similarity_surface_generators

Initialize self. See help(type(self)) for accurate signature.

class flatsurf.geometry.similarity_surface_generators.DilationSurfaceGenerators

static basic_dilation_torus(a)

Return a dilation torus built from a \(1 \times 1\) square and a \(a \times 1\) rectangle. Each edge of the square is glued to the opposite edge of the rectangle. This results in horizontal edges glued by a dilation with a scaling factor of a, and vertical edges being glued by translation:

    b       a
  +----+---------+
  | 0  | 1       |
c |    |         | c
  +----+---------+
    a       b

EXAMPLES:

sage: from flatsurf import dilation_surfaces
sage: ds = dilation_surfaces.basic_dilation_torus(AA(sqrt(2)))
sage: ds
Genus 1 Positive Dilation Surface built from a square and a rectangle
sage: from flatsurf.geometry.categories import DilationSurfaces
sage: ds in DilationSurfaces().Positive()
True
sage: TestSuite(ds).run()

static genus_two_square(a, b, c, d)

A genus two dilation surface is returned.

The unit square is made into an octagon by marking a point on each of its edges. Then opposite sides of this octagon are glued together by translation. (Since we currently require strictly convex polygons, we subdivide the square into a hexagon and two triangles as depicted below.) The parameters a, b, c, and d should be real numbers strictly between zero and one. These represent the lengths of an edge of the resulting octagon, as below:

        c
  +--+-------+
d |2/        |
  |/         |
  +    0     +
  |         /|
  |        /1| b
  +-------+--+
     a

The other edges will have length \(1-a\), \(1-b\), \(1-c\), and \(1-d\). Dilations used to glue edges will be by factors \(c/a\), \(d/b\), \((1-c)/(1-a)\) and \((1-d)/(1-b)\).

EXAMPLES:

sage: from flatsurf import dilation_surfaces
sage: ds = dilation_surfaces.genus_two_square(1/2, 1/3, 1/4, 1/5)
sage: ds
Genus 2 Positive Dilation Surface built from 2 right triangles and a hexagon
sage: from flatsurf.geometry.categories import DilationSurfaces
sage: ds in DilationSurfaces().Positive()
True
sage: TestSuite(ds).run()

class flatsurf.geometry.similarity_surface_generators.EInfinitySurface(lambda_squared=None, field=None)

The surface based on the \(E_\infinity\) graph.

The biparite graph is shown below, with edges numbered:

  0   1   2  -2   3  -3   4  -4
*---o---*---o---*---o---*---o---*...
        |
        |-1
        o

Here, black vertices are colored *, and white o. Black nodes represent vertical cylinders and white nodes represent horizontal cylinders.

get_black(n)

Get the weight of the black endpoint of edge n.

get_white(n)

Get the weight of the white endpoint of edge n.

is_compact()

Return whether this surface is compact as a topological space, i.e., return False.

This implements flatsurf.geometry.categories.topological_surfaces.TopologicalSurfaces.ParentMethods.is_compact().

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: S = translation_surfaces.e_infinity_surface()
sage: S.is_compact()
False

is_mutable()

Return whether this surface is mutable, i.e., return False.

This implements flatsurf.geometry.categories.topological_surfaces.TopologicalSurfaces.ParentMethods.is_mutable().

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: S = translation_surfaces.e_infinity_surface()
sage: S.is_mutable()
False

labels()

Return the labels of this surface.

This implements flatsurf.geometry.categories.polygonal_surfaces.PolygonalSurfaces.ParentMethods.labels().

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: S = translation_surfaces.e_infinity_surface()
sage: S.labels()
(0, 1, -1, 2, -2, 3, -3, 4, -4, 5, -5, 6, -6, 7, -7, 8, …)

opposite_edge(p, e)

Return the pair (pp,ee) to which the edge (p,e) is glued to.

polygon(lab)

Return the polygon labeled by lab.

roots()

Return root labels for the polygons forming the connected components of this surface.

This implements flatsurf.geometry.categories.polygonal_surfaces.PolygonalSurfaces.ParentMethods.roots().

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: S = translation_surfaces.e_infinity_surface()
sage: S.roots()
(0,)

class flatsurf.geometry.similarity_surface_generators.HalfTranslationSurfaceGenerators

static step_billiard(w, h)

Return a (finite) step billiard associated to the given widths w and heights h.

EXAMPLES:

sage: from flatsurf import half_translation_surfaces
sage: S = half_translation_surfaces.step_billiard([1,1,1,1], [1,1/2,1/3,1/5])
sage: S
StepBilliard(w=[1, 1, 1, 1], h=[1, 1/2, 1/3, 1/5])
sage: from flatsurf.geometry.categories import DilationSurfaces
sage: S in DilationSurfaces()
True
sage: TestSuite(S).run()

class flatsurf.geometry.similarity_surface_generators.SimilaritySurfaceGenerators

Examples of similarity surfaces.

static billiard(P, rational=None)

Return the ConeSurface associated to the billiard in the polygon P.

INPUT:

• P – a polygon

• rational – a boolean or None (default: None) – whether to assume that all the angles of P are a rational multiple of π.

EXAMPLES:

sage: from flatsurf import Polygon, similarity_surfaces

sage: P = Polygon(vertices=[(0,0), (1,0), (0,1)])
sage: Q = similarity_surfaces.billiard(P, rational=True)
doctest:warning
...
UserWarning: the rational keyword argument of billiard() has been deprecated and will be removed in a future version of sage-flatsurf; rationality checking is now faster so this is not needed anymore
sage: Q
Genus 0 Rational Cone Surface built from 2 isosceles triangles
sage: from flatsurf.geometry.categories import ConeSurfaces
sage: Q in ConeSurfaces().Rational()
True
sage: M = Q.minimal_cover(cover_type="translation")
sage: M
Minimal Translation Cover of Genus 0 Rational Cone Surface built from 2 isosceles triangles
sage: TestSuite(M).run()
sage: from flatsurf.geometry.categories import TranslationSurfaces
sage: M in TranslationSurfaces()
True

A non-convex examples (L-shape polygon):

sage: P = Polygon(vertices=[(0,0), (2,0), (2,1), (1,1), (1,2), (0,2)])
sage: Q = similarity_surfaces.billiard(P)
sage: TestSuite(Q).run()
sage: M = Q.minimal_cover(cover_type="translation")
sage: TestSuite(M).run()
sage: M.stratum()
H_2(2, 0^5)

A quadrilateral from Eskin-McMullen-Mukamel-Wright:

sage: from flatsurf import Polygon
sage: P = Polygon(angles=(1, 1, 1, 7))
sage: S = similarity_surfaces.billiard(P)
sage: TestSuite(S).run()
sage: S = S.minimal_cover(cover_type="translation")
sage: TestSuite(S).run()
sage: S = S.erase_marked_points() # optional: pyflatsurf
sage: TestSuite(S).run()
sage: S, _ = S.normalized_coordinates()
sage: TestSuite(S).run()

Unfolding a triangle with non-algebraic lengths:

sage: from flatsurf import EuclideanPolygonsWithAngles
sage: E = EuclideanPolygonsWithAngles((3, 3, 5))
sage: from pyexactreal import ExactReals # optional: pyexactreal
sage: R = ExactReals(E.base_ring()) # optional: pyexactreal
sage: angles = (3, 3, 5)
sage: slopes = EuclideanPolygonsWithAngles(*angles).slopes()
sage: P = Polygon(angles=angles, edges=[R.random_element() * slopes[0]])  # optional: pyexactreal
sage: S = similarity_surfaces.billiard(P); S # optional: pyexactreal
Genus 0 Rational Cone Surface built from 2 isosceles triangles
sage: TestSuite(S).run() # long time (6s), optional: pyexactreal
sage: from flatsurf.geometry.categories import ConeSurfaces
sage: S in ConeSurfaces()
True

static example()

Construct a SimilaritySurface from a pair of triangles.

EXAMPLES:

sage: from flatsurf import similarity_surfaces
sage: ex = similarity_surfaces.example()
sage: ex
Genus 1 Surface built from 2 isosceles triangles

static polygon_double(P)

Return the ConeSurface associated to the billiard in the polygon P. Differs from billiard(P) only in the graphical display. Here, we display the polygons separately.

static right_angle_triangle(w, h)

static self_glued_polygon(P)

Return the HalfTranslationSurface formed by gluing all edges of P to themselves.

EXAMPLES:

sage: from flatsurf import Polygon, similarity_surfaces
sage: p = Polygon(edges=[(2,0),(-1,3),(-1,-3)])
sage: s = similarity_surfaces.self_glued_polygon(p)
sage: s
Half-Translation Surface in Q_0(-1^4) built from an isosceles triangle
sage: TestSuite(s).run()

class flatsurf.geometry.similarity_surface_generators.TFractalSurface(w=1, r=2, h1=1, h2=1)

The TFractal surface.

The TFractal surface is a translation surface of finite area built from infinitely many polygons. The basic building block is the following polygon:

 w/r    w     w/r
+---+------+---+
| 1 |   2  | 3 | h2
+---+------+---+
    |   0  | h1
    +------+
        w

where w, h1, h2, r are some positive numbers. Default values are w=h1=h2=1 and r=2.

is_mutable()

Return whether this surface is mutable, i.e., return False.

This implements flatsurf.geometry.categories.topological_surfaces.TopologicalSurfaces.ParentMethods.is_mutable().

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: S = translation_surfaces.t_fractal()
sage: S.is_mutable()
False

labels()

Return the labels of this surface.

This implements flatsurf.geometry.categories.polygonal_surfaces.PolygonalSurfaces.ParentMethods.labels().

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: S = translation_surfaces.t_fractal()
sage: S.labels()
((word: , 0), (word: , 2), (word: , 3), (word: , 1), (word: R, 2), (word: R, 0), (word: L, 2), (word: L, 0), (word: R, 3), (word: R, 1), (word: L, 3), (word: L, 1), (word: RR, 2), (word: RR, 0), (word: RL, 2), (word: RL, 0), …)

opposite_edge(p, e)

Labeling of polygons:

 wl,0             wr,0
+-----+---------+------+
|     |         |      |
| w,1 |   w,2   |  w,3 |
|     |         |      |
+-----+---------+------+
      |         |
      |   w,0   |
      |         |
      +---------+
           w

and we always have: bot->0, right->1, top->2, left->3

EXAMPLES:

sage: import flatsurf.geometry.similarity_surface_generators as sfg
sage: T = sfg.tfractal_surface()
sage: W = T._words
sage: w = W('LLRLRL')
sage: T.opposite_edge((w,0),0)
((word: LLRLR, 1), 2)
sage: T.opposite_edge((w,0),1)
((word: LLRLRL, 0), 3)
sage: T.opposite_edge((w,0),2)
((word: LLRLRL, 2), 0)
sage: T.opposite_edge((w,0),3)
((word: LLRLRL, 0), 1)

polygon(lab)

Return the polygon with label lab:

 w/r         w/r
+---+------+---+
| 1 |  2   | 3 |
|   |      |   |  h2
+---+------+---+
    |  0   | h1
    +------+
    w

EXAMPLES:

sage: import flatsurf.geometry.similarity_surface_generators as sfg
sage: T = sfg.tfractal_surface()
sage: T.polygon(('L',0))
Polygon(vertices=[(0, 0), (1/2, 0), (1/2, 1/2), (0, 1/2)])
sage: T.polygon(('LRL',0))
Polygon(vertices=[(0, 0), (1/8, 0), (1/8, 1/8), (0, 1/8)])

roots()

Return root labels for the polygons forming the connected components of this surface.

This implements flatsurf.geometry.categories.polygonal_surfaces.PolygonalSurfaces.ParentMethods.roots().

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: S = translation_surfaces.t_fractal()
sage: S.roots()
((word: , 0),)

class flatsurf.geometry.similarity_surface_generators.TranslationSurfaceGenerators

Common and less common translation surfaces.

static arnoux_yoccoz(genus)

Construct the Arnoux-Yoccoz surface of genus 3 or greater.

This presentation of the surface follows Section 2.3 of Joshua P. Bowman’s paper “The Complete Family of Arnoux-Yoccoz Surfaces.”

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s = translation_surfaces.arnoux_yoccoz(4)
sage: s
Translation Surface in H_4(3^2) built from 16 triangles
sage: TestSuite(s).run()
sage: s.is_delaunay_decomposed()
True
sage: s = s.canonicalize()
sage: s
Translation Surface in H_4(3^2) built from 16 triangles
sage: field=s.base_ring()
sage: a = field.gen()
sage: from sage.matrix.constructor import Matrix
sage: m = Matrix([[a,0],[0,~a]])
sage: ss = m*s
sage: ss = ss.canonicalize()
sage: s.cmp(ss) == 0
True

The Arnoux-Yoccoz pseudo-Anosov are known to have (minimal) invariant foliations with SAF=0:

sage: S3 = translation_surfaces.arnoux_yoccoz(3)
sage: Jxx, Jyy, Jxy = S3.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0  2  0]
[ 2 -2  0]
[ 0  0  2]

sage: S4 = translation_surfaces.arnoux_yoccoz(4)
sage: Jxx, Jyy, Jxy = S4.j_invariant()
sage: Jxx.is_zero() and Jyy.is_zero()
True
sage: Jxy
[ 0  2  0  0]
[ 2 -2  0  0]
[ 0  0  2  2]
[ 0  0  2  0]

static cathedral(a, b)

Return the cathedral surface with parameters a and b.

For any parameter a and b, the cathedral surface belongs to the so-called Gothic locus described in McMullen, Mukamel, Wright “Cubic curves and totally geodesic subvarieties of moduli space” (2017):

         1
       <--->

        /\           2a
       /  \      +------+
   a  b|   | a  /        \
 +----+    +---+          +
 |    |    |   |          |
1| P0 |P1  |P2 |  P3      |
 |    |    |   |          |
 +----+    +---+          +
     b|    |    \        /
       \  /      +------+
        \/

If a and b satisfies

\[a = x + y \sqrt(d) \qquad b = -3x -3/2 + 3y \sqrt(d)\]

for some rational x,y and d >= 0 then it is a Teichmüller curve.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: C = translation_surfaces.cathedral(1,2)
sage: C
Translation Surface in H_4(2^3) built from 2 squares, a hexagon with 4 marked vertices and an octagon
sage: TestSuite(C).run()

sage: from pyexactreal import ExactReals # optional: pyexactreal
sage: K = QuadraticField(5, embedding=AA(5).sqrt())
sage: R = ExactReals(K) # optional: pyexactreal
sage: C = translation_surfaces.cathedral(K.gen(), R.random_element([0.1, 0.2])) # optional: pyexactreal
sage: C  # optional: pyexactreal
Translation Surface in H_4(2^3) built from 2 rectangles, a hexagon with 4 marked vertices and an octagon
sage: C.stratum() # optional: pyexactreal
H_4(2^3)
sage: TestSuite(C).run() # long time (6s), optional: pyexactreal

static chamanara(alpha)

Return the Chamanara surface with parameter alpha.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: C = translation_surfaces.chamanara(1/2)
sage: C
Minimal Translation Cover of Chamanara surface with parameter 1/2

static e_infinity_surface(lambda_squared=None, field=None)

The translation surface based on the \(E_\infinity\) graph.

The biparite graph is shown below, with edges numbered:

  0   1   2  -2   3  -3   4  -4
*---o---*---o---*---o---*---o---*...
        |
        |-1
        o

Here, black vertices are colored *, and white o. Black nodes represent vertical cylinders and white nodes represent horizontal cylinders.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s = translation_surfaces.e_infinity_surface()
sage: TestSuite(s).run()  # long time (1s)

static from_flipper(h)

Build a (half-)translation surface from a flipper pseudo-Anosov.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: import flipper                             # optional - flipper

A torus example:

sage: t1 = (0r,1r,2r)                            # optional - flipper
sage: t2 = (~0r,~1r,~2r)                         # optional - flipper
sage: T = flipper.create_triangulation([t1,t2])  # optional - flipper
sage: L1 = T.lamination([1r,0r,1r])              # optional - flipper
sage: L2 = T.lamination([0r,1r,1r])              # optional - flipper
sage: h1 = L1.encode_twist()                     # optional - flipper
sage: h2 = L2.encode_twist()                     # optional - flipper
sage: h = h1*h2^(-1r)                            # optional - flipper
sage: f = h.flat_structure()                     # optional - flipper
sage: ts = translation_surfaces.from_flipper(h)  # optional - flipper
sage: ts                                         # optional - flipper; computation of the stratum fails here, see #227
Half-Translation Surface built from 2 triangles
sage: TestSuite(ts).run()                        # optional - flipper
sage: from flatsurf.geometry.categories import HalfTranslationSurfaces  # optional: flipper
sage: ts in HalfTranslationSurfaces()  # optional: flipper
True

A non-orientable example:

sage: T = flipper.load('SB_4')                   # optional - flipper
sage: h = T.mapping_class('s_0S_1s_2S_3s_1S_2')  # optional - flipper
sage: h.is_pseudo_anosov()                       # optional - flipper
True
sage: S = translation_surfaces.from_flipper(h)   # optional - flipper
sage: TestSuite(S).run()                         # optional - flipper
sage: len(S.polygons())                          # optional - flipper
4
sage: from flatsurf.geometry.similarity_surface_generators import flipper_nf_element_to_sage
sage: a = flipper_nf_element_to_sage(h.dilatation())  # optional - flipper

static infinite_staircase()

Return the infinite staircase built as an origami.

EXAMPLES:

sage: from flatsurf import translation_surfaces

sage: S = translation_surfaces.infinite_staircase()
sage: S
The infinite staircase
sage: TestSuite(S).run()

static lanneau_nguyen_genus3_prototype(w, h, t, e)

Return the Lanneau–Ngyuen prototype in the stratum H(4) with parameters w, h, t, and e.

The associated discriminant is \(e^2 + 8 wh\).

INPUT:

• w – a positive integer

• h – a positive integer

• t – a non-negative integer strictly less than the gcd of w and h

• e – an integer strictly less than w - 2h

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: T = translation_surfaces.lanneau_nguyen_genus3_prototype(2,1,0,-1)
sage: T.stratum()
H_3(4)

REFERENCES:

• Erwan Lanneau, and Duc-Manh Nguyen, “Teichmüller curves generated by Weierstrass Prym eigenforms in genus 3 and genus 4”, Journal of Topology 7.2, 2014, pp.475-522

static lanneau_nguyen_genus4_prototype(w, h, t, e)

Return the Lanneau–Ngyuen prototype in the stratum H(6) with parameters w, h, t, and e.

The associated discriminant is \(D = e^2 + 4 wh\).

INPUT:

• w – a positive integer

• h – a positive integer

• t – a non-negative integer strictly smaller than the gcd of w and h

• e – a positive integer strictly smaller than \(2w - \sqrt{D}\)

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: T = translation_surfaces.lanneau_nguyen_genus4_prototype(1,1,0,-2)
sage: T.stratum()
H_4(6)

REFERENCES:

• Erwan Lanneau, and Duc-Manh Nguyen, “Weierstrass Prym eigenforms in genus four”, Journal of the Institute of Mathematics of Jussieu, 19.6, 2020, pp.2045-2085

static mcmullen_L(l1, l2, l3, l4)

Return McMullen’s L shaped surface with parameters l1, l2, l3, l4.

Polygon labels and lengths are marked below:

+-----+
|     |
|  1  |l1
|     |
|     |    l4
+-----+---------+
|     |         |
|  0  |    2    |l2
|     |         |
+-----+---------+
  l3

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s = translation_surfaces.mcmullen_L(1,1,1,1)
sage: s
Translation Surface in H_2(2) built from 3 squares
sage: TestSuite(s).run()

static mcmullen_genus2_prototype(w, h, t, e, rel=0, base_ring=None)

McMullen prototypes in the stratum H(2).

These prototype appear at least in McMullen “Teichmüller curves in genus two: Discriminant and spin” (2004). The notation from that paper are quadruple (a, b, c, e) which translates in our notation as w = b, h = c, t = a (and e = e).

The associated discriminant is \(D = e^2 + 4 wh\).

If rel is a positive parameter (less than w-lambda) the surface belongs to the eigenform locus in H(1,1).

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: from surface_dynamics import Stratum

sage: prototypes = {
....:      5: [(1,1,0,-1)],
....:      8: [(1,1,0,-2), (2,1,0,0)],
....:      9: [(2,1,0,-1)],
....:     12: [(1,2,0,-2), (2,1,0,-2), (3,1,0,0)],
....:     13: [(1,1,0,-3), (3,1,0,-1), (3,1,0,1)],
....:     16: [(3,1,0,-2), (4,1,0,0)],
....:     17: [(1,2,0,-3), (2,1,0,-3), (2,2,0,-1), (2,2,1,-1), (4,1,0,-1), (4,1,0,1)],
....:     20: [(1,1,0,-4), (2,2,1,-2), (4,1,0,-2), (4,1,0,2)],
....:     21: [(1,3,0,-3), (3,1,0,-3)],
....:     24: [(1,2,0,-4), (2,1,0,-4), (3,2,0,0)],
....:     25: [(2,2,0,-3), (2,2,1,-3), (3,2,0,-1), (4,1,0,-3)]}

sage: for D in sorted(prototypes):  # long time (.5s)
....:     for w,h,t,e in prototypes[D]:
....:          T = translation_surfaces.mcmullen_genus2_prototype(w,h,t,e)
....:          assert T.stratum() == Stratum([2], 1)
....:          assert (D.is_square() and T.base_ring() is QQ) or (T.base_ring().polynomial().discriminant() == D)

An example with some relative homology:

sage: U8 = translation_surfaces.mcmullen_genus2_prototype(2,1,0,0,1/4)    # discriminant 8
sage: U8
Translation Surface in H_2(1^2) built from a rectangle and a quadrilateral
sage: U12 = translation_surfaces.mcmullen_genus2_prototype(3,1,0,0,3/10)   # discriminant 12
sage: U12
Translation Surface in H_2(1^2) built from a rectangle and a quadrilateral

sage: U8.stratum()
H_2(1^2)
sage: U8.base_ring().polynomial().discriminant()
8
sage: U8.j_invariant()
(
          [4 0]
(0), (0), [0 2]
)

sage: U12.stratum()
H_2(1^2)
sage: U12.base_ring().polynomial().discriminant()
12
sage: U12.j_invariant()
(
          [6 0]
(0), (0), [0 2]
)

static octagon_and_squares()

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: os = translation_surfaces.octagon_and_squares()
sage: os
Translation Surface in H_3(4) built from 2 squares and a regular octagon
sage: TestSuite(os).run()
sage: from flatsurf.geometry.categories import TranslationSurfaces
sage: os in TranslationSurfaces()
True

static origami(r, u, rr=None, uu=None, domain=None)

Return the origami defined by the permutations r and u.

EXAMPLES:

sage: from flatsurf import translation_surfaces

sage: S = SymmetricGroup(3)
sage: r = S('(1,2)')
sage: u = S('(1,3)')
sage: o = translation_surfaces.origami(r,u)
sage: o
Origami defined by r=(1,2) and u=(1,3)
sage: o.stratum()
H_2(2)
sage: TestSuite(o).run()

static regular_octagon()

Return the translation surface built from the regular octagon by identifying opposite sides.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: T = translation_surfaces.regular_octagon()
sage: T
Translation Surface in H_2(2) built from a regular octagon
sage: TestSuite(T).run()
sage: from flatsurf.geometry.categories import TranslationSurfaces
sage: T in TranslationSurfaces()
True

static square_torus(a=1)

Return flat torus obtained by identification of the opposite sides of a square.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: T = translation_surfaces.square_torus()
sage: T
Translation Surface in H_1(0) built from a square
sage: from flatsurf.geometry.categories import TranslationSurfaces
sage: T in TranslationSurfaces()
True
sage: TestSuite(T).run()

Rational directions are completely periodic:

sage: v = T.tangent_vector(0, (1/33, 1/257), (13,17))
sage: L = v.straight_line_trajectory()
sage: L.flow(13+17)
sage: L.is_closed()
True

static t_fractal(w=1, r=2, h1=1, h2=1)

Return the T-fractal with parameters w, r, h1, h2.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: tf = translation_surfaces.t_fractal()  # long time (.4s)
sage: tf  # long time (see above)
The T-fractal surface with parameters w=1, r=2, h1=1, h2=1
sage: TestSuite(tf).run()  # long time (see above)

static torus(u, v)

Return the flat torus obtained as the quotient of the plane by the pair of vectors u and v.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: T = translation_surfaces.torus((1, AA(2).sqrt()), (AA(3).sqrt(), 3))
sage: T
Translation Surface in H_1(0) built from a quadrilateral
sage: T.polygon(0)
Polygon(vertices=[(0, 0), (1, 1.414213562373095?), (2.732050807568878?, 4.414213562373095?), (1.732050807568878?, 3)])
sage: from flatsurf.geometry.categories import TranslationSurfaces
sage: T in TranslationSurfaces()
True

static veech_2n_gon(n)

The regular 2n-gon with opposite sides identified.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s = translation_surfaces.veech_2n_gon(5)
sage: s
Translation Surface in H_2(1^2) built from a regular decagon
sage: TestSuite(s).run()

static veech_double_n_gon(n)

A pair of regular n-gons with each edge of one identified to an edge of the other to make a translation surface.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s=translation_surfaces.veech_double_n_gon(5)
sage: s
Translation Surface in H_2(2) built from 2 regular pentagons
sage: TestSuite(s).run()

static ward(n)

Return the surface formed by gluing a regular 2n-gon to two regular n-gons. These surfaces have Veech’s lattice property due to work of Ward.

EXAMPLES:

sage: from flatsurf import translation_surfaces
sage: s = translation_surfaces.ward(3)
sage: s
Translation Surface in H_1(0^3) built from 2 equilateral triangles and a regular hexagon
sage: TestSuite(s).run()
sage: s = translation_surfaces.ward(7)
sage: s
Translation Surface in H_6(10) built from 2 regular heptagons and a regular tetradecagon
sage: TestSuite(s).run()

flatsurf.geometry.similarity_surface_generators.flipper_nf_element_to_sage(x, K=None)

Convert a flipper number field element into Sage

EXAMPLES:

sage: from flatsurf.geometry.similarity_surface_generators import flipper_nf_element_to_sage
sage: import flipper                               # optional - flipper
sage: T = flipper.load('SB_6')                     # optional - flipper
sage: h = T.mapping_class('s_0S_1S_2s_3s_4s_3S_5') # optional - flipper
sage: flipper_nf_element_to_sage(h.dilatation())   # optional - flipper
a
sage: AA(_)                                        # optional - flipper
6.45052513748511?

flatsurf.geometry.similarity_surface_generators.flipper_nf_to_sage(K, name='a')

Convert a flipper number field into a Sage number field

Note

Currently, the code is not careful at all with root isolation.

EXAMPLES:

sage: import flipper  # optional - flipper  # random output due to matplotlib warnings with some combinations of setuptools and matplotlib
sage: import realalg  # optional - flipper
sage: from flatsurf.geometry.similarity_surface_generators import flipper_nf_to_sage
sage: K = realalg.RealNumberField([-2r] + [0r]*5 + [1r])   # optional - flipper
sage: K_sage = flipper_nf_to_sage(K)                       # optional - flipper
sage: K_sage                                               # optional - flipper
Number Field in a with defining polynomial x^6 - 2 with a = 1.122462048309373?
sage: AA(K_sage.gen())                                     # optional - flipper
1.122462048309373?

flatsurf.geometry.similarity_surface_generators.tfractal_surface(w=1, r=2, h1=1, h2=1)