generators

generators#

List of common origamis

Usage#

To see a list of all origami constructors, type “origamis.” or “pillowcases.” and then press the tab key. The documentation for each constructor includes information about each origami, which provides a useful reference.

Organization#

The constructors available in this database are organized as follows

class surface_dynamics.flat_surfaces.origamis.generators.OrigamiGenerators#

Examples of origamis.

static CyclicCover(a, M=None)#

Return the Abelian differential associated to the quadruple a

INPUT:

  • a - quadruple of integers

  • M - positive integer (default: None)

EXAMPLES:

sage: from surface_dynamics import *

There are two well known cyclic covers with completely degenerated Lyapunov spectrum:

sage: o = origamis.CyclicCover([1,1,1,1])
sage: o.sum_of_lyapunov_exponents()
1

sage: o = origamis.CyclicCover([1,1,1,3])
sage: o.sum_of_lyapunov_exponents()
1

Some examples with Veech group SL(2,Z) (this happen if and only if the entries of the vector is of the form [x,x,x,y]):

sage: for a in [[1,1,1,5],[1,1,1,7],[1,3,3,3],[1,1,1,9],[1,1,1,11],[3,3,3,5]]:
....:     o = origamis.CyclicCover(a)
....:     assert o.veech_group().index() == 1
....:     print(o.nb_squares(), o.stratum_component())
16 H_7(3^4)^c
20 H_9(4^4)^odd
20 H_9(4^4)^odd
24 H_10(5^3, 1^3)^c
28 H_13(6^4)^odd
28 H_13(6^4)^odd
static EierlegendeWollmilchsau()#

Eierlegende Wollmilchsau origami.

This origami is associated to the quaternion group and was discovered by Gaby Schmithüsen.

EXAMPLES:

sage: from surface_dynamics import *

sage: o = origamis.EierlegendeWollmilchsau()
sage: o
Eierlegende Wollmilchsau
sage: print(str(o))
(1,2,3,4)(5,6,7,8)
(1,5,3,7)(2,8,4,6)
sage: o.veech_group().index()
1
sage: o.sum_of_lyapunov_exponents()
1

sage: oo = origamis.CyclicCover([1,1,1,1])
sage: o.is_isomorphic(oo)
True
static Escalator(n)#

Escalator origamis

The escalator origami has 2n squares and is defined by

r = (1 2)(3 4) … (2n-1 2n) u = (2 3)(4 5) … (2n 1)

EXAMPLES:

sage: from surface_dynamics import *
sage: o = origamis.Escalator(3)
sage: o
Escalator with 3 steps
sage: print(str(o))
(1,2)(3,4)(5,6)
(1,6)(2,3)(4,5)
sage: o.veech_group().index()
3
static Heisenberg(p)#

Return the Heisenberg origami.

EXAMPLES:

sage: from surface_dynamics import *

sage: h2 = origamis.Heisenberg(2)
sage: h2.stratum_component()
H_3(1^4)^c
sage: h2.veech_group().index()
3
sage: h2.sum_of_lyapunov_exponents()
2

sage: h3 = origamis.Heisenberg(3)
sage: h3.stratum_component()
H_10(2^9)^even
sage: h3.veech_group().index()
1
sage: h3.sum_of_lyapunov_exponents()
5

sage: h5 = origamis.Heisenberg(5)
sage: h5.stratum_component()
H_51(4^25)^even
sage: h5.veech_group().index()
1
sage: h5.sum_of_lyapunov_exponents()
15
static Podium(data)#

If data is an integer than the standard podium with data steps is returned. Otherwise, data should be a weakly decreasing list of integers (i.e. a integer partition).

EXAMPLES:

sage: from surface_dynamics import *

sage: o = origamis.Podium([3,3,2,1])
sage: o
Podium origami with partition [3, 3, 2, 1]
sage: print(o)
(1,2,3)(4,5,6)(7,8)(9)
(1,4,7,9)(2,5,8)(3,6)
static ProjectiveLine(p, r=None, u=None)#

Return the projective line with action given by the matrices r and u on the projective line over the field with p elements.

If r and u are None, then r and u are chosen to be

\[r: z \mapsto z+1 \qquad u: z \mapsto 1/z\]

The monodromy of this origami is included in PGL(2,ZZ/pZZ).

EXAMPLES:

sage: from surface_dynamics import *

By default, the matrix used to generate the projective line origami are given by z -> z+1 and z -> 1/z which gives a family of origamis which belongs to the strata H(2^k):

sage: o = origamis.ProjectiveLine(3); o
Projective line origami on GF(3)
 r = [1 1]    u = [0 1]
     [0 1]        [1 0]
sage: o.stratum()
H_2(2)
sage: o.veech_group().index()
9
sage: o.sum_of_lyapunov_exponents()
4/3

sage: o = origamis.ProjectiveLine(5); o
Projective line origami on GF(5)
 r = [1 1]    u = [0 1]
     [0 1]        [1 0]
sage: o.stratum()
H_3(2^2)
sage: o.veech_group().index()
9
sage: o.sum_of_lyapunov_exponents()
5/3

sage: o = origamis.ProjectiveLine(7); o
Projective line origami on GF(7)
 r = [1 1]    u = [0 1]
     [0 1]        [1 0]
sage: o.stratum()
H_3(2^2)
sage: o.veech_group().index()
45
sage: o.sum_of_lyapunov_exponents()
5/3

Any invertible matrix mod p can be chosen to generate the origami:

sage: r = matrix([[1,3],[0,1]])
sage: u = matrix([[1,0],[3,1]])
sage: o = origamis.ProjectiveLine(5,r,u); o
Projective line origami on GF(5)
 r = [1 3]    u = [1 0]
     [0 1]        [3 1]
sage: o.veech_group().index()
10
sage: o.sum_of_lyapunov_exponents()
9/5
sage: o.stratum_component()
H_3(4)^hyp
ShresthaWang()#

An origami with Veech group SL(2,Z) that is not normal.

This is the second example page 9 of https://arxiv.org/abs/1902.08214

EXAMPLES:

sage: from surface_dynamics import origamis
sage: o = origamis.ShresthaWang()
sage: o.nb_squares()
16
sage: o
(1,2,3,4,5,6)(7,8,9,10,11,12)(13,14)(15,16)
(1,8,5,15,13,12)(2,16,14,9,6,7)(3,10)(4,11)
sage: o.stratum_component()
H_4(2^3)^odd
sage: o.veech_group().index()
1
sage: o.automorphism_group().cardinality()
1

And this example is not a cyclic cover:

sage: o.is_isomorphic(origamis.CyclicCover([1,1,1,5]))
False
static Stair(n)#

Stair origamis

The stair origami with n=2k squares is defined by the permutations

r = (1 2)(3 4) … (n-1 n) u = (1)(2 3) … (n-2 n-1)(n)

The stair origamis with n=2k+1 squares is defined by the permutations

r = (1 2)(3 4) … (n-2 n-1)(n) u = (1)(2 3) … (n-1 n-2)

EXAMPLES:

sage: from surface_dynamics import *

sage: o = origamis.Stair(4)
sage: o
Stair origami with 4 squares
sage: print(str(o))
(1,2)(3,4)
(1)(2,3)(4)
sage: o = origamis.Stair(5)
sage: o
Stair origami with 5 squares
sage: print(str(o))
(1,2)(3,4)(5)
(1)(2,3)(4,5)
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