$$ \def\CC{\bf C} \def\QQ{\bf Q} \def\RR{\bf R} \def\ZZ{\bf Z} \def\NN{\bf N} $$

Demo of EkEkstar -- Milton Minervino -- LaBRI Sage Thursdays -- March 22, 2018

Notes taken by Sébastien Labbé

k-Faces:



In [1]:

    
from EkEkstar import kFace,kPatch,GeoSub
F1 = kFace((0,0,0),(1,2))
F1









    Out[1]:





[(0, 0, 0), (1, 2)]



In [2]:

    
F1.vector()









    Out[2]:





(0, 0, 0)



In [3]:

    
F1.type()









    Out[3]:





(1, 2)

k-Patches are formal sums of k-faces:



In [4]:

    
F3 = kFace((0,0,0,6,7),(1,4,5,2))
F3









    Out[4]:





[(0, 0, 0, 6, 7), (1, 4, 5, 2)]



In [6]:

    
F3 + F1









    Out[6]:





Patch: 1[(0, 0, 0), (1, 2)] + 1[(0, 0, 0, 6, 7), (1, 2, 4, 5)]

Adding k-faces may cancel depending on their multiplicity and the permutation sign of their type:



In [7]:

    
F2 = kFace((0,0,0),(2,1))
F2









    Out[7]:





[(0, 0, 0), (2, 1)]



In [8]:

    
F1 + F2









    Out[8]:





Empty patch

Creation of k-Patches:



In [9]:

    
F1









    Out[9]:





[(0, 0, 0), (1, 2)]



In [10]:

    
kPatch([F1,F1])









    Out[10]:





Patch: 2[(0, 0, 0), (1, 2)]



In [11]:

    
F1 + F1









    Out[11]:





Patch: 2[(0, 0, 0), (1, 2)]



In [12]:

    
F2









    Out[12]:





[(0, 0, 0), (2, 1)]



In [13]:

    
kPatch([F2])









    Out[13]:





Patch: -1[(0, 0, 0), (1, 2)]

BUG:



In [14]:

    
kFace((0,0,0),(1,2)) + kFace((0,0,1),(3,1)) + kFace((13,23,34),(1,1))









    



---------------------------------------------------------------------------
AttributeError                            Traceback (most recent call last)
<ipython-input-14-9115324a1fa8> in <module>()
----> 1 kFace((Integer(0),Integer(0),Integer(0)),(Integer(1),Integer(2))) + kFace((Integer(0),Integer(0),Integer(1)),(Integer(3),Integer(1))) + kFace((Integer(13),Integer(23),Integer(34)),(Integer(1),Integer(1)))

/home/slabbe/GitBox/EkEkstar/EkEkstar/EkEkstar.py in __add__(self, other)
    440         for (f,m) in self._faces.items():
    441             C[f] += m
--> 442         for (f,m) in other._faces.items():
    443             C[f] += m
    444         return kPatch(C)

AttributeError: 'kFace' object has no attribute '_faces'

Dual k-faces:



In [15]:

    
F = kFace((0,0,0,0),(1,3,2))
F+F









    Out[15]:





Patch: -2[(0, 0, 0, 0), (1, 2, 3)]



In [16]:

    
F._dual









    Out[16]:





False



In [18]:

    
#Fd = F.dual()
Fd = kFace((0,0,0,0),(1,3,2),dual=True)
Fd.is_dual()









    Out[18]:





True



In [19]:

    
Fd









    Out[19]:





[(0, 0, 0, 0), (1, 3, 2)]*

The EkEkStar module generalizes what is already in Sage:



In [20]:

    
from sage.combinat.e_one_star import Face, Patch, E1Star

Substitutions are already in Sage:



In [21]:

    
sub = {1:[1,2], 2:[1,3], 3:[1]}
sub[1]









    Out[21]:





[1, 2]



In [22]:

    
s = WordMorphism(sub)
s









    Out[22]:





WordMorphism: 1->12, 2->13, 3->1



In [23]:

    
s(1)









    Out[23]:





word: 12



In [24]:

    
s([2])









    Out[24]:





word: 13



In [25]:

    
s([1,1,1,1])









    Out[25]:





word: 12121212



In [26]:

    
s([2,1,3])









    Out[26]:





word: 13121

Iteration of the Tribonacci substitution on letter 1:



In [28]:

    
s(1,1)









    Out[28]:





word: 12



In [29]:

    
s(1,2)









    Out[29]:





word: 1213



In [30]:

    
s(1,3)









    Out[30]:





word: 1213121



In [31]:

    
s(1,4)









    Out[31]:





word: 1213121121312

The word 121312112131212131211213 defines a path in \$ZZ^3\$. This can be formalized by a geometric substitution (GeoSub). This is useless for 3d-path coded by words since concatenation of words $uv$ is very easy and the geometric position of the start of the word $v$ is simply the end position of the word $u$ seen as a path. But this concatenation rule works only for words and it can't help for applying a substitution on a set of faces. Therefore, we need to consider k-Patches of faces as formal sums.

Here we consider the GeoSub in the easy case of faces of dimension 1 which can be seen just as concatenation of paths:



In [32]:

    
E1 = GeoSub(sub,1,dual=False)
E1









    Out[32]:





E_1(1->12, 2->13, 3->1)



In [33]:

    
P = kPatch([kFace((0,0,0),(1,))])
P









    Out[33]:





Patch: 1[(0, 0, 0), (1,)]



In [34]:

    
E1(P)









    Out[34]:





Patch: 1[(1, 0, 0), (2,)] + 1[(0, 0, 0), (1,)]



In [35]:

    
E1(P,2)









    Out[35]:





Patch: 1[(1, 0, 0), (2,)] + 1[(2, 1, 0), (3,)] + 1[(0, 0, 0), (1,)] + 1[(1, 1, 0), (1,)]



In [36]:

    
E1(P,7)









    Out[36]:





Patch of 81 faces



In [37]:

    
E1(P,5).plot(E1)









    Out[37]:

Now we consider dual faces which are not segments anymore and for which the formal sums formalism is necessary (this corresponds to the E1star that is already in Sage):



In [38]:

    
E1star = GeoSub(sub,1,dual=True)
E1star









    Out[38]:





E*_1(1->12, 2->13, 3->1)



In [39]:

    
P









    Out[39]:





Patch: 1[(0, 0, 0), (1,)]



In [41]:

    
#Pstar = P.dual()
Pstar = kPatch([kFace((0,0,0),(1,),dual=True)])
Pstar









    Out[41]:





Patch: 1[(0, 0, 0), (1,)]*



In [42]:

    
E1(P)









    Out[42]:





Patch: 1[(0, 0, 0), (1,)] + 1[(1, 0, 0), (2,)]



In [43]:

    
E1star(Pstar)









    Out[43]:





Patch: 1[(0, 0, 0), (1,)]* + -1[(0, 0, 0), (2,)]* + 1[(0, 0, 0), (3,)]*

In the EkEkstar module, the projection is done in the contracting plane:



In [44]:

    
E1star(Pstar).plot(E1star)









    Out[44]:



In [45]:

    
E1star(Pstar,5).plot(E1star)









    Out[45]:



In [46]:

    
E1star(Pstar,7).plot(E1star)









    Out[46]:

The module allows more general geometric substitution like \$E\_2^\*\$ which computes the boundary of the fractal:



In [47]:

    
E2star = GeoSub(sub,2,dual=True)
E2star









    Out[47]:





E*_2(1->12, 2->13, 3->1)



In [48]:

    
E2star.base_iter()









    Out[48]:





{(1, 2): [[(0, 0, 0), (1,)],
  [(0, 0, 0), (2,)],
  [(0, 0, 0), (3,)],
  [(0, 0, -1), (1, 1)],
  [(0, 0, -1), (2, 1)],
  [(0, 0, -1), (3, 1)]],
 (1, 3): [[(0, 0, 0), (1,)],
  [(0, 0, 0), (2,)],
  [(0, 0, 0), (3,)],
  [(0, 0, -1), (1, 2)],
  [(0, 0, -1), (2, 2)],
  [(0, 0, -1), (3, 2)]],
 (2, 3): [[(0, 0, -1), (1,)], [(0, 0, -2), (1, 2)]]}



In [49]:

    
P = kPatch([kFace((0,0,0),(1,2),dual=True)])
E2star(P,6).plot(E2star)









    Out[49]:



In [ ]: