but I haven't tried it out.

Okay, so I'll get rid of the \newcommand in any case, and use either the \foreach or the \def solution; I admit I also like the idea of outputing a tikzpicture that works out of the box''. Thanks for the suggestions and comments.

### comment:23 follow-up: ↓ 24 Changed 7 years ago by tjolivet

• Description modified (diff)

Hello, I uploaded a new patch (that applies over e_one_star and  and review-sl).

• It changes the plot_tikz function. There are no more \newcommand's, but only \def's that are inside the tikzpicture environment. (No more global annoyance, and it should work fine with sagetex now.) Also, RGB colors a printed with two decimals only.
• In response to "the domain alphabet of the substitution used must correspond to the type of faces": now, every substitution defining an E1Star object is converted to an equivalent substitution on alphabet '1', ..., 'd'. The type of a face can be specified by a positive integer or a string representing a positive integer, and is converted to a string when the Face object is created. This allows us to assume that the type of a face in 3D is in {'1', '2', '3'}, which is very useful for plotting functions.
• An option to color face according to their type has been added.
• It is possible to choose whether to (or not to) print the tikzpicture environment definition, and the \def macros.
• Some other minor fixes, and small documentation fixes.

### comment:24 in reply to: ↑ 23 Changed 7 years ago by slabbe

Salut Timo,

I look at the patch and still have comments.

• It changes the plot_tikz function. There are no more \newcommand's, but only \def's that are inside the tikzpicture environment. (No more global annoyance, and it should work fine with sagetex now.) Also, RGB colors a printed with two decimals only.

Great for the \def instead of \newcommand.

• In response to "the domain alphabet of the substitution used must correspond to the type of faces":

When I wrote that comment, I was meaning that I fixed it in my patch. In fact, I removed the coercion of the face type into the integer ring which allows the user to use the type of object he wants for the face types.

now, every substitution defining an E1Star object is converted to an equivalent substitution on alphabet '1', ..., 'd'. The type of a face can be specified by a positive integer or a string representing a positive integer, and is converted to a string when the Face object is created.

I do not agree with this. I do not want the type face to be converted to an integer nor to a str. I want it to stays as it is given by the user. I want the correspondance between the faces type and the domain alphabet of the morphism to be clean. I do not want the correspondance appear by coincidence. Here are three examples to clarify my point :

GOOD (integer faces type with integers for the domain of the substitution) :

    sage: from sage.combinat.e_one_star import E1Star, Face, Patch
sage: x = [Face((0,0,0),1), Face((0,0,0),2), Face((0,0,0),3)]
sage: P = Patch(x)
sage: sigma = WordMorphism({1:[1,2], 2:[1,3], 3:[1]})
sage: E = E1Star(sigma)
sage: E(P)
Patch: [[(1, 0, -1), 1]*, [(0, 1, -1), 2]*, [(0, 0, 0), 3]*, [(0, 0, 0), 1]*, [(0, 0, 0), 2]*]


GOOD (str faces type with str for the domain of the substitution) :

    sage: from sage.combinat.e_one_star import E1Star, Face, Patch
sage: x = [Face((0,0,0),'1'), Face((0,0,0),'2'), Face((0,0,0),'3')]
sage: P = Patch(x)
sage: sigma = WordMorphism('1->12,2->13,3->1')
sage: E = E1Star(sigma)
sage: E(P)
Patch: [[(1, 0, -1), 1]*, [(0, 1, -1), 2]*, [(0, 0, 0), 3]*, [(0, 0, 0), 1]*, [(0, 0, 0), 2]*]


BAD (integer faces type with str for the domain of the substitution). Correspondance is lost. If the following is suppported, it makes the code less usable for potential future uses by others.

    sage: from sage.combinat.e_one_star import E1Star, Face, Patch
sage: x = [Face((0,0,0),1), Face((0,0,0),2), Face((0,0,0),3)]
sage: P = Patch(x)
sage: sigma = WordMorphism('1->12,2->13,3->1')
sage: E = E1Star(sigma)
sage: E(P)
Patch: [[(1, 0, -1), 1]*, [(0, 1, -1), 2]*, [(0, 0, 0), 3]*, [(0, 0, 0), 1]*, [(0, 0, 0), 2]*]


The reason is that the code we write is *always* used differently by the others. And the coercion of the face type to a str object is an unneccessary restriction. Do you agree with me?

This allows us to assume that the type of a face in 3D is in {'1', '2', '3'}, which is very useful for plotting functions.

Can you explain me why it is useful for plotting functions? Is it because digits can not be used in a definition like \def\loz1{...} ? But then, I don't understand why '1' is better than 1. I am sure we can find a solution for the plotting problem that will not restrict the usability of the code.

• An option to color face according to their type has been added.

Thanks for fixing this in the __call__ method. Small suggestions : coloring='color_from_type' is redundant. I suggest that the following

-  color_from_type - boolean (default: False). Colors the faces
according to their type.


becomes

-  'face_type' - boolean (default: False). Colors the faces
according to their type.


For the repaint method. I dislike the addition of a new argument to the function. The reason is that the user should never use the argument cmap at the same time as the argument color_from_type. Hence, inconstencies in the arguments can happen and should be considered. This tells us that it should be only one argument that plays the role for either one or the other. Python is great for that because it is not a typed language. Moreover, I was disappointed that it is not possible for the user to specify which color he can give to each faces type. Hence, I suggest to do the following which should fix all the problems I mention::

    def repaint(self, cmap='hsv'):
r"""
Repaint all the faces of self from the given color map.

This changes self.

INPUT:

-  cmap - color map (default: 'hsv'). It can be one of the
following :

- string - A color map. For available color map names type: import
matplotlib.cm; matplotlib.cm.datad.keys(). Each face will be
given a color according to their rank in the patch.
- list - a list of color to assign to the faces in order.
- dict - a dict of faces type mapped to colors.
- {}, the empty dict - the dict {'1':'red', '2':'green', '3':'blue'}              is used.

• It is possible to choose whether to (or not to) print the tikzpicture environment definition, and the \def macros.

Ok. And by default, they are printed : great.

• Some other minor fixes, and small documentation fixes.

Great.

Finally, lines like

if print_macros == True:


should be replaced by

if print_macros:


That is it for now.

If you rework on it, you can reload your last patch or add a new one. It is as you wish. I do not have any local modifications.

Cheers!

### comment:25 follow-up: ↓ 26 Changed 7 years ago 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 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 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!

### comment:26 in reply to: ↑ 25 ; follow-up: ↓ 27 Changed 7 years ago by abmasse

Hi, Timo and Sébastien !

Hi Sébastien and the others,

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]})


I agree with you on that one. This is something I've noticed too when using WordMorphism's. The thing is: the problem comes from the object WordMorphism?, and it's where it should be solved, in my opinion, and not in the class E1Star.

Since all the face types are integers and the 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 disagree. If I'm not mistaken, the integer meaning of 1, 2, 3 is never used in the construction of patches and discrete plane. I mean, we could use a,b,c and everything would work the same, there is no gain from encoding it with integers, since no additive group structure is used (correct me if I'm wrong, though, that could be a good argument).

In fact, I think it's quite the opposite. It would be better to allow the user to have any alphabet, so that he can use any structure he wants (additive group, for instance, or any other operation that might encode interesting discrete plane construction).

I strongly insist on this last remark!

What I propose you is that either Sébastien or I work on the improvement of the construction of WordMorphism in order to offer something like

sigma = WordMorphism('1->12,2->13,3->1', type='integer')


Does that seem reasonable?

P.S. I'm following your discussion and I see the progress... Keep the good work!

### comment:27 in reply to: ↑ 26 Changed 7 years ago by tjolivet

Thanks for you answer and hello!

I disagree. If I'm not mistaken, the integer meaning of 1, 2, 3 is never used in the construction of patches and discrete plane. I mean, we could use a,b,c and everything would work the same, there is no gain from encoding it with integers, since no additive group structure is used (correct me if I'm wrong, though, that could be a good argument). In fact, I think it's quite the opposite. It would be better to allow the user to have any alphabet, so that he can use any structure he wants (additive group, for instance, or any other operation that might encode interesting discrete plane construction).

Yes, the types are just names, so anything else could be used. But I'm not sure that it would be very useful, it's not comparable to allowing any choice of letters for words, for example. If you (or anybody else) want to implement it, it's OK, but please make sure that if integer faces are given, it works as in the current patch by default (to make it "easy-usable").

What I propose you is that either Sébastien or I work on the improvement of the construction of WordMorphism in order to offer something like  sigma = WordMorphism('1->12,2->13,3->1', type='integer')  Does that seem reasonable? P.S. I'm following your discussion and I see the progress... Keep the good work!

This a good idea, even with respect to WordMorphism only. I'm just a bit saddened that we will have to add "type='integer'") when specifying a substitution to work with the default integers, but I can live with it!

---

I think that apart from the possible option to leave the choice of the alphabet, and the "type='integer'" feature to be added to WordMorphism, the work on this patch is over. Do you agree? I leave these two last points up to you.

Thanks for all your help, work and comments on this (it's my first patch, as you know), I'm learning a lot. And I'm ready to work more if there is something more that I forgot!

### comment:28 Changed 7 years ago by slabbe

Hi Timo,

• 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!

Great idea! Very very good. That is really upgrading the quality of the code. The design is converging!

• I took care of the above remarks concerning color_from_preimage, and used the clever solution suggested by Sébastien: dictionaries {types:colors}.

Great. I am just wondering if we could avoid to use a list of one color to change the color of every faces. If their is no intersection between the list of available color and the list of available color map, then simply a color string could be used for coloring every faces the same color.

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]})


I agree.

Since all the face types are integers and the 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!

One solution is to allow face type '1', '2' and '3' which would work easily with such expressions : WordMorphism('1->12,2->13,3->1'). But I know you don't like this solution.

On the other side, as I already explained, I dislike the solution you propose because (1) it is getting the code less efficient for those who specify a WordMorphism? already defined on integers, (2) because those translation of object gets the code less versatile and reusable by others and (3) it may leads to conceptual problems for the user when letters gets translated not in the way he expected.

Personnaly, I can not give a positive review with this translation of alphabet.

I think Alexandre's ideas for improving the definition of WordMorphism might be a good compromise.

Moreover, in the code of plot_tikz, I suggest to use the following ideas :

sage: P = Patch([Face((0,0,0),t) for t in [1,2,3]])
sage: d = P._face_contour
sage: ' -- '.join(map(str, d[1])) + ' -- cycle;\n'
'(0, 0, 0) -- (0, 1, 0) -- (0, 1, 1) -- (0, 0, 1) -- cycle;\n'


rather than hardcoding the face contour points:

        if print_macros:
s += '\\def\\loza#1#2#3#4#5#6{\n'
s += '  \\definecolor{facecolor}{rgb}{#4,#5,#6}\n'
s += '  \\fill[fill=facecolor, draw=black, shift={(#1, #2, #3)}]\n'
s += '  (0,0,0) -- (0,1,0) -- (0,1,1) -- (0,0,1) -- cycle;\n'
s += '}\n'


### comment:29 Changed 7 years ago by slabbe

Just added a patch that improves the color map manipulations and tikz code. I removed the Face._plot_tikz method that was not really usable on its own and moved its code into the Patch.plot_tikz.

Patches apply in this order :

1. trac_8431_e_one_star.patch
2. trac_8431_review-sl.patch
3. trac_8431_might_be_final_tj.patch
4. trac_8431_colors_tikz-sl.patch

Earlier, I wrote :

Great. I am just wondering if we could avoid to use a list of one color to change the color of every faces. If their is no intersection between the list of available color and the list of available color map, then simply a color string could be used for coloring every faces the same color.

Forget about that, I was wrong. I like the way it is.

### Changed 7 years ago by slabbe

Applies over the precedent 3 patches

### comment:30 Changed 7 years ago by slabbe

So, now, I can say that I give a positive review to this ticket if the following 5 lines of E1Star.__init__ are removed.

        # changing the alphabet of sigma to [1, ..., d]
A = sigma.domain().alphabet()
change_alphabet = WordMorphism(dict([(A[i], [Integer(i+1)]) for i in range(len(A))]))
D = dict([(change_alphabet(a)[0], list(change_alphabet(sigma(a)))) for a in A])
sigma = WordMorphism(D)


Of course, my two patches also have to get a positive review by Timo or Alexandre (preferably both) before the ticket be ready to be merged into Sage.

### comment:31 follow-up: ↓ 33 Changed 7 years ago by tjolivet

OK, I uploaded a last patch to fix the type of the WordMorphism. The __init__ method of E1Star accepts only integer substitutions. I have changed the doc accordingly. I hope that the type='integer' option of WordMosphim will be available soon!

The patch seems to be ready. How do we give positive review of an individual patch, and not to the whole ticket? How long will it take before it is integrated to sage?

To recap, the patches apply in this order:

1. trac_8431_e_one_star.patch
2. trac_8431_review-sl.patch
3. trac_8431_might_be_final_tj.patch
4. trac_8431_colors_tikz-sl.patch
5. trac_8431_sigma_type_fix.patch

(Can we remove the useless trac_8431_E1Star.patch patch?)

sigma type fix

### comment:33 in reply to: ↑ 31 Changed 7 years ago by abmasse

The patch seems to be ready. How do we give positive review of an individual patch, and not to the whole ticket? How long will it take before it is integrated to sage?

Normally, when two people work on the same ticket and both of them provide patches, the last person cannot make the final positive review, so in that case, it's Sébastien that should do it. However, I will probably do it this week since both you and Sébastien worked on it. This way, it will give a different point of view. The patch probably won't be merged in sage-4.6 as it is already in its alpha stage, but should be included in the next version.

To recap, the patches apply in this order:

1. trac_8431_e_one_star.patch
2. trac_8431_review-sl.patch
3. trac_8431_might_be_final_tj.patch
4. trac_8431_colors_tikz-sl.patch
5. trac_8431_sigma_type_fix.patch

(Can we remove the useless trac_8431_E1Star.patch patch?)

Unfortunately, you can't.

### Changed 7 years ago by slabbe

Applies over the precedent 5 patches

### comment:34 Changed 7 years ago by slabbe

• Reviewers set to Sébastien Labbé, Alexandre Blondin-Massé
• Status changed from needs_work to needs_review

Just added a new patch that adds two methods that might be useful : E1Star.__mul__ and WordMorphism.dual_map.

Patches need to be applied in this order:

1. trac_8431_e_one_star.patch
2. trac_8431_review-sl.patch
3. trac_8431_might_be_final_tj.patch
4. trac_8431_colors_tikz-sl.patch
5. trac_8431_sigma_type_fix.patch
6. trac_8431-wordmorphism-sl.patch

### comment:35 Changed 7 years ago by slabbe

• Reviewers changed from Sébastien Labbé, Alexandre Blondin-Massé to Sébastien Labbé, Alexandre Blondin Massé

small fixes

### comment:36 follow-up: ↓ 37 Changed 7 years ago by tjolivet

Hi,

This patch (trac_8431-smallfixes-tj.patch) corrects three bugs and adds three minor features:

• in .plot_tikz, the option edgecolor='facecolor' didnt't work
• in .plot_tikz, the option print_tikz_env=False caused an error message
• in .plot3d, the plot was rotated uselessly (it was confusing)
• patches can now be added (which gives their union)
• E1Star morphisms can now be multiplied (according to the rule E1Star(s*t) = E1Star(t)*E1Star(s))

Patches need to be applied in this order:

1. trac_8431_e_one_star.patch
2. trac_8431_review-sl.patch
3. trac_8431_might_be_final_tj.patch
4. trac_8431_colors_tikz-sl.patch
5. trac_8431_sigma_type_fix.patch
6. trac_8431-wordmorphism-sl.patch
7. trac_8431-smallfixes-tj.patch

### comment:37 in reply to: ↑ 36 Changed 7 years ago by tjolivet

• in .plot3d, the plot was rotated uselessly (it was confusing)

A precision about the .plot3d fix, the removal of these two lines:

911 P.aspect_ratio((1,1,1))
912 P = P.rotateY(pi/2).rotateX(pi/2)

The first line doesn't change anything. The only way to control the aspect ratio of the rendered object in Jmol is to include it into a cube that contains everything, because Jmol will automatically take the smallest 3D rectangle that contains the object, and deform it to a cube (hence changing the aspect ratio). If a cube bounds everything, we get an actual aspect ratio of (1,1,1).

The second line was initally written to "turn" the patch so that it faces the viewer when Jmol opens. The problem is that rotate does not change the point of view, but rotates the object itself, which is very confusing when we want to plot something else with the Patch (normal vectors or contracting planes, for example).

### comment:38 Changed 7 years ago by tjolivet

I just added a new patch (trac_8431-alphaset-tj.patch) that applies over the previous ones, and that allows to set the alpha parameter in the .plot method. It is useful if we want to plot fractals without seeing the unit cube edges.

### comment:39 Changed 7 years ago by abmasse

Hi, Timo !

I tried applying the various patches on a sage-4.6 clone and the patch trac_8431-smallfixes-tj.patch fails. Here's what I get.

labo [~/Applications/sage/devel/sage-t8431/sage/combinat]
$hg qseries trac_8431_e_one_star.patch trac_8431_review-sl.patch trac_8431_might_be_final_tj.patch trac_8431_colors_tikz-sl.patch trac_8431_sigma_type_fix.patch trac_8431-wordmorphism-sl.patch trac_8431-smallfixes-tj.patch labo [~/Applications/sage/devel/sage-t8431/sage/combinat]$ hg qtop
trac_8431-smallfixes-tj.patch

labo [~/Applications/sage/devel/sage-t8431/sage/combinat]
$hg qpop now at: trac_8431-wordmorphism-sl.patch labo [~/Applications/sage/devel/sage-t8431/sage/combinat]$ hg qpush
applying trac_8431-smallfixes-tj.patch
patching file sage/combinat/e_one_star.py
Hunk #6 FAILED at 1337
1 out of 7 hunks FAILED -- saving rejects to file sage/combinat/e_one_star.py.rej
patch failed, unable to continue (try -v)
patch failed, rejects left in working dir
errors during apply, please fix and refresh trac_8431-smallfixes-tj.patch


Could you please fix it? Or is it because the order is not correct? Here's the output for the rejected hunk:

labo [~/Applications/sage/devel/sage-t8431/sage/combinat]
$cat e_one_star.py.rej --- e_one_star.py +++ e_one_star.py @@ -1230,49 +1338,6 @@ X[letter].append((v, k)) return X - def __call__(self, patch, iterations=1): - r""" - Apply a generalized substitution to a Patch; this returns a new - object. - - The color of every new face in the image is given the same color as its preimage. - - INPUT: - - - patch - a patch - - iterations - integer (optional, default: 1) number of iterations - - OUTPUT: - - a patch - - EXAMPLES:: - - sage: from sage.combinat.e_one_star import E1Star, Face, Patch - sage: P = Patch([Face((0,0,0),t) for t in [1,2,3]]) - sage: sigma = WordMorphism({1:[1,2], 2:[1,3], 3:[1]}) - sage: E = E1Star(sigma) - sage: R = E(P) - sage: len(R) - 5 - - :: - - sage: x = (0,0,0) - sage: p = Patch([Face(x, 1, 'red'), Face(x, 2, 'yellow'), Face(x, 3, 'green')]) - sage: p = E(p, iterations=4) - sage: p - Patch of 31 faces - """ - old_faces = patch - for i in xrange(iterations): - new_faces = [] - for f in old_faces: - new_faces.extend(self._call_on_face(f, color=f.color())) - old_faces = new_faces - - return Patch(new_faces) - def _call_on_face(self, face, color=None): r""" Returns an iterator of faces obtained by applying self on the face.  ### comment:40 Changed 7 years ago by tjolivet Hi ! Well, that's very strange: I just compiled 4.6, made a fresh clone, and I could apply all the patches without any problem: timo@ukko:~/sage-4.6/devel/sage-haha$ ../../sage -hg qseries
trac_8431_e_one_star.patch
trac_8431_review-sl.patch
trac_8431_might_be_final_tj.patch
trac_8431_colors_tikz-sl.patch
trac_8431_sigma_type_fix.patch
trac_8431-wordmorphism-sl.patch
trac_8431-smallfixes-tj.patch
trac_8431-alphaset-tj.patch
timo@ukko:~/sage-4.6/devel/sage-haha$../../sage -hg qtop trac_8431-alphaset-tj.patch timo@ukko:~/sage-4.6/devel/sage-haha$ ../../sage -hg qpop
now at: trac_8431-smallfixes-tj.patch
timo@ukko:~/sage-4.6/devel/sage-haha$../../sage -hg qpop now at: trac_8431-wordmorphism-sl.patch timo@ukko:~/sage-4.6/devel/sage-haha$ ../../sage -hg qpush
applying trac_8431-smallfixes-tj.patch
now at: trac_8431-smallfixes-tj.patch
timo@ukko:~/sage-4.6/devel/sage-haha\$ ../../sage -hg qpush
applying trac_8431-alphaset-tj.patch
now at: trac_8431-alphaset-tj.patch


Could you try it again? There's no reason that it works for me but not for you!

(And don't forget the last patch trac_8431-alphaset-tj.patch !)

To recap, the order is:

1. trac_8431_e_one_star.patch
2. trac_8431_review-sl.patch
3. trac_8431_might_be_final_tj.patch
4. trac_8431_colors_tikz-sl.patch
5. trac_8431_sigma_type_fix.patch
6. trac_8431-wordmorphism-sl.patch
7. trac_8431-smallfixes-tj.patch
8. trac_8431-alphaset-tj.patch

[BTW, a question: when I need to apply such a sequence of patches, do I have to do "import, qpush, import, qpush, import, qpush, ...", with an import and a qpush for each patch ?]

### comment:41 Changed 7 years ago by tjolivet

Hi. Sorry, this error message was because of a manipulation error from my part. (I misapplied the wordmorphism-sl patch, and my two last patches were based on this misapplying.)

I made a new patch that applies correctly over wordmorphism-sl, and that takes into account my two last patches (which should now be ignored; it's a shame that we can't delete patches from the ticket!).

Patches should be applied in this order:

1. trac_8431_e_one_star.patch
2. trac_8431_review-sl.patch
3. trac_8431_might_be_final_tj.patch
4. trac_8431_colors_tikz-sl.patch
5. trac_8431_sigma_type_fix.patch
6. trac_8431-wordmorphism-sl.patch
7. trac_new-fixes-final-tj.patch

(Sorry for misnaming the last patch, I forgot "8431"... And sorry if this made you lose some time!)

### comment:42 Changed 7 years ago by tjolivet

Thanks for you final remarks. The last patch is trac_8431_typos-docfix-tj.patch, and the sequence is:

1. trac_8431_e_one_star.patch
2. trac_8431_review-sl.patch
3. trac_8431_might_be_final_tj.patch
4. trac_8431_colors_tikz-sl.patch
5. trac_8431_sigma_type_fix.patch
6. trac_8431-wordmorphism-sl.patch
7. trac_new-fixes-final-tj.patch
8. trac_8431_typos-docfix-tj.patch

### Changed 7 years ago by abmasse

Apply on top of preceding patches

### comment:43 follow-up: ↓ 45 Changed 7 years ago by abmasse

Hi Timo and Sébastien !

I just uploaded a (I hope last) patch that fixes a minor doctest failure (decimal number, on my computer but not Timo's).

I tested all of it on sage-4.6. I'm satisfied with the code and the documentation looks good (no warning neither).

Before setting this ticket to "positive review", I want to know if Sébastien is ok with it, since he reviewed big parts, and if Timo agrees with my last patch (Timo, just make sure it still passes on your machine).

### comment:44 Changed 7 years ago by tjolivet

Yes, the patch applies fine, and the doctest passes. Everything seems to be fine now.

Just a recap:

1. trac_8431_e_one_star.patch
2. trac_8431_review-sl.patch
3. trac_8431_might_be_final_tj.patch
4. trac_8431_colors_tikz-sl.patch
5. trac_8431_sigma_type_fix.patch
6. trac_8431-wordmorphism-sl.patch
7. trac_new-fixes-final-tj.patch
8. trac_8431_typos-docfix-tj.patch
9. trac_8431_doctest_fix-abm.patch

### comment:45 in reply to: ↑ 43 Changed 7 years ago by slabbe

• Status changed from needs_review to positive_review

Before setting this ticket to "positive review", I want to know if Sébastien is ok with it, since he reviewed big parts, and if Timo agrees with my last patch (Timo, just make sure it still passes on your machine).

All test pass on my machine. Coverage is 100%. Documentation builds fine. Positive review!

I just folded the 9 patches into one : trac_8431_folded.patch. It might be easier for the release manager.

Great work!

### comment:46 follow-up: ↓ 47 Changed 7 years ago by jdemeyer

• Status changed from positive_review to needs_work

1) Please update the commit message of the folded patch (use hg qrefresh -e for that). Make sure the first line contains the ticket number and a short summary of the whole patch. The next lines can contain a longer description.

2) The copyright message in the files MUST be changed to state

#  Distributed under the terms of the GNU General Public License (GPL)
#  as published by the Free Software Foundation; either version 2 of
#*****************************************************************************


### Changed 7 years ago by slabbe

Folded all the 9 appropriate patches.

### comment:47 in reply to: ↑ 46 Changed 7 years ago by slabbe

• Status changed from needs_work to needs_review

I just re-uploaded the folded patch.

1) Please update the commit message of the folded patch (use hg qrefresh -e for that).

Done.

2) The copyright message in the files MUST be changed to state

Done. I did not know the copyright message changed.

Needs review.

### comment:48 Changed 7 years ago by abmasse

• Status changed from needs_review to positive_review

I just checked that the fixes required by the release manager have indeed been done.

All tests still pass.

Positive review.

### comment:49 Changed 7 years ago by jdemeyer

• Merged in set to sage-4.6.1.alpha2
• Resolution set to fixed
• Status changed from positive_review to closed

### comment:50 Changed 9 months ago by chapoton

• Authors changed from Vincent Delecroix, Timo Jolivet, Franco Saliola, Stepan Starosta to Vincent Delecroix, Timo Jolivet, Franco Saliola, Štěpán Starosta
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