# Word paths#

This module implements word paths, which is an application of Combinatorics on Words to Discrete Geometry. A word path is the representation of a word as a discrete path in a vector space using a one-to-one correspondence between the alphabet and a set of vectors called steps. Many problems surrounding 2d lattice polygons (such as questions of self-intersection, area, inertia moment, etc.) can be solved in linear time (linear in the length of the perimeter) using theory from Combinatorics on Words.

On the square grid, the encoding of a path using a four-letter alphabet (for East, North, West and South directions) is also known as the Freeman chain code [1,2] (see [3] for further reading).

AUTHORS:

• Arnaud Bergeron (2008) : Initial version, path on the square grid

• Sebastien Labbe (2009-01-14) : New classes and hierarchy, doc and functions.

EXAMPLES:

The combinatorial class of all paths defined over three given steps:

sage: P = WordPaths('abc', steps=[(1,2), (-3,4), (0,-3)]); P
Word Paths over 3 steps


This defines a one-to-one correspondence between alphabet and steps:

sage: d = P.letters_to_steps()
sage: sorted(d.items())
[('a', (1, 2)), ('b', (-3, 4)), ('c', (0, -3))]


Creation of a path from the combinatorial class P defined above:

sage: p = P('abaccba'); p
Path: abaccba


Many functions can be used on p: the coordinates of its trajectory, ask whether p is a closed path, plot it and many other:

sage: list(p.points())
[(0, 0), (1, 2), (-2, 6), (-1, 8), (-1, 5), (-1, 2), (-4, 6), (-3, 8)]
sage: p.is_closed()
False
sage: p.plot()                                                                      # optional - sage.plot
Graphics object consisting of 3 graphics primitives


To obtain a list of all the available word path specific functions, use help(p):

sage: help(p)
Help on FiniteWordPath_2d_str in module sage.combinat.words.paths object:
...
Methods inherited from FiniteWordPath_2d:
...
Methods inherited from FiniteWordPath_all:
...


Since p is a finite word, many functions from the word library are available:

sage: p.crochemore_factorization()
(a, b, a, c, c, ba)
sage: p.is_palindrome()
False
sage: p[:3]
Path: aba
sage: len(p)
7


P also herits many functions from Words:

sage: P = WordPaths('rs', steps=[(1,2), (-1,4)]); P
Word Paths over 2 steps
sage: P.alphabet()
{'r', 's'}
sage: list(P.iterate_by_length(3))
[Path: rrr,
Path: rrs,
Path: rsr,
Path: srr,
Path: srs,
Path: ssr,
Path: sss]


When the number of given steps is half the size of alphabet, the opposite of vectors are used:

sage: P = WordPaths('abcd', [(1,0), (0,1)])
sage: sorted(P.letters_to_steps().items())
[('a', (1, 0)), ('b', (0, 1)), ('c', (-1, 0)), ('d', (0, -1))]


Some built-in combinatorial classes of paths:

sage: P = WordPaths('abAB', steps='square_grid'); P
Word Paths on the square grid

sage: D = WordPaths('()', steps='dyck'); D
Finite Dyck paths
sage: d = D('()()()(())'); d
Path: ()()()(())
sage: d.plot()                                                                      # optional - sage.plot
Graphics object consisting of 3 graphics primitives

sage: P = WordPaths('abcdef', steps='triangle_grid')
sage: p.plot()                                                                      # optional - sage.plot
Graphics object consisting of 3 graphics primitives


Vector steps may be in more than 2 dimensions:

sage: d = [(1,0,0), (0,1,0), (0,0,1)]
sage: P = WordPaths(alphabet='abc', steps=d); P
Word Paths over 3 steps
sage: p = P('abcabcabcabcaabacabcababcacbabacacabcaccbcac')
sage: p.plot()                                                                      # optional - sage.plot
Graphics3d Object

sage: d = [(1,3,5,1), (-5,1,-6,0), (0,0,1,9), (4,2,-1,0)]
sage: P = WordPaths(alphabet='rstu', steps=d); P
Word Paths over 4 steps
sage: p = P('rtusuusususuturrsust'); p
Path: rtusuusususuturrsust
sage: p.end_point()
(5, 31, -26, 30)

sage: CubePaths = WordPaths('abcABC', steps='cube_grid'); CubePaths
Word Paths on the cube grid
sage: CubePaths('abcabaabcabAAAAA').plot()                                          # optional - sage.plot
Graphics3d Object


The input data may be a str, a list, a tuple, a callable or a finite iterator:

sage: P = WordPaths([0, 1, 2, 3])
sage: P([0,1,2,3,2,1,2,3,2])
Path: 012321232
sage: P((0,1,2,3,2,1,2,3,2))
Path: 012321232
sage: P(lambda n:n%4, length=10)
Path: 0123012301
sage: P(iter([0,3,2,1]), length='finite')
Path: 0321


REFERENCES:

• [1] Freeman, H.: On the encoding of arbitrary geometric configurations. IRE Trans. Electronic Computer 10 (1961) 260-268.

• [2] Freeman, H.: Boundary encoding and processing. In Lipkin, B., Rosenfeld, A., eds.: Picture Processing and Psychopictorics, Academic Press, New York (1970) 241-266.

• [3] Braquelaire, J.P., Vialard, A.: Euclidean paths: A new representation of boundary of discrete regions. Graphical Models and Image Processing 61 (1999) 16-43.

class sage.combinat.words.paths.FiniteWordPath_2d#
animate()#

Returns an animation object illustrating the path growing step by step.

EXAMPLES:

sage: P = WordPaths('abAB')
sage: p = P('aaababbb')
sage: a = p.animate(); print(a)                                             # optional - sage.plot
Animation with 9 frames
sage: show(a)                           # long time  # optional -- ImageMagick sage.plot
sage: show(a, delay=35, iterations=3)   # long time  # optional -- ImageMagick sage.plot

sage: P = WordPaths('abcdef',steps='triangle')
sage: p =  P('abcdef')
sage: a = p.animate(); print(a)                                             # optional - sage.plot
Animation with 8 frames
sage: show(a)                           # long time  # optional -- ImageMagick sage.plot


If the path is closed, the plain polygon is added at the end of the animation:

sage: P = WordPaths('abAB')
sage: p = P('ababAbABABaB')
sage: a = p.animate(); print(a)                                             # optional - sage.plot
Animation with 14 frames
sage: show(a)                           # long time  # optional -- ImageMagick sage.plot


Another example illustrating a Fibonacci tile:

sage: w = words.fibonacci_tile(2)
sage: a = w.animate(); print(a)                                             # optional - sage.plot
Animation with 54 frames
sage: show(a)                           # long time  # optional -- ImageMagick sage.plot


The first 4 Fibonacci tiles in an animation:

sage: a = words.fibonacci_tile(0).animate()                                 # optional - sage.plot
sage: b = words.fibonacci_tile(1).animate()                                 # optional - sage.plot
sage: c = words.fibonacci_tile(2).animate()                                 # optional - sage.plot
sage: d = words.fibonacci_tile(3).animate()                                 # optional - sage.plot
sage: print(a*b*c*d)                                                        # optional - sage.plot
Animation with 296 frames
sage: show(a*b*c*d)                     # long time  # optional -- ImageMagick sage.plot


Note

If ImageMagick is not installed, you will get an error message like this:

convert: not found

Error: ImageMagick does not appear to be installed. Saving an
animation to a GIF file or displaying an animation requires
ImageMagick, so please install it and try again.


See www.imagemagick.org, for example.

area()#

Returns the area of a closed path.

INPUT:

• self - a closed path

EXAMPLES:

sage: P = WordPaths('abcd',steps=[(1,1),(-1,1),(-1,-1),(1,-1)])
sage: p = P('abcd')
sage: p.area()          #todo: not implemented
2

height()#

Returns the height of self.

The height of a $$2d$$-path is merely the difference between the highest and the lowest $$y$$-coordinate of each points traced by it.

OUTPUT:

non negative real number

EXAMPLES:

sage: Freeman = WordPaths('abAB')
sage: Freeman('aababaabbbAA').height()
5


The function is well-defined if self is not simple or close:

sage: Freeman('aabAAB').height()
1
sage: Freeman('abbABa').height()
2


This works for any $$2d$$-paths:

sage: Paths = WordPaths('ab', steps=[(1,0),(1,1)])
sage: p = Paths('abbaa')
sage: p.height()
2
sage: DyckPaths = WordPaths('ab', steps='dyck')
sage: p = DyckPaths('abaabb')
sage: p.height()
2
sage: w = WordPaths('abcABC', steps='triangle')('ababcaaBC')
sage: w.height()
2.59807621135332

height_vector()#

Return the height at each point.

EXAMPLES:

sage: Paths = WordPaths('ab', steps=[(1,0),(0,1)])
sage: p = Paths('abbba')
sage: p.height_vector()
[0, 0, 1, 2, 3, 3]

plot(pathoptions={'rgbcolor': 'red', 'thickness': 3}, fill=True, filloptions={'rgbcolor': 'red', 'alpha': 0.2}, startpoint=True, startoptions={'rgbcolor': 'red', 'pointsize': 100}, endarrow=True, arrowoptions={'rgbcolor': 'red', 'arrowsize': 20, 'width': 3}, gridlines=False, gridoptions={})#

Returns a 2d Graphics illustrating the path.

INPUT:

• pathoptions - (dict, default:dict(rgbcolor=’red’,thickness=3)), options for the path drawing

• fill - (boolean, default: True), if fill is True and if the path is closed, the inside is colored

• filloptions - (dict, default:dict(rgbcolor=’red’,alpha=0.2)), options for the inside filling

• startpoint - (boolean, default: True), draw the start point?

• startoptions - (dict, default:dict(rgbcolor=’red’,pointsize=100)) options for the start point drawing

• endarrow - (boolean, default: True), draw an arrow end at the end?

• arrowoptions - (dict, default:dict(rgbcolor=’red’,arrowsize=20, width=3)) options for the end point arrow

• gridlines- (boolean, default: False), show gridlines?

• gridoptions - (dict, default: {}), options for the gridlines

EXAMPLES:

A non closed path on the square grid:

sage: P = WordPaths('abAB')
sage: P('abababAABAB').plot()                                               # optional - sage.plot
Graphics object consisting of 3 graphics primitives


A closed path on the square grid:

sage: P('abababAABABB').plot()                                              # optional - sage.plot
Graphics object consisting of 4 graphics primitives


A Dyck path:

sage: P = WordPaths('()', steps='dyck')
sage: P('()()()((()))').plot()                                              # optional - sage.plot
Graphics object consisting of 3 graphics primitives


A path in the triangle grid:

sage: P = WordPaths('abcdef', steps='triangle_grid')
sage: P('abcdedededefab').plot()                                            # optional - sage.plot
Graphics object consisting of 3 graphics primitives


A polygon of length 220 that tiles the plane in two ways:

sage: P = WordPaths('abAB')
sage: P('aBababAbabaBaBABaBabaBaBABAbABABaBabaBaBABaBababAbabaBaBABaBabaBaBABAbABABaBABAbAbabAbABABaBABAbABABaBabaBaBABAbABABaBABAbAbabAbABAbAbabaBababAbABAbAbabAbABABaBABAbAbabAbABAbAbabaBababAbabaBaBABaBababAbabaBababAbABAbAbab').plot()  # optional - sage.plot
Graphics object consisting of 4 graphics primitives


With gridlines:

sage: P('ababababab').plot(gridlines=True)                                  # optional - sage.plot

plot_directive_vector(options={'rgbcolor': 'blue'})#

Returns an arrow 2d graphics that goes from the start of the path to the end.

INPUT:

• options - dictionary, default: {‘rgbcolor’: ‘blue’} graphic options for the arrow

If the start is the same as the end, a single point is returned.

EXAMPLES:

sage: P = WordPaths('abcd'); P
Word Paths on the square grid
sage: p = P('aaaccaccacacacaccccccbbdd'); p
Path: aaaccaccacacacaccccccbbdd
sage: R = p.plot() + p.plot_directive_vector()                              # optional - sage.plot
sage: R.axes(False)                                                         # optional - sage.plot
sage: R.set_aspect_ratio(1)                                                 # optional - sage.plot
sage: R.plot()                                                              # optional - sage.plot
Graphics object consisting of 4 graphics primitives

width()#

Returns the width of self.

The height of a $$2d$$-path is merely the difference between the rightmost and the leftmost $$x$$-coordinate of each points traced by it.

OUTPUT:

non negative real number

EXAMPLES:

sage: Freeman = WordPaths('abAB')
sage: Freeman('aababaabbbAA').width()
5


The function is well-defined if self is not simple or close:

sage: Freeman('aabAAB').width()
2
sage: Freeman('abbABa').width()
1


This works for any $$2d$$-paths:

sage: Paths = WordPaths('ab', steps=[(1,0),(1,1)])
sage: p = Paths('abbaa')
sage: p.width()
5
sage: DyckPaths = WordPaths('ab', steps='dyck')
sage: p = DyckPaths('abaabb')
sage: p.width()
6
sage: w = WordPaths('abcABC', steps='triangle')('ababcaaBC')
sage: w.width()
4.50000000000000

width_vector()#

Return the width at each point.

EXAMPLES:

sage: Paths = WordPaths('ab', steps=[(1,0),(0,1)])
sage: p = Paths('abbba')
sage: p.width_vector()
[0, 1, 1, 1, 1, 2]

xmax()#

Returns the maximum of the x-coordinates of the path.

EXAMPLES:

sage: P = WordPaths('0123')
sage: p = P('0101013332')
sage: p.xmax()
3


This works for any $$2d$$-paths:

sage: Paths = WordPaths('ab', steps=[(1,-1),(-1,1)])
sage: p = Paths('ababa')
sage: p.xmax()
1
sage: DyckPaths = WordPaths('ab', steps='dyck')
sage: p = DyckPaths('abaabb')
sage: p.xmax()
6
sage: w = WordPaths('abcABC', steps='triangle')('ababcaaBC')
sage: w.xmax()
4.50000000000000

xmin()#

Returns the minimum of the x-coordinates of the path.

EXAMPLES:

sage: P = WordPaths('0123')
sage: p = P('0101013332')
sage: p.xmin()
0


This works for any $$2d$$-paths:

sage: Paths = WordPaths('ab', steps=[(1,0),(-1,1)])
sage: p = Paths('abbba')
sage: p.xmin()
-2
sage: DyckPaths = WordPaths('ab', steps='dyck')
sage: p = DyckPaths('abaabb')
sage: p.xmin()
0
sage: w = WordPaths('abcABC', steps='triangle')('ababcaaBC')
sage: w.xmin()
0.000000000000000

ymax()#

Returns the maximum of the y-coordinates of the path.

EXAMPLES:

sage: P = WordPaths('0123')
sage: p = P('0101013332')
sage: p.ymax()
3


This works for any $$2d$$-paths:

sage: Paths = WordPaths('ab', steps=[(1,-1),(-1,1)])
sage: p = Paths('ababa')
sage: p.ymax()
0
sage: DyckPaths = WordPaths('ab', steps='dyck')
sage: p = DyckPaths('abaabb')
sage: p.ymax()
2
sage: w = WordPaths('abcABC', steps='triangle')('ababcaaBC')
sage: w.ymax()
2.59807621135332

ymin()#

Returns the minimum of the y-coordinates of the path.

EXAMPLES:

sage: P = WordPaths('0123')
sage: p = P('0101013332')
sage: p.ymin()
0


This works for any $$2d$$-paths:

sage: Paths = WordPaths('ab', steps=[(1,-1),(-1,1)])
sage: p = Paths('ababa')
sage: p.ymin()
-1
sage: DyckPaths = WordPaths('ab', steps='dyck')
sage: p = DyckPaths('abaabb')
sage: p.ymin()
0
sage: w = WordPaths('abcABC', steps='triangle')('ababcaaBC')
sage: w.ymin()
0.000000000000000

class sage.combinat.words.paths.FiniteWordPath_2d_callable(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_2d_callable_with_caching(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_2d_iter(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_2d_iter_with_caching(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_2d_list#
class sage.combinat.words.paths.FiniteWordPath_2d_str#
class sage.combinat.words.paths.FiniteWordPath_2d_tuple#
class sage.combinat.words.paths.FiniteWordPath_3d#
plot(pathoptions={'rgbcolor': 'red', 'arrow_head': True, 'thickness': 3}, startpoint=True, startoptions={'rgbcolor': 'red', 'size': 10})#

INPUT:

• pathoptions - (dict, default:dict(rgbcolor=’red’,arrow_head=True, thickness=3)), options for the path drawing

• startpoint - (boolean, default: True), draw the start point?

• startoptions - (dict, default:dict(rgbcolor=’red’,size=10))

options for the start point drawing

EXAMPLES:

sage: d = ( vector((1,3,2)), vector((2,-4,5)) )
sage: P = WordPaths(alphabet='ab', steps=d); P
Word Paths over 2 steps
sage: p = P('ababab'); p
Path: ababab
sage: p.plot()                                                              # optional - sage.plot
Graphics3d Object

sage: P = WordPaths('abcABC', steps='cube_grid')
sage: p = P('abcabcAABBC')
sage: p.plot()                                                              # optional - sage.plot
Graphics3d Object

class sage.combinat.words.paths.FiniteWordPath_3d_callable(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_3d_callable_with_caching(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_3d_iter(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_3d_iter_with_caching(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_3d_list#
class sage.combinat.words.paths.FiniteWordPath_3d_str#
class sage.combinat.words.paths.FiniteWordPath_3d_tuple#
class sage.combinat.words.paths.FiniteWordPath_all#

Bases: SageObject

directive_vector()#

Returns the directive vector of self.

The directive vector is the vector starting at the start point and ending at the end point of the path self.

EXAMPLES:

sage: WordPaths('abcdef')('abababab').directive_vector()
(6, 2*sqrt3)
sage: WordPaths('abAB')('abababab').directive_vector()
(4, 4)
sage: P = WordPaths('abcABC', steps='cube_grid')
sage: P('ababababCC').directive_vector()
(4, 4, -2)
sage: WordPaths('abcdef')('abcdef').directive_vector()
(0, 0)
sage: P = WordPaths('abc', steps=[(1,3,7,9),(-4,1,0,0),(0,32,1,8)])
sage: P('abcabababacaacccbbcac').directive_vector()
(-16, 254, 63, 128)

end_point()#

Returns the end point of the path.

EXAMPLES:

sage: WordPaths('abcdef')('abababab').end_point()
(6, 2*sqrt3)
sage: WordPaths('abAB')('abababab').end_point()
(4, 4)
sage: P = WordPaths('abcABC', steps='cube_grid')
sage: P('ababababCC').end_point()
(4, 4, -2)
sage: WordPaths('abcdef')('abcdef').end_point()
(0, 0)
sage: P = WordPaths('abc', steps=[(1,3,7,9),(-4,1,0,0),(0,32,1,8)])
sage: P('abcabababacaacccbbcac').end_point()
(-16, 254, 63, 128)

is_closed()#

Returns True if the path is closed, i.e. if the origin and the end of the path are equal.

EXAMPLES:

sage: P = WordPaths('abcd', steps=[(1,0),(0,1),(-1,0),(0,-1)])
sage: P('abcd').is_closed()
True
sage: P('abc').is_closed()
False
sage: P().is_closed()
True
sage: P('aacacc').is_closed()
True

is_simple()#

Returns True if the path is simple, i.e. if all its points are distincts.

If the path is closed, the last point is not considered.

EXAMPLES:

sage: P = WordPaths('abcdef',steps='triangle_grid');P
Word Paths on the triangle grid
sage: P('abc').is_simple()
True
sage: P('abcde').is_simple()
True
sage: P('abcdef').is_simple()
True
True
sage: P('aabdee').is_simple()
False

is_tangent()#

The is_tangent() method, which is implemented for words, has an extended meaning for word paths, which is not implemented yet.

AUTHOR:

• Thierry Monteil

plot_projection(v=None, letters=None, color=None, ring=None, size=12, kind='right')#

Return an image of the projection of the successive points of the path into the space orthogonal to the given vector.

INPUT:

• self - a word path in a 3 or 4 dimension vector space

• v - vector (optional, default: None) If None, the directive vector (i.e. the end point minus starting point) of the path is considered.

• letters - iterable (optional, default: None) of the letters to be projected. If None, then all the letters are considered.

• color - dictionary (optional, default: None) of the letters mapped to colors. If None, automatic colors are chosen.

• ring - ring (optional, default: None) where to do the computations. If None, RealField(53) is used.

• size - number (optional, default: 12) size of the points.

• kind - string (optional, default 'right') either 'right' or 'left'. The color of a letter is given to the projected prefix to the right or the left of the letter.

OUTPUT:

2d or 3d Graphic object.

EXAMPLES:

The Rauzy fractal:

sage: s = WordMorphism('1->12,2->13,3->1')
sage: D = s.fixed_point('1')
sage: v = s.pisot_eigenvector_right()
sage: P = WordPaths('123',[(1,0,0),(0,1,0),(0,0,1)])
sage: w = P(D[:200])
sage: w.plot_projection(v)  # long time (2s)
Graphics object consisting of 200 graphics primitives


In this case, the abelianized vector doesn’t give a good projection:

sage: w.plot_projection()  # long time (2s)
Graphics object consisting of 200 graphics primitives


You can project only the letters you want:

sage: w.plot_projection(v, letters='12')  # long time (2s)
Graphics object consisting of 168 graphics primitives


You can increase or decrease the precision of the computations by changing the ring of the projection matrix:

sage: w.plot_projection(v, ring=RealField(20))  # long time (2s)
Graphics object consisting of 200 graphics primitives


You can change the size of the points:

sage: w.plot_projection(v, size=30)  # long time (2s)
Graphics object consisting of 200 graphics primitives


You can assign the color of a letter to the projected prefix to the right or the left of the letter:

sage: w.plot_projection(v, kind='left')  # long time (2s)
Graphics object consisting of 200 graphics primitives


To remove the axis, do like this:

sage: r = w.plot_projection(v)                                              # optional - sage.plot
sage: r.axes(False)                                                         # optional - sage.plot
sage: r               # long time (2s)                                      # optional - sage.plot
Graphics object consisting of 200 graphics primitives


You can assign different colors to each letter:

sage: color = {'1': 'purple', '2': (.2,.3,.4), '3': 'magenta'}
sage: w.plot_projection(v, color=color)  # long time (2s)                   # optional - sage.plot
Graphics object consisting of 200 graphics primitives


The 3d-Rauzy fractal:

sage: s = WordMorphism('1->12,2->13,3->14,4->1')
sage: D = s.fixed_point('1')
sage: v = s.pisot_eigenvector_right()
sage: P = WordPaths('1234',[(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)])
sage: w = P(D[:200])
sage: w.plot_projection(v)                                                  # optional - sage.plot
Graphics3d Object


The dimension of vector space of the parent must be 3 or 4:

sage: P = WordPaths('ab', [(1, 0), (0, 1)])
sage: p = P('aabbabbab')
sage: p.plot_projection()                                                   # optional - sage.plot
Traceback (most recent call last):
...
TypeError: The dimension of the vector space (=2) must be 3 or 4

points(include_last=True)#

Returns an iterator yielding a list of points used to draw the path represented by this word.

INPUT:

• include_last - bool (default: True) whether to include the last point

EXAMPLES:

A simple closed square:

sage: P = WordPaths('abAB')
sage: list(P('abAB').points())
[(0, 0), (1, 0), (1, 1), (0, 1), (0, 0)]


A simple closed square without the last point:

sage: list(P('abAB').points(include_last=False))
[(0, 0), (1, 0), (1, 1), (0, 1)]

sage: list(P('abaB').points())
[(0, 0), (1, 0), (1, 1), (2, 1), (2, 0)]

projected_path(v=None, ring=None)#

Return the path projected into the space orthogonal to the given vector.

INPUT:

• v - vector (optional, default: None) If None, the directive vector (i.e. the end point minus starting point) of the path is considered.

• ring - ring (optional, default: None) where to do the computations. If None, RealField(53) is used.

OUTPUT:

word path

EXAMPLES:

The projected path of the tribonacci word:

sage: s = WordMorphism('1->12,2->13,3->1')
sage: D = s.fixed_point('1')
sage: v = s.pisot_eigenvector_right()
sage: P = WordPaths('123',[(1,0,0),(0,1,0),(0,0,1)])
sage: w = P(D[:1000])
sage: p = w.projected_path(v)
sage: p
Path: 1213121121312121312112131213121121312121...
sage: p[:20].plot()                                                         # optional - sage.plot
Graphics object consisting of 3 graphics primitives


The ring argument allows to change the precision of the projected steps:

sage: p = w.projected_path(v, RealField(10))
sage: p
Path: 1213121121312121312112131213121121312121...
sage: p.parent().letters_to_steps()
{'1': (-0.53, 0.00), '2': (0.75, -0.48), '3': (0.41, 0.88)}

projected_point_iterator(v=None, ring=None)#

Return an iterator of the projection of the orbit points of the path into the space orthogonal to the given vector.

INPUT:

• v - vector (optional, default: None) If None, the directive vector (i.e. the end point minus starting point) of the path is considered.

• ring - ring (optional, default: None) where to do the computations. If None, RealField(53) is used.

OUTPUT:

iterator of points

EXAMPLES:

Projected points of the Rauzy fractal:

sage: s = WordMorphism('1->12,2->13,3->1')
sage: D = s.fixed_point('1')
sage: v = s.pisot_eigenvector_right()
sage: P = WordPaths('123',[(1,0,0),(0,1,0),(0,0,1)])
sage: w = P(D[:200])
sage: it = w.projected_point_iterator(v)
sage: for i in range(6): next(it)
(0.000000000000000, 0.000000000000000)
(-0.526233343362516, 0.000000000000000)
(0.220830337618112, -0.477656250512816)
(-0.305403005744404, -0.477656250512816)
(0.100767309386062, 0.400890564600664)
(-0.425466033976454, 0.400890564600664)


Projected points of a 2d path:

sage: P = WordPaths('ab','ne')
sage: p = P('aabbabbab')
sage: it = p.projected_point_iterator(ring=RealField(20))
sage: for i in range(8): next(it)
(0.00000)
(0.78087)
(1.5617)
(0.93704)
(0.31235)
(1.0932)
(0.46852)
(-0.15617)

start_point()#

Return the starting point of self.

OUTPUT:

vector

EXAMPLES:

sage: WordPaths('abcdef')('abcdef').start_point()
(0, 0)
sage: WordPaths('abcdef', steps='cube_grid')('abcdef').start_point()
(0, 0, 0)
sage: P = WordPaths('ab', steps=[(1,0,0,0),(0,1,0,0)])
sage: P('abbba').start_point()
(0, 0, 0, 0)

tikz_trajectory()#

Returns the trajectory of self as a tikz str.

EXAMPLES:

sage: P = WordPaths('abcdef')
sage: p = P('abcde')
sage: p.tikz_trajectory()
'(0.000, 0.000) -- (1.00, 0.000) -- (1.50, 0.866) -- (1.00, 1.73) -- (0.000, 1.73) -- (-0.500, 0.866)'

class sage.combinat.words.paths.FiniteWordPath_all_callable(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_all_callable_with_caching(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_all_iter(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_all_iter_with_caching(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_all_list#
class sage.combinat.words.paths.FiniteWordPath_all_str#
class sage.combinat.words.paths.FiniteWordPath_all_tuple#
class sage.combinat.words.paths.FiniteWordPath_cube_grid#
class sage.combinat.words.paths.FiniteWordPath_cube_grid_callable(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_cube_grid_callable_with_caching(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_cube_grid_iter(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_cube_grid_iter_with_caching(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_cube_grid_list#
class sage.combinat.words.paths.FiniteWordPath_cube_grid_str#
class sage.combinat.words.paths.FiniteWordPath_cube_grid_tuple#
class sage.combinat.words.paths.FiniteWordPath_dyck#
class sage.combinat.words.paths.FiniteWordPath_dyck_callable(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_dyck_callable_with_caching(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_dyck_iter(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_dyck_iter_with_caching(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_dyck_list#
class sage.combinat.words.paths.FiniteWordPath_dyck_str#
class sage.combinat.words.paths.FiniteWordPath_dyck_tuple#
class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid(parent, *args, **kwds)#

INPUT:

• parent - a parent object inheriting from Words_all that has the alphabet attribute defined

• *args, **kwds - arguments accepted by AbstractWord

EXAMPLES:

sage: F = WordPaths('abcdef', steps='hexagon'); F
Word Paths on the hexagonal grid
sage: f = F('aaabbbccddef'); f
Path: aaabbbccddef

sage: f == loads(dumps(f))
True

class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_callable(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_callable_with_caching(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_iter(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_iter_with_caching(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_list#
class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_str#
class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_tuple#
class sage.combinat.words.paths.FiniteWordPath_north_east#
class sage.combinat.words.paths.FiniteWordPath_north_east_callable(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_north_east_callable_with_caching(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_north_east_iter(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_north_east_iter_with_caching(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_north_east_list#
class sage.combinat.words.paths.FiniteWordPath_north_east_str#
class sage.combinat.words.paths.FiniteWordPath_north_east_tuple#
class sage.combinat.words.paths.FiniteWordPath_square_grid#
area()#

Returns the area of a closed path.

INPUT:

• self - a closed path

EXAMPLES:

sage: P = WordPaths('abAB', steps='square_grid')
sage: P('abAB').area()
1
sage: P('aabbAABB').area()
4
sage: P('aabbABAB').area()
3


The area of the Fibonacci tiles:

sage: [words.fibonacci_tile(i).area() for i in range(6)]
[1, 5, 29, 169, 985, 5741]
sage: [words.dual_fibonacci_tile(i).area() for i in range(6)]
[1, 5, 29, 169, 985, 5741]
sage: oeis(_)[0]                            # optional -- internet
A001653: Numbers k such that 2*k^2 - 1 is a square.
sage: _.first_terms()                       # optional -- internet
(1,
5,
29,
169,
985,
5741,
33461,
195025,
1136689,
6625109,
38613965,
225058681,
1311738121,
7645370045,
44560482149,
259717522849,
1513744654945,
8822750406821,
51422757785981,
299713796309065,
1746860020068409,
10181446324101389,
59341817924539925)

is_closed()#

Returns True if self represents a closed path and False otherwise.

EXAMPLES:

sage: P = WordPaths('abAB', steps='square_grid')
sage: P('aA').is_closed()
True
sage: P('abAB').is_closed()
True
sage: P('ababAABB').is_closed()
True
sage: P('aaabbbAABB').is_closed()
False
sage: P('ab').is_closed()
False

is_simple()#

Returns True if the path is simple, i.e. if all its points are distincts.

If the path is closed, the last point is not considered.

Note

The linear algorithm described in the thesis of Xavier Provençal should be implemented here.

EXAMPLES:

sage: P = WordPaths('abAB', steps='square_grid')
sage: P('abab').is_simple()
True
sage: P('abAB').is_simple()
True
sage: P('abA').is_simple()
True
sage: P('aabABB').is_simple()
False
sage: P().is_simple()
True
sage: P('A').is_simple()
True
sage: P('aA').is_simple()
True
sage: P('aaA').is_simple()
False


REFERENCES:

• Provençal, X., Combinatoires des mots, géometrie discrète et pavages, Thèse de doctorat en Mathématiques, Montréal, UQAM, septembre 2008, 115 pages.

tikz_trajectory()#

Returns the trajectory of self as a tikz str.

EXAMPLES:

sage: f = words.fibonacci_tile(1)
sage: f.tikz_trajectory()
'(0, 0) -- (0, -1) -- (-1, -1) -- (-1, -2) -- (0, -2) -- (0, -3) -- (1, -3) -- (1, -2) -- (2, -2) -- (2, -1) -- (1, -1) -- (1, 0) -- (0, 0)'

class sage.combinat.words.paths.FiniteWordPath_square_grid_callable(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_square_grid_callable_with_caching(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_square_grid_iter(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_square_grid_iter_with_caching(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_square_grid_list#
class sage.combinat.words.paths.FiniteWordPath_square_grid_str#
class sage.combinat.words.paths.FiniteWordPath_square_grid_tuple#
class sage.combinat.words.paths.FiniteWordPath_triangle_grid#
xmax()#

Returns the maximum of the x-coordinates of the path.

EXAMPLES:

sage: w = WordPaths('abcABC', steps='triangle')('ababcaaBC')
sage: w.xmax()
4.50000000000000
sage: w = WordPaths('abcABC', steps='triangle')('ABAcacacababababcbcbAC')
sage: w.xmax()
4.00000000000000

xmin()#

Returns the minimum of the x-coordinates of the path.

EXAMPLES:

sage: w = WordPaths('abcABC', steps='triangle')('ababcaaBC')
sage: w.xmin()
0.000000000000000
sage: w = WordPaths('abcABC', steps='triangle')('ABAcacacababababcbcbAC')
sage: w.xmin()
-3.00000000000000

ymax()#

Returns the maximum of the y-coordinates of the path.

EXAMPLES:

sage: w = WordPaths('abcABC', steps='triangle')('ababcaaBC')
sage: w.ymax()
2.59807621135332
sage: w = WordPaths('abcABC', steps='triangle')('ABAcacacababababcbcbAC')
sage: w.ymax()
8.66025403784439

ymin()#

Returns the minimum of the y-coordinates of the path.

EXAMPLES:

sage: w = WordPaths('abcABC', steps='triangle')('ababcaaBC')
sage: w.ymin()
0.000000000000000
sage: w = WordPaths('abcABC', steps='triangle')('ABAcacacababababcbcbAC')
sage: w.ymin()
-0.866025403784439

class sage.combinat.words.paths.FiniteWordPath_triangle_grid_callable(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_triangle_grid_callable_with_caching(parent, callable, length=None)#
class sage.combinat.words.paths.FiniteWordPath_triangle_grid_iter(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_triangle_grid_iter_with_caching(parent, iter, length=None)#
class sage.combinat.words.paths.FiniteWordPath_triangle_grid_list#
class sage.combinat.words.paths.FiniteWordPath_triangle_grid_str#
class sage.combinat.words.paths.FiniteWordPath_triangle_grid_tuple#
sage.combinat.words.paths.WordPaths(alphabet, steps=None)#

Returns the combinatorial class of paths of the given type of steps.

INPUT:

• alphabet - ordered alphabet

• steps - (default is None). It can be one of the following:

• an iterable ordered container of as many vectors as there are letters in the alphabet. The vectors are associated to the letters according to their order in steps. The vectors can be a tuple or anything that can be passed to vector function.

• an iterable ordered container of k vectors where k is half the size of alphabet. The vectors and their opposites are associated to the letters according to their order in steps (given vectors first, opposite vectors after).

• None: In this case, the type of steps are guessed from the length of alphabet.

• ‘square_grid’ or ‘square’: (default when size of alphabet is 4) The order is : East, North, West, South.

• ‘triangle_grid’ or ‘triangle’:

• ‘hexagonal_grid’ or ‘hexagon’: (default when size of alphabet is 6)

• ‘cube_grid’ or ‘cube’:

• ‘north_east’, ‘ne’ or ‘NE’: (the default when size of alphabet is 2)

• ‘dyck’:

OUTPUT:

• The combinatorial class of all paths of the given type.

EXAMPLES:

The steps can be given explicitly:

sage: WordPaths('abc', steps=[(1,2), (-1,4), (0,-3)])
Word Paths over 3 steps


Different type of input alphabet:

sage: WordPaths(range(3), steps=[(1,2), (-1,4), (0,-3)])
Word Paths over 3 steps
sage: WordPaths(['cric','crac','croc'], steps=[(1,2), (1,4), (0,3)])
Word Paths over 3 steps


Directions can be in three dimensions as well:

sage: WordPaths('ab', steps=[(1,2,2),(-1,4,2)])
Word Paths over 2 steps


When the number of given steps is half the size of alphabet, the opposite of vectors are used:

sage: P = WordPaths('abcd', [(1,0), (0,1)])
sage: P
Word Paths over 4 steps
sage: sorted(P.letters_to_steps().items())
[('a', (1, 0)), ('b', (0, 1)), ('c', (-1, 0)), ('d', (0, -1))]


When no steps are given, default classes are returned:

sage: WordPaths('ab')
Word Paths in North and East steps
sage: WordPaths(range(4))
Word Paths on the square grid
sage: WordPaths(range(6))
Word Paths on the hexagonal grid


There are many type of built-in steps…

On a two letters alphabet:

sage: WordPaths('ab', steps='north_east')
Word Paths in North and East steps
sage: WordPaths('()', steps='dyck')
Finite Dyck paths


On a four letters alphabet:

sage: WordPaths('ruld', steps='square_grid')
Word Paths on the square grid


On a six letters alphabet:

sage: WordPaths('abcdef', steps='hexagonal_grid')
Word Paths on the hexagonal grid
sage: WordPaths('abcdef', steps='triangle_grid')
Word Paths on the triangle grid
sage: WordPaths('abcdef', steps='cube_grid')
Word Paths on the cube grid

class sage.combinat.words.paths.WordPaths_all(alphabet, steps)#

Bases: FiniteWords

The combinatorial class of all paths, i.e of all words over an alphabet where each letter is mapped to a step (a vector).

letters_to_steps()#

Returns the dictionary mapping letters to vectors (steps).

EXAMPLES:

sage: d = WordPaths('ab').letters_to_steps()
sage: sorted(d.items())
[('a', (0, 1)), ('b', (1, 0))]
sage: d = WordPaths('abcd').letters_to_steps()
sage: sorted(d.items())
[('a', (1, 0)), ('b', (0, 1)), ('c', (-1, 0)), ('d', (0, -1))]
sage: d = WordPaths('abcdef').letters_to_steps()
sage: sorted(d.items())
[('a', (1, 0)),
('b', (1/2, 1/2*sqrt3)),
('c', (-1/2, 1/2*sqrt3)),
('d', (-1, 0)),
('e', (-1/2, -1/2*sqrt3)),
('f', (1/2, -1/2*sqrt3))]

vector_space()#

Return the vector space over which the steps of the paths are defined.

EXAMPLES:

sage: WordPaths('ab',steps='dyck').vector_space()
Ambient free module of rank 2 over the principal ideal domain Integer Ring
sage: WordPaths('ab',steps='north_east').vector_space()
Ambient free module of rank 2 over the principal ideal domain Integer Ring
sage: WordPaths('abcd',steps='square_grid').vector_space()
Ambient free module of rank 2 over the principal ideal domain Integer Ring
sage: WordPaths('abcdef',steps='hexagonal_grid').vector_space()
Vector space of dimension 2 over Number Field in sqrt3 with defining polynomial x^2 - 3 with sqrt3 = 1.732050807568878?
sage: WordPaths('abcdef',steps='cube_grid').vector_space()
Ambient free module of rank 3 over the principal ideal domain Integer Ring
sage: WordPaths('abcdef',steps='triangle_grid').vector_space()
Vector space of dimension 2 over Number Field in sqrt3 with defining polynomial x^2 - 3 with sqrt3 = 1.732050807568878?

class sage.combinat.words.paths.WordPaths_cube_grid(alphabet)#

Bases: WordPaths_all

The combinatorial class of all paths on the cube grid.

class sage.combinat.words.paths.WordPaths_dyck(alphabet)#

Bases: WordPaths_all

The combinatorial class of all Dyck paths.

class sage.combinat.words.paths.WordPaths_hexagonal_grid(alphabet)#

The combinatorial class of all paths on the hexagonal grid.

class sage.combinat.words.paths.WordPaths_north_east(alphabet)#

Bases: WordPaths_all

The combinatorial class of all paths using North and East directions.

class sage.combinat.words.paths.WordPaths_square_grid(alphabet)#

Bases: WordPaths_all

The combinatorial class of all paths on the square grid.

class sage.combinat.words.paths.WordPaths_triangle_grid(alphabet)#

Bases: WordPaths_all

The combinatorial class of all paths on the triangle grid.