origami_database

origami_database#

Database of reduced origamis.

Origamis are cover of the torus over one point. The group SL(2,ZZ) act on them (as their mapping class group). This file contains the implementation of a database of these SL(2,Z)-orbits. To do concrete computations with origamis see ?.

The database of origamis contain information about SL(2,ZZ) orbits of the origamis that are arithmetic Teichmueller curves.

EXAMPLES:

sage: from surface_dynamics import *

The most useful way to explore the database through queries. We develop several examples below and refer to the documentation in the method OrigamiDatabase.query() for the complete specifications.

Let us start by finding the two exceptional surfaces whose Teichmueller curves have a complete degenerate Lyapunov spectrum: the Eierlegende Wollmilchsau and the Ornythorinque:

sage: D = OrigamiDatabase()

sage: q = D.query(sum_of_L_exp=1)
sage: q.number_of()
2
sage: o0, o1 = q.list()
sage: o0
(1,2,3,4)(5,6,7,8)
(1,5,3,7)(2,8,4,6)
sage: o1
(1,2,3,4,5,6)(7,8,9,10,11,12)
(1,7,5,9,3,11)(2,8,4,12,6,10)
sage: o0.is_isomorphic(origamis.CyclicCover([1,1,1,1]))
True
sage: o1.is_isomorphic(origamis.CyclicCover([1,1,1,3]))
True

We show two methods to get information on a given query: number_of() and list(). To simply display the result on the screen in a table, one can use show() of OrigamiQuery:

sage: q.show()
Origami
--------------------------------
r=1234 5678  u=1537 2846
r=123456 789abc  u=17593b 284c6a

We can display much more information by modifying the displayed columns:

sage: q.cols('nb_squares', 'stratum', 'component', 'regular', 'quasi_regular')
sage: q.show()
Nb squares  Stratum   Comp.  Regular   Quasi regular
----------------------------------------------------
8           H_3(1^4)  c      True      True
12          H_4(2^3)  even   False     True

And the complete list of columns in the database is obtained through:

sage: D.cols()
['representative',
 'stratum',
 'component',
 'primitive',
 'quasi_primitive',
 'orientation_cover',
 'hyperelliptic',
 'regular',
 'quasi_regular',
...
 'relative_monodromy_nilpotent',
 'relative_monodromy_gap_primitive_id',
 'orientation_stratum',
 'orientation_genus',
 'pole_partition',
 'automorphism_group_order',
 'automorphism_group_name']

Now, we do some simple counting of the number of primitive arithmetic Teichmueller curves. Let us first check the classification of arithmetic Teichmueller curves in H(2) from Hubert-Lelievre and McMullen:

sage: A = Stratum([2], k=1)
sage: for n in range(3, 17):
....:     q = D.query(stratum=A, nb_squares=n)
....:     print("%2d %d"%(n, q.number_of()))
 3 1
 4 1
 5 2
 6 1
 7 2
 8 1
 9 2
10 1
11 2
12 1
13 2
14 1
15 2
16 1

And look at the conjecture of Delecroix-Lelievre in the stratum H(1,1):

sage: A = Stratum([1,1], k=1)
sage: for n in range(4,20):
....:     q = D.query(stratum=A, nb_squares=n, primitive=True)
....:     print("%2d %d"%(n, q.number_of()))
 4 1
 5 1
 6 2
 7 2
 8 2
 9 2
10 2
11 2
12 2
13 2
14 2
15 2
16 2
17 2
18 2
19 2

You can get an overview of the content of the database:

sage: D.info()
genus 2
=======
 H_2(2)^hyp        : 205 T. curves (up to 139 squares)
 H_2(1^2)^hyp      : 452 T. curves (up to 80 squares)
<BLANKLINE>
genus 3
=======
 H_3(4)^hyp        : 163 T. curves (up to 51 squares)
 H_3(4)^odd        : 118 T. curves (up to 41 squares)
 ...
genus 6
=======
 H_6(10)^hyp       :  46 T. curves (up to 15 squares)
 H_6(10)^odd       :   4 T. curves (up to 11 squares)
 H_6(10)^even      :  33 T. curves (up to 12 squares)
<BLANKLINE>
<BLANKLINE>
Total: 4688 Teichmueller curves

Here is a last example of the list of regular origamis (i.e. such that their group of translation acts transitively on the set of squares):

sage: q = D.query(regular=True)
sage: q.cols('nb_squares', 'stratum', 'automorphism_group_name')
sage: q.show()
Nb squares  Stratum   Automorphism
----------------------------------
6           H_3(2^2)  S3
8           H_3(1^4)  D8
8           H_3(1^4)  Q8
10          H_5(4^2)  D10
12          H_4(1^6)  A4

AUTHOR:

  • Vincent Delecroix (2011-2014): initial version

class surface_dynamics.flat_surfaces.origamis.origami_database.EnhancedSQLQuery(query_string, database=None)#

A query for a OrigamiDatabase.

INPUT:

  • database - the OrigamiDatabase instance to query (if None then a new instance is created)

  • query_string - a string representing the SQL query

surface_dynamics.flat_surfaces.origamis.origami_database.L_exp_approx_to_data(t)#

Convert a tuple of real numbers with same precision into a string.

The output string is a list of numbers written in base 36 (0, 1, …, 9, a, b, …, z) separated by space ‘ ‘. The first number is the precision of the real field. Then each real number consists of three numbers as sign, mantissa, exponent.

EXAMPLES:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: odb.real_tuple_to_data((1.23,-4.0))
'1h 1 1ijk9vqiyzy -1g -1 18ce53un18g -1e'

We may check that the first part consists of the precision:

sage: Integer('1h', 36)
53
sage: RR.precision()
53

And then of the two real numbers we input:

sage: sign = Integer('1', 36)
sage: mantissa = Integer('1ijk9vqiyzy', 36)
sage: exponent = Integer('-1g', 36)
sage: RR(sign * mantissa * 2 ** exponent)
1.23000000000000

sage: sign = Integer('-1', 36)
sage: mantissa = Integer('18ce53un18g', 36)
sage: exponent = Integer('-1e', 36)
sage: RR(sign * mantissa * 2 ** exponent)
-4.00000000000000

TESTS:

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: t = (RR(5.0), RR(pi))
sage: t
(5.00000000000000, 3.14159265358979)
sage: s = odb.real_tuple_to_data(t)
sage: isinstance(s,str)
True
sage: t == odb.data_to_real_tuple(s)
True
class surface_dynamics.flat_surfaces.origamis.origami_database.OrigamiDatabase(dblocation=None, read_only=True, force_creation=False, old_version=False)#

Database of arithmetic Teichmueller curves.

EXAMPLES:

sage: from surface_dynamics import *

To query the database the main method is meth:query:

sage: D = OrigamiDatabase()
sage: q = D.query(genus=3, nb_squares=12)
sage: q.number_of()
146
sage: l = q.list()
sage: o = l[0]
sage: o.genus()
3
sage: o.nb_squares()
12

For the precise usage of the method query, see the documentation of that function. Here is an example to show how to found the Eierlegende Wollmilchsau in the database from its property of complete degenerate spectrum:

sage: q = D.query(sum_of_L_exp=1, stratum=Stratum([1,1,1,1],k=1))
sage: len(q)
1
sage: o = q.list()[0]
sage: o.is_isomorphic(origamis.EierlegendeWollmilchsau())
True

We check the classification of arithmetic Teichmueller curves in H(2) from Hubert-Lelievre and McMullen:

sage: A = Stratum([2], k=1)
sage: for n in range(3, 15):
....:     q = D.query(stratum=A, nb_squares=n)
....:     print("%2d %d"%(n, q.number_of()))
 3 1
 4 1
 5 2
 6 1
 7 2
 8 1
 9 2
10 1
11 2
12 1
13 2
14 1

To know all entries of the database look at meth:cols, meth:info or meth:help. The latter also provides a summary description of the columns.

build(comp, N, force_computation=False, verbose=False)#

Update the database for the given component comp up to N squares.

Note that in order to work the database should not be in read only mode.

INPUT:

  • comp - stratum or component of stratum

  • N - integer

  • force_computation - force computation of data which may be yet in the database.

  • verbose - boolean - if True, print useful interactive information during the process.

EXAMPLES:

sage: from surface_dynamics import *

sage: import os
sage: import tempfile
sage: with tempfile.TemporaryDirectory() as tmpdir:
....:     db_name = os.path.join(tmpdir, 'my_db.db')
....:     D = OrigamiDatabase(db_name, read_only=False)
....:     D.build(Stratum([4],k=1).odd_component(), 8)  # optional: gap_packages
....:     D.info()                                       # optional: gap_packages
genus 2
=======
<BLANKLINE>
genus 3
=======
 H_3(4)^odd   :  8 T. curves (up to  7 squares)
<BLANKLINE>
<BLANKLINE>
 Total: 8 Teichmueller curves
cols()#

Returns the skeleton of self (which is the list of possible entries).

EXAMPLES:

sage: from surface_dynamics import *

sage: O = OrigamiDatabase()
sage: cols = O.cols()
sage: "representative" in cols
True
sage: len(cols)
45
help(cols=None)#

Print some help relative to the columns of the database.

If cols is provided then gives help only for these columns.

info(genus=None, dimension=None, print_all=False)#

Print the list of connected components and the number of squares up to which the database is filled.

INPUT:

  • genus - integer (default: None) - if not None, print only info for that genus.

  • dimension - Integer (default: None) - if not None, print only info for that dimension.

  • print_all - boolean (default: False) - print also the components for which nothing has been computed yet.

EXAMPLES:

sage: from surface_dynamics import *

sage: O = OrigamiDatabase()
sage: O.info()
genus 2
=======
...
max_nb_squares(comp=None)#

Returns the maximum number of squares for which Teichmueller curves have been computed.

If comp is None (default), then returns the biggest integer for which the database contains all data up to that integer. If comp is a stratum or a component of stratum, then returns the maximum number of squares computed for that stratum.

EXAMPLES:

sage: from surface_dynamics import *

sage: O = OrigamiDatabase()
sage: O.max_nb_squares(Stratum([2],k=1))
139
sage: O.max_nb_squares()
11
query(*query_list, **kwds)#

From a list of restriction, returns a list of possible entry in the database. By default, returns only the found origamis.

Where to find the possible entries self.cols() then a sign among ‘=’ (equality), ‘<>’ (difference), ‘<’, ‘>’, ‘<=’, ‘>=’ (comparisons).

EXAMPLES:

sage: from surface_dynamics import *

sage: D = OrigamiDatabase()
sage: for o in D.query(stratum=Stratum([1,1], k=1), nb_squares=6):
....:     print("%s\n---------------" % o)
(1)(2)(3,4,5,6)
(1,2,3)(4,5,6)
---------------
(1)(2)(3)(4,5,6)
(1,2,3,4)(5,6)
---------------
(1)(2)(3,4)(5)(6)
(1,2,3)(4,5,6)
---------------
(1)(2)(3)(4,5)(6)
(1,2,3,4)(5,6)
---------------
(1)(2,3)(4,5)(6)
(1,2,4)(3,5,6)
---------------
rebuild(q=None, local_data=False, lyapunov_exponents=False, global_data=False, nb_iterations=4096, nb_experiments=5, verbose=False)#

Rebuild some of the data for the origami in the query q.

INPUT:

  • q - a query

  • local_data - boolean - whether or not we rebuild local data.

  • lyapunov_exponents - boolean - whether or not rebuild lyapunov exponents approximation (time consuming).

  • global_data - boolean - whether or not rebuild global data (time and memory consuming).

  • nb_experiments, nb_iterations - integers - option for the computation of Lyapunov exponents.

  • verbose - boolean - if True displays nice information in real time.

update(other, replace=False, verbose=False)#

Update the content of this database with the content of another one.

INPUT:

  • other - a query, an origami database or a path to an origami database

  • replace - boolean - whether or not replace entries which are yet in the database.

  • verbose - boolean - if True, displays information during the transfer.

EXAMPLES:

sage: from surface_dynamics import *

sage: import os
sage: import tempfile
sage: with tempfile.TemporaryDirectory() as tmpdir:
....:     db1_name = os.path.join(tmpdir, 'the_first_one.db')
....:     db2_name = os.path.join(tmpdir, 'the_second_one.db')
....:     D1 = OrigamiDatabase(db1_name, read_only=False)
....:     D2 = OrigamiDatabase(db2_name, read_only=False)
....:     D1.build(Stratum([1,1],k=1).unique_component(), 7)  # optional: gap_packages
....:     D2.build(Stratum([2],k=1).unique_component(), 5)    # optional: gap_packages
....:     D2.update(D1)                                        # optional: gap_packages
....:     D2.info()                                            # optional: gap_packages
genus 2
=======
 H_2(2)^hyp  :   2 T. curves (up to  4 squares)
 H_2(1^2)^hyp:   8 T. curves (up to  6 squares)
<BLANKLINE>
<BLANKLINE>
  Total: 10 Teichmueller curves
class surface_dynamics.flat_surfaces.origamis.origami_database.OrigamiQuery(db, query_string, cols, **kwds)#

Origami database query.

A query for an instance of OrigamiDatabase. This class nicely wraps the SQLQuery class located in surface_dynamics.databases.database.py to make the query constraints intuitive and with as many pre-definitions as possible. (i.e.: since it has to be a OrigamiDatabase, we already know the table structure and types; and since it is immutable, we can treat these as a guarantee).

cols(*cols)#

Get or modify columns.

In a sql query, it corresponds to the clause ‘SELECT’.

EXAMPLES:

sage: from surface_dynamics import *

sage: O = OrigamiDatabase()
sage: q = O.query()
sage: q.cols()
['representative']
sage: q.cols("nb_squares")
sage: q.cols()
['nb_squares']
sage: q.cols("veech_group_index","primitive")
sage: q.cols()
['veech_group_index', 'primitive']
database()#

Returns the database of that query.

EXAMPLES:

sage: from surface_dynamics import *

sage: D = OrigamiDatabase()
sage: q = D.query(stratum=Stratum([6], k=1))
sage: q.database()
Database of origamis
sage: q.database() is D
True
dict()#

Returns a list of dictionaries: col -> value.

EXAMPLES:

sage: from surface_dynamics import *

sage: D = OrigamiDatabase()
sage: q = D.query(stratum=Stratum([1,1],k=1), nb_squares=8)
sage: q.cols('teich_curve_genus')
sage: q.dict()
[{'teich_curve_genus': 1},
 {'teich_curve_genus': 1},
 {'teich_curve_genus': 0},
 {'teich_curve_genus': 0}]

sage: q.cols('pole_partition', 'primitive')
sage: q.dict()
[{'pole_partition': (0, 2, 2, 2), 'primitive': True},
 {'pole_partition': (2, 0, 2, 2), 'primitive': True},
 {'pole_partition': (0, 0, 2, 4), 'primitive': False},
 {'pole_partition': (0, 0, 2, 4), 'primitive': False}]
get_query_string()#

Output the query string in sql format.

This is the method where the attribute of this object are translated into a sql query.

EXAMPLES:

sage: from surface_dynamics import *

sage: D = OrigamiDatabase()
sage: q = D.query(('stratum','=',Stratum([1,1], k=1)), ('nb_squares','<', 13))
sage: q.get_query_string()
"SELECT representative FROM origamis WHERE stratum='1 1' AND nb_squares<13 ORDER BY nb_squares ASC"
sage: q.order(("nb_squares",1),("pole_partition",-1))
sage: q.get_query_string()
"SELECT representative FROM origamis WHERE stratum='1 1' AND nb_squares<13 ORDER BY nb_squares ASC,pole_partition DESC"
list()#

Returns the list of entries of the query.

The output is either a list of objects if there is only one column for that query. Otherwise, it is a list of lists where each item is the entries of columns.

See also meth:dict to get a dictionary output.

EXAMPLES:

sage: from surface_dynamics import *

sage: S = OrigamiDatabase(read_only=False)
sage: q = S.query(('stratum','=',Stratum([1,1],k=1)),('nb_squares','=',6))
sage: q.list()
[(1)(2)(3,4,5,6)
(1,2,3)(4,5,6), (1)(2)(3)(4,5,6)
(1,2,3,4)(5,6), (1)(2)(3,4)(5)(6)
(1,2,3)(4,5,6), (1)(2)(3)(4,5)(6)
(1,2,3,4)(5,6), (1)(2,3)(4,5)(6)
(1,2,4)(3,5,6)]
number_of()#

Returns the number of entries in the database that satisfy the query.

order(*args)#

Get or modify order.

Order should be a list (col_name, +1) or (col_name, -1). First one means ascending and the second one descending. In a sql query, it corresponds to the clause ‘ORDER BY’.

EXAMPLES:

sage: from surface_dynamics import *

sage: O = OrigamiDatabase()
sage: q = O.query()
sage: q.order(("nb_squares",1),("pole_partition",1))
sage: q.get_query_string()
'SELECT representative FROM origamis ORDER BY nb_squares ASC,pole_partition ASC'
sage: q.order(("nb_squares",-1))
sage: q.get_query_string()
'SELECT representative FROM origamis ORDER BY nb_squares DESC'
show(**opts)#

Output a text array with the results of that query.

EXAMPLES:

sage: from surface_dynamics import *

There is a problem with show method the SQLQuery:

sage: O = OrigamiDatabase()
sage: q = O.query(("nb_squares","=",6))
sage: q.cols(("stratum","sum_of_L_exp","L_exp_approx"))
sage: q.show()
Stratum    Sum of L exp  L exp approx
---------------------------------------------------------------
...
sql_query()#

Returns the SQLQuery.

surface_dynamics.flat_surfaces.origamis.origami_database.are_skeleton_equal(sk1, sk2)#

Test whether the two skeleton sk1 and sk2 are equals.

EXAMPLES:

sage: from surface_dynamics.flat_surfaces.origamis.origami_database import are_skeleton_equal
sage: sk1 = {'table1': {'col1': {'sql': 'TEXT', 'unique': True}, 'col2': {'sql': 'INTEGER'}}}
sage: sk2 = {'table1': {'col1': {'sql': 'TEXT', 'unique': True}, 'col2': {'sql': 'INTEGER', 'unique': False}}}
sage: are_skeleton_equal(sk1,sk2)
True
surface_dynamics.flat_surfaces.origamis.origami_database.build_global_data(o, c=None)#

Compute the global data that are obtained from the Teichmueller curve of the origami o. If the Teichmueller curve c is not provided, then it is recomputed from scratch (and may be long).

EXAMPLES:

sage: from surface_dynamics import *

sage: o = Origami('(1,2)', '(1,3)')
sage: from surface_dynamics.flat_surfaces.origamis.origami_database import build_global_data
sage: data = build_global_data(o)
sage: for item in sorted(data): print("%-25s %s" % (item, data[item]))
max_hom_dim               2
max_nb_of_cyls            2
min_hom_dim               1
min_nb_of_cyls            1
minus_identity_invariant  True
sum_of_L_exp              4/3
teich_curve_genus         0
teich_curve_ncusps        2
teich_curve_nu2           1
teich_curve_nu3           0
veech_group_congruence    True
veech_group_index         3
veech_group_level         2
surface_dynamics.flat_surfaces.origamis.origami_database.build_local_data(o)#

Build local data for the origami o.

The output is a dictionary that is intended to be used to feed the database of origamis. The local data are geometrical aspects (stratum, genus, …) the monodromy group (primitivity, orientation cover, …) and the automorphisms of o.

See also build_lyapunov_exponents() to construct Lyapunov exponents and build_global_data().

EXAMPLES:

sage: from surface_dynamics import *

sage: o = Origami('(1,2)','(1,3)')
sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: data = odb.build_local_data(o)  # optional: gap_packages
sage: data['stratum']                 # optional: gap_packages
H_2(2)
sage: data['nb_squares']              # optional: gap_packages
3
surface_dynamics.flat_surfaces.origamis.origami_database.build_lyapunov_exponents(o, nb_iterations=65536, nb_experiments=10)#

Compute the lyapunov exponents for the origami o and update the database.

EXAMPLES:

sage: from surface_dynamics import *

sage: o = Origami('(1,2)', '(1,3)')
sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: data = odb.build_lyapunov_exponents(o)
sage: data       # abs tol .1
{'L_exp_approx': [0.333348091]}
surface_dynamics.flat_surfaces.origamis.origami_database.data_to_L_exp_approx(s)#

Convert a string into a tuple of real numbers.

For the encoding convention, see meth:real_tuple_to_data.

EXAMPLES:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: R = RealField(22)
sage: t = (R(1), R(pi))
sage: s = odb.real_tuple_to_data(t)
sage: tt = odb.data_to_real_tuple(s)
sage: tt == t
True
sage: tt[0].parent()
Real Field with 22 bits of precision
sage: tt[1].parent()
Real Field with 22 bits of precision
surface_dynamics.flat_surfaces.origamis.origami_database.data_to_integer_tuple(s)#

Convert a string into a tuple of integers.

For the encoding convention, see meth:integer_tuple_to_data.

EXAMPLES:

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: t = (-12, 351234123, 45)
sage: s = odb.integer_tuple_to_data(t)
sage: odb.data_to_integer_tuple(s) == t
True
surface_dynamics.flat_surfaces.origamis.origami_database.data_to_nb_cyls_spectrum(s)#

Convert a string into a tuple of integers.

For the encoding convention, see meth:integer_tuple_to_data.

EXAMPLES:

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: t = (-12, 351234123, 45)
sage: s = odb.integer_tuple_to_data(t)
sage: odb.data_to_integer_tuple(s) == t
True
surface_dynamics.flat_surfaces.origamis.origami_database.data_to_orientation_stratum(s)#
surface_dynamics.flat_surfaces.origamis.origami_database.data_to_pole_partition(s)#

TESTS:

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: odb.data_to_pole_partition('0aFe')
(0, 10, 15, 14)
sage: odb.data_to_pole_partition('') is None
True
surface_dynamics.flat_surfaces.origamis.origami_database.data_to_real_tuple(s)#

Convert a string into a tuple of real numbers.

For the encoding convention, see meth:real_tuple_to_data.

EXAMPLES:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: R = RealField(22)
sage: t = (R(1), R(pi))
sage: s = odb.real_tuple_to_data(t)
sage: tt = odb.data_to_real_tuple(s)
sage: tt == t
True
sage: tt[0].parent()
Real Field with 22 bits of precision
sage: tt[1].parent()
Real Field with 22 bits of precision
surface_dynamics.flat_surfaces.origamis.origami_database.data_to_representative(s)#

Convert data to representative.

For encoding convention, see meth:representative_to_data.

surface_dynamics.flat_surfaces.origamis.origami_database.data_to_small_positive_integer_tuple(s)#

Convert a string into a tuple of Integer.

For encoding convention, see meth:small_positive_integer_tuple_to_data.

EXAMPLES:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: odb.data_to_small_positive_integer_tuple('14m')
(1, 4, 22)
sage: odb.data_to_small_positive_integer_tuple('')
()
surface_dynamics.flat_surfaces.origamis.origami_database.data_to_stratum(s)#

Convert a string into a stratum.

For encoding convention, see meth:stratum_to_data.

EXAMPLES:

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: odb.data_to_stratum('f f 2')
H_17(15^2, 2)
surface_dynamics.flat_surfaces.origamis.origami_database.format_pole_partition(p)#

Format into a nice readable string the pole partition.

EXAMPLES:

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: s = odb.pole_partition_to_data((0,3,5,1))
sage: odb.format_pole_partition(s)
'0 (3,5,1)'
surface_dynamics.flat_surfaces.origamis.origami_database.format_representative(s)#

Convert a string that encodes an origami into a human readable string.

The output form consists of the concatenation of the cycles without the parenthesis. In other words the permutation (1,5)(2)(3,7,4) will be convert into ‘15 374’.

EXAMPLES:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: o = Origami('(1,2)','(1,3)')
sage: s = odb.representative_to_data(o)
sage: odb.format_representative(s)
'r=12  u=13'
surface_dynamics.flat_surfaces.origamis.origami_database.integer_tuple_to_data(t)#

Convert a tuple of arbitrary integers into a string.

The encoding consists in a string that are representation of the integers in base 36 separated by space.

EXAMPLES:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: t = (0,-12435123,3)
sage: s = odb.integer_tuple_to_data(t)
sage: s
'0 -7ej03 3'
sage: list(map(lambda x: Integer(x,36), s.split(' ')))
[0, -12435123, 3]
surface_dynamics.flat_surfaces.origamis.origami_database.nb_cyls_spectrum_to_data(t)#

Convert a tuple of arbitrary integers into a string.

The encoding consists in a string that are representation of the integers in base 36 separated by space.

EXAMPLES:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: t = (0,-12435123,3)
sage: s = odb.integer_tuple_to_data(t)
sage: s
'0 -7ej03 3'
sage: list(map(lambda x: Integer(x,36), s.split(' ')))
[0, -12435123, 3]
surface_dynamics.flat_surfaces.origamis.origami_database.orientation_stratum_to_data(q)#

TESTS:

sage: from surface_dynamics import *
sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: q = Stratum([2,2,0,-1,-1,-1,-1], k=2)
sage: s = odb.orientation_stratum_to_data(q)
sage: isinstance(s,str)
True
sage: q == odb.data_to_orientation_stratum(s)
True
surface_dynamics.flat_surfaces.origamis.origami_database.pole_partition_to_data(t)#

Convert a tuple of integers between 0 and 35 to a string.

The encoding consists of a string of the same length as t where each character is the representation of the number in base 36 (0, 1, …, 9, a, b, …, z).

EXAMPLES:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: t = (0, 35, 25, 2, 12)
sage: s = odb.small_positive_integer_tuple_to_data(t)
sage: s
'0zp2c'
sage: list(map(lambda x: Integer(x, 36), s))
[0, 35, 25, 2, 12]
surface_dynamics.flat_surfaces.origamis.origami_database.real_tuple_to_data(t)#

Convert a tuple of real numbers with same precision into a string.

The output string is a list of numbers written in base 36 (0, 1, …, 9, a, b, …, z) separated by space ‘ ‘. The first number is the precision of the real field. Then each real number consists of three numbers as sign, mantissa, exponent.

EXAMPLES:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: odb.real_tuple_to_data((1.23,-4.0))
'1h 1 1ijk9vqiyzy -1g -1 18ce53un18g -1e'

We may check that the first part consists of the precision:

sage: Integer('1h', 36)
53
sage: RR.precision()
53

And then of the two real numbers we input:

sage: sign = Integer('1', 36)
sage: mantissa = Integer('1ijk9vqiyzy', 36)
sage: exponent = Integer('-1g', 36)
sage: RR(sign * mantissa * 2 ** exponent)
1.23000000000000

sage: sign = Integer('-1', 36)
sage: mantissa = Integer('18ce53un18g', 36)
sage: exponent = Integer('-1e', 36)
sage: RR(sign * mantissa * 2 ** exponent)
-4.00000000000000

TESTS:

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: t = (RR(5.0), RR(pi))
sage: t
(5.00000000000000, 3.14159265358979)
sage: s = odb.real_tuple_to_data(t)
sage: isinstance(s,str)
True
sage: t == odb.data_to_real_tuple(s)
True
surface_dynamics.flat_surfaces.origamis.origami_database.representative_to_data(o)#

The maximum number of squares is 95. The encoding consists of the concatenation of the two permutations that define the origami.

TESTS:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: o = Origami('(1,2,3)','(3,2,1)')
sage: odb.representative_to_data(o)
'120201'
sage: o = Origami('(1,2,4,3)(5,7,6)(10,9)','(10,8,6,2,4,3,1)')
sage: s = odb.representative_to_data(o)
sage: isinstance(s,str)
True
sage: o == odb.data_to_representative(s)
True

Try examples in the critical range 75-85:

sage: for n in range(75,85): ….: r = list(range(1,n)) + [0] ….: u = list(range(n)) ….: shuffle(u) ….: o = Origami(r, u, as_tuple=True) ….: s = odb.representative_to_data(o) ….: assert o == odb.data_to_representative(s)

surface_dynamics.flat_surfaces.origamis.origami_database.small_positive_integer_tuple_to_data(t)#

Convert a tuple of integers between 0 and 35 to a string.

The encoding consists of a string of the same length as t where each character is the representation of the number in base 36 (0, 1, …, 9, a, b, …, z).

EXAMPLES:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: t = (0, 35, 25, 2, 12)
sage: s = odb.small_positive_integer_tuple_to_data(t)
sage: s
'0zp2c'
sage: list(map(lambda x: Integer(x, 36), s))
[0, 35, 25, 2, 12]
surface_dynamics.flat_surfaces.origamis.origami_database.square_num_to_str(i)#
surface_dynamics.flat_surfaces.origamis.origami_database.stratum_to_data(h)#

Encode the stratum entry.

TESTS:

sage: from surface_dynamics import *

sage: import surface_dynamics.flat_surfaces.origamis.origami_database as odb
sage: h = Stratum([2,0], k=1)
sage: s = odb.stratum_to_data(h)
sage: isinstance(s,str)
True
sage: h == odb.data_to_stratum(s)
True
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