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: rss,
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 = P('babaddefadabcadefaadfafabacdefa')
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#
Bases:
FiniteWordPath_all
- 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 drawingfill
- (boolean, default: True), if fill is True and if the path is closed, the inside is coloredfilloptions
- (dict, default:dict(rgbcolor=’red’,alpha=0.2)), options for the inside fillingstartpoint
- (boolean, default: True), draw the start point?startoptions
- (dict, default:dict(rgbcolor=’red’,pointsize=100)) options for the start point drawingendarrow
- (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 arrowgridlines
- (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)#
Bases:
WordDatatype_callable
,FiniteWordPath_2d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_2d_callable_with_caching(parent, callable, length=None)#
Bases:
WordDatatype_callable_with_caching
,FiniteWordPath_2d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_2d_iter(parent, iter, length=None)#
Bases:
WordDatatype_iter
,FiniteWordPath_2d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_2d_iter_with_caching(parent, iter, length=None)#
Bases:
WordDatatype_iter_with_caching
,FiniteWordPath_2d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_2d_list#
Bases:
WordDatatype_list
,FiniteWordPath_2d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_2d_str#
Bases:
WordDatatype_str
,FiniteWordPath_2d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_2d_tuple#
Bases:
WordDatatype_tuple
,FiniteWordPath_2d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_3d#
Bases:
FiniteWordPath_all
- 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 drawingstartpoint
- (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)#
Bases:
WordDatatype_callable
,FiniteWordPath_3d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_3d_callable_with_caching(parent, callable, length=None)#
Bases:
WordDatatype_callable_with_caching
,FiniteWordPath_3d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_3d_iter(parent, iter, length=None)#
Bases:
WordDatatype_iter
,FiniteWordPath_3d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_3d_iter_with_caching(parent, iter, length=None)#
Bases:
WordDatatype_iter_with_caching
,FiniteWordPath_3d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_3d_list#
Bases:
WordDatatype_list
,FiniteWordPath_3d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_3d_str#
Bases:
WordDatatype_str
,FiniteWordPath_3d
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_3d_tuple#
Bases:
WordDatatype_tuple
,FiniteWordPath_3d
,FiniteWord_class
- 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 sage: P('ad').is_simple() 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 spacev
- 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)#
Bases:
WordDatatype_callable
,FiniteWordPath_all
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_all_callable_with_caching(parent, callable, length=None)#
Bases:
WordDatatype_callable_with_caching
,FiniteWordPath_all
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_all_iter(parent, iter, length=None)#
Bases:
WordDatatype_iter
,FiniteWordPath_all
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_all_iter_with_caching(parent, iter, length=None)#
Bases:
WordDatatype_iter_with_caching
,FiniteWordPath_all
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_all_list#
Bases:
WordDatatype_list
,FiniteWordPath_all
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_all_str#
Bases:
WordDatatype_str
,FiniteWordPath_all
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_all_tuple#
Bases:
WordDatatype_tuple
,FiniteWordPath_all
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_cube_grid#
Bases:
FiniteWordPath_3d
- class sage.combinat.words.paths.FiniteWordPath_cube_grid_callable(parent, callable, length=None)#
Bases:
WordDatatype_callable
,FiniteWordPath_cube_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_cube_grid_callable_with_caching(parent, callable, length=None)#
Bases:
WordDatatype_callable_with_caching
,FiniteWordPath_cube_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_cube_grid_iter(parent, iter, length=None)#
Bases:
WordDatatype_iter
,FiniteWordPath_cube_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_cube_grid_iter_with_caching(parent, iter, length=None)#
Bases:
WordDatatype_iter_with_caching
,FiniteWordPath_cube_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_cube_grid_list#
Bases:
WordDatatype_list
,FiniteWordPath_cube_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_cube_grid_str#
Bases:
WordDatatype_str
,FiniteWordPath_cube_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_cube_grid_tuple#
Bases:
WordDatatype_tuple
,FiniteWordPath_cube_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_dyck#
Bases:
FiniteWordPath_2d
- class sage.combinat.words.paths.FiniteWordPath_dyck_callable(parent, callable, length=None)#
Bases:
WordDatatype_callable
,FiniteWordPath_dyck
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_dyck_callable_with_caching(parent, callable, length=None)#
Bases:
WordDatatype_callable_with_caching
,FiniteWordPath_dyck
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_dyck_iter(parent, iter, length=None)#
Bases:
WordDatatype_iter
,FiniteWordPath_dyck
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_dyck_iter_with_caching(parent, iter, length=None)#
Bases:
WordDatatype_iter_with_caching
,FiniteWordPath_dyck
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_dyck_list#
Bases:
WordDatatype_list
,FiniteWordPath_dyck
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_dyck_str#
Bases:
WordDatatype_str
,FiniteWordPath_dyck
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_dyck_tuple#
Bases:
WordDatatype_tuple
,FiniteWordPath_dyck
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid(parent, *args, **kwds)#
Bases:
FiniteWordPath_triangle_grid
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)#
Bases:
WordDatatype_callable
,FiniteWordPath_hexagonal_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_callable_with_caching(parent, callable, length=None)#
Bases:
WordDatatype_callable_with_caching
,FiniteWordPath_hexagonal_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_iter(parent, iter, length=None)#
Bases:
WordDatatype_iter
,FiniteWordPath_hexagonal_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_iter_with_caching(parent, iter, length=None)#
Bases:
WordDatatype_iter_with_caching
,FiniteWordPath_hexagonal_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_list#
Bases:
WordDatatype_list
,FiniteWordPath_hexagonal_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_str#
Bases:
WordDatatype_str
,FiniteWordPath_hexagonal_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_hexagonal_grid_tuple#
Bases:
WordDatatype_tuple
,FiniteWordPath_hexagonal_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_north_east#
Bases:
FiniteWordPath_2d
- class sage.combinat.words.paths.FiniteWordPath_north_east_callable(parent, callable, length=None)#
Bases:
WordDatatype_callable
,FiniteWordPath_north_east
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_north_east_callable_with_caching(parent, callable, length=None)#
Bases:
WordDatatype_callable_with_caching
,FiniteWordPath_north_east
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_north_east_iter(parent, iter, length=None)#
Bases:
WordDatatype_iter
,FiniteWordPath_north_east
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_north_east_iter_with_caching(parent, iter, length=None)#
Bases:
WordDatatype_iter_with_caching
,FiniteWordPath_north_east
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_north_east_list#
Bases:
WordDatatype_list
,FiniteWordPath_north_east
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_north_east_str#
Bases:
WordDatatype_str
,FiniteWordPath_north_east
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_north_east_tuple#
Bases:
WordDatatype_tuple
,FiniteWordPath_north_east
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_square_grid#
Bases:
FiniteWordPath_2d
- 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)#
Bases:
WordDatatype_callable
,FiniteWordPath_square_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_square_grid_callable_with_caching(parent, callable, length=None)#
Bases:
WordDatatype_callable_with_caching
,FiniteWordPath_square_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_square_grid_iter(parent, iter, length=None)#
Bases:
WordDatatype_iter
,FiniteWordPath_square_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_square_grid_iter_with_caching(parent, iter, length=None)#
Bases:
WordDatatype_iter_with_caching
,FiniteWordPath_square_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_square_grid_list#
Bases:
WordDatatype_list
,FiniteWordPath_square_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_square_grid_str#
Bases:
WordDatatype_str
,FiniteWordPath_square_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_square_grid_tuple#
Bases:
WordDatatype_tuple
,FiniteWordPath_square_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_triangle_grid#
Bases:
FiniteWordPath_2d
- 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)#
Bases:
WordDatatype_callable
,FiniteWordPath_triangle_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_triangle_grid_callable_with_caching(parent, callable, length=None)#
Bases:
WordDatatype_callable_with_caching
,FiniteWordPath_triangle_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_triangle_grid_iter(parent, iter, length=None)#
Bases:
WordDatatype_iter
,FiniteWordPath_triangle_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_triangle_grid_iter_with_caching(parent, iter, length=None)#
Bases:
WordDatatype_iter_with_caching
,FiniteWordPath_triangle_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_triangle_grid_list#
Bases:
WordDatatype_list
,FiniteWordPath_triangle_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_triangle_grid_str#
Bases:
WordDatatype_str
,FiniteWordPath_triangle_grid
,FiniteWord_class
- class sage.combinat.words.paths.FiniteWordPath_triangle_grid_tuple#
Bases:
WordDatatype_tuple
,FiniteWordPath_triangle_grid
,FiniteWord_class
- sage.combinat.words.paths.WordPaths(alphabet, steps=None)#
Returns the combinatorial class of paths of the given type of steps.
INPUT:
alphabet
- ordered alphabetsteps
- (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)#
Bases:
WordPaths_triangle_grid
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.