Interface to polymake#

polymake (https://polymake.org) is a mature open source package for research in polyhedral geometry and related fields, developed since 1997 by Ewgenij Gawrilow and Michael Joswig and various contributors.

polymake has been described in [GJ1997], [GJ2006], [JMP2009], [GJRW2010], [GHJ2016], and [AGHJLPR2017].

sage.interfaces.polymake.Polymake#: alias of PolymakeJuPyMake

class sage.interfaces.polymake.PolymakeAbstract(seed=None)#

Bases: ExtraTabCompletion, Interface

Abstract interface to the polymake interpreter.

This class should not be instantiated directly, but through its subclasses Polymake (Pexpect interface) or PolymakeJuPyMake (JuPyMake interface).

EXAMPLES:

sage: from sage.interfaces.polymake import PolymakeAbstract, polymake_jupymake

We test the verbosity management with very early doctests because messages will not be repeated.

Testing the JuPyMake interface:

sage: isinstance(polymake_jupymake, PolymakeAbstract)
True
sage: p = polymake_jupymake.rand_sphere(4, 20, seed=5)       # optional - jupymake
sage: p                                             # optional - jupymake
Random spherical polytope of dimension 4; seed=5...
sage: set_verbose(3)
sage: p.H_VECTOR                                    # optional - jupymake
polymake: used package ppl
  The Parma Polyhedra Library ...
1 16 40 16 1
sage: set_verbose(0)
sage: p.F_VECTOR                                    # optional - jupymake
20 94 148 74

application(app)#

Change to a given polymake application.

INPUT:

app, a string, one of “common”, “fulton”, “group”, “matroid”, “topaz”, “fan”, “graph”, “ideal”, “polytope”, “tropical”

EXAMPLES:

We expose a computation that uses both the ‘polytope’ and the ‘fan’ application of polymake. Let us start by defining a polytope \(q\) in terms of inequalities. Polymake knows to compute the f- and h-vector and finds that the polytope is very ample:

sage: # optional - jupymake
sage: q = polymake.new_object("Polytope", INEQUALITIES=[[5,-4,0,1],[-3,0,-4,1],[-2,1,0,0],[-4,4,4,-1],[0,0,1,0],[8,0,0,-1],[1,0,-1,0],[3,-1,0,0]])
sage: q.H_VECTOR
1 5 5 1
sage: q.F_VECTOR
8 14 8
sage: q.VERY_AMPLE
true

In the application ‘fan’, polymake can now compute the normal fan of \(q\) and its (primitive) rays:

sage: # optional - jupymake
sage: polymake.application('fan')
sage: g = q.normal_fan()
sage: g.RAYS
-1 0 1/4
0 -1 1/4
1 0 0
1 1 -1/4
0 1 0
0 0 -1
0 -1 0
-1 0 0
sage: g.RAYS.primitive()
-4 0 1
0 -4 1
1 0 0
4 4 -1
0 1 0
0 0 -1
0 -1 0
-1 0 0

Note that the list of functions available by tab completion depends on the application.

clear(var)#: Clear the variable named var.

Note

This is implicitly done when deleting an element in the interface.

console()#

Raise an error, pointing to interact() and polymake_console().

EXAMPLES:

sage: polymake.console()
Traceback (most recent call last):
...
NotImplementedError: Please use polymake_console() function or the .interact() method

function_call(function, args=None, kwds=None)#

EXAMPLES:

sage: polymake.rand_sphere(4, 30, seed=15)           # optional - jupymake  # indirect doctest
Random spherical polytope of dimension 4; seed=15...

get(cmd)#

Return the string representation of an object in the polymake interface.

EXAMPLES:

sage: polymake.get('cube(3)')                     # optional - jupymake
'Polymake::polytope::Polytope__Rational=ARRAY(...)'

Note that the above string representation is what polymake provides. In our interface, we use what polymake calls a “description”:

sage: polymake('cube(3)')                         # optional - jupymake
cube of dimension 3

help(topic, pager=True)#

Displays polymake’s help on a given topic, as a string.

INPUT:

topic, a string
pager, optional bool, default True: When True, display help, otherwise return as a string.

EXAMPLES:

sage: print(polymake.help('Polytope', pager=False))         # optional - jupymake # random
objects/Polytope:
 Not necessarily bounded or unbounded polyhedron.
 Nonetheless, the name "Polytope" is used for two reasons:
 Firstly, combinatorially we always deal with polytopes; see the description of VERTICES_IN_FACETS for details.
 The second reason is historical.
 We use homogeneous coordinates, which is why Polytope is derived from Cone.
 Note that a pointed polyhedron is projectively equivalent to a polytope.
 Scalar is the numeric data type used for the coordinates.

In some cases, polymake expects user interaction to choose from different available help topics. In these cases, a warning is given, and the available help topics are displayed resp. printed, without user interaction:

sage: polymake.help('TRIANGULATION')                        # optional - jupymake # random
doctest:warning
...
UserWarning: Polymake expects user interaction. We abort and return the options that Polymake provides.
There are 5 help topics matching 'TRIANGULATION':
1: objects/Visualization/Visual::Polytope/methods/TRIANGULATION
2: objects/Visualization/Visual::PointConfiguration/methods/TRIANGULATION
3: objects/Cone/properties/Triangulation and volume/TRIANGULATION
4: objects/PointConfiguration/properties/Triangulation and volume/TRIANGULATION
5: objects/Polytope/properties/Triangulation and volume/TRIANGULATION

If an unknown help topic is requested, a PolymakeError results:

sage: polymake.help('Triangulation')      # optional - jupymake
Traceback (most recent call last):
...
PolymakeError: unknown help topic 'Triangulation'

new_object(name, *args, **kwds)#

Return a new instance of a given polymake type, with given positional or named arguments.

INPUT:

name of a polymake class (potentially templated), as string.
further positional or named arguments, to be passed to the constructor.

EXAMPLES:

sage: # optional - jupymake
sage: q = polymake.new_object("Polytope<Rational>", INEQUALITIES=[[4,-4,0,1],[-4,0,-4,1],[-2,1,0,0],[-4,4,4,-1],[0,0,1,0],[8,0,0,-1]])
sage: q.N_VERTICES
4
sage: q.BOUNDED
true
sage: q.VERTICES
1 2 0 4
1 3 0 8
1 2 1 8
1 3 1 8
sage: q.full_typename()
'Polytope<Rational>'

set(var, value)#

Set the variable var to the given value.

Eventually, var is a reference to value.

Warning

This method, although it doesn’t start with an underscore, is an internal method and not part of the interface. So, please do not try to call it explicitly. Instead, use the polymake interface as shown in the examples.

REMARK:

Polymake’s user language is Perl. In Perl, if one wants to assign the return value of a function to a variable, the syntax to do so depends on the type of the return value. While this is fine in compiled code, it seems quite awkward in user interaction.

To make this polymake pexpect interface a bit more user friendly, we treat all variables as arrays. A scalar value (most typically a reference) is thus interpreted as the only item in an array of length one. It is, of course, possible to use the interface without knowing these details.

EXAMPLES:

sage: c = polymake('cube(3)')                       # optional - jupymake # indirect doctest
sage: d = polymake.cube(3)                          # optional - jupymake

Equality is, for “big” objects such as polytopes, comparison by identity:

sage: c == d                                        # optional - jupymake
False

However, the list of vertices is equal:

sage: c.VERTICES == d.VERTICES                      # optional - jupymake
True

version()#

Version of the polymake installation.

EXAMPLES:

sage: polymake.version()               # optional - jupymake # random
'4...'

class sage.interfaces.polymake.PolymakeElement(parent, value, is_name=False, name=None)#

Bases: ExtraTabCompletion, InterfaceElement

Elements in the polymake interface.

EXAMPLES:

We support all “big” polymake types, Perl arrays of length different from one, and Perl scalars:

sage: p = polymake.rand_sphere(4, 20, seed=5)            # optional - jupymake
sage: p.typename()                                      # optional - jupymake
'Polytope'
sage: p                                                 # optional - jupymake
Random spherical polytope of dimension 4; seed=5...

Now, one can work with that element in Python syntax, for example:

sage: p.VERTICES[2][2]                                  # optional - jupymake
1450479926727001/2251799813685248

full_typename()#

The name of the specialised type of this element.

EXAMPLES:

sage: c = polymake.cube(4)          # optional - jupymake
sage: c.full_typename()             # optional - jupymake
'Polytope<Rational>'
sage: c.VERTICES.full_typename()    # optional - jupymake
'Matrix<Rational, NonSymmetric>'

get_member(attrname)#

Get a member/property of this element.

Note

Normally, it should be possible to just access the property in the usual Python syntax for attribute access. However, if that fails, one can request the member explicitly.

EXAMPLES:

sage: p = polymake.rand_sphere(4, 20, seed=5)    # optional - jupymake

Normally, a property would be accessed as follows:

sage: p.F_VECTOR                                # optional - jupymake
20 94 148 74

However, explicit access is possible as well:

sage: p.get_member('F_VECTOR')                  # optional - jupymake
20 94 148 74

In some cases, the explicit access works better:

sage: p.type                                    # optional - jupymake
Member function 'type' of Polymake::polytope::Polytope__Rational object
sage: p.get_member('type')                      # optional - jupymake
Polytope<Rational>[SAGE...]
sage: p.get_member('type').get_member('name')   # optional - jupymake
Polytope

Note that in the last example calling the erroneously constructed member function type still works:

sage: p.type()                                  # optional - jupymake
Polytope<Rational>[SAGE...]

get_member_function(attrname)#

Request a member function of this element.

Note

It is not checked whether a member function with the given name exists.

EXAMPLES:

sage: # optional - jupymake
sage: c = polymake.cube(2)
sage: c.contains
Member function 'contains' of Polymake::polytope::Polytope__Rational object
sage: V = polymake.new_object('Vector', [1,0,0])
sage: V
1 0 0
sage: c.contains(V)
true

Whether a member function of the given name actually exists for that object will only be clear when calling it:

sage: c.get_member_function("foo")                          # optional - jupymake
Member function 'foo' of Polymake::polytope::Polytope__Rational object
sage: c.get_member_function("foo")()                        # optional - jupymake
Traceback (most recent call last):
...
TypeError: Can't locate object method "foo" via package "Polymake::polytope::Polytope__Rational"

known_properties()#

List the names of properties that have been computed so far on this element.

Note

This is in many cases equivalent to use polymake’s list_properties, which returns a blank separated string representation of the list of properties. However, on some elements, list_properties would simply result in an error.

EXAMPLES:

sage: c = polymake.cube(4)                      # optional - jupymake
sage: c.known_properties()                      # optional - jupymake
['AFFINE_HULL',
 'BOUNDED',
 'CONE_AMBIENT_DIM',
 'CONE_DIM',
...
 'VERTICES_IN_FACETS']
sage: c.list_properties()                       # optional - jupymake
CONE_AMBIENT_DIM, CONE_DIM, FACETS, AFFINE_HULL, VERTICES_IN_FACETS,
BOUNDED...

A computation can change the list of known properties:

sage: c.F_VECTOR                                # optional - jupymake
16 32 24 8
sage: c.known_properties()                      # optional - jupymake
['AFFINE_HULL',
 'BOUNDED',
 'COMBINATORIAL_DIM',
 'CONE_AMBIENT_DIM',
 'CONE_DIM',
...
 'VERTICES_IN_FACETS']

qualified_typename()#

The qualified name of the type of this element.

EXAMPLES:

sage: c = polymake.cube(4)              # optional - jupymake
sage: c.qualified_typename()            # optional - jupymake
'polytope::Polytope<Rational>'
sage: c.VERTICES.qualified_typename()   # optional - jupymake
'common::Matrix<Rational, NonSymmetric>'

typename()#

The name of the underlying base type of this element in polymake.

EXAMPLES:

sage: c = polymake.cube(4)          # optional - jupymake
sage: c.typename()                  # optional - jupymake
'Polytope'
sage: c.VERTICES.typename()         # optional - jupymake
'Matrix'

typeof()#

Return the type of a polymake “big” object, and its underlying Perl type.

Note

This is mainly for internal use.

EXAMPLES:

sage: # optional - jupymake
sage: p = polymake.rand_sphere(3, 13, seed=12)
sage: p.typeof()
('Polymake::polytope::Polytope__Rational', 'ARRAY')
sage: p.VERTICES.typeof()
('Polymake::common::Matrix_A_Rational_I_NonSymmetric_Z', 'ARRAY')
sage: p.get_schedule('"F_VECTOR"').typeof()
('Polymake::Core::Scheduler::RuleChain', 'ARRAY')

On “small” objects, it just returns empty strings:

sage: p.N_VERTICES.typeof()                                 # optional - jupymake
('', '')
sage: p.list_properties().typeof()                          # optional - jupymake
('', '')

exception sage.interfaces.polymake.PolymakeError#

Bases: RuntimeError

Raised if polymake yields an error message.

class sage.interfaces.polymake.PolymakeFunctionElement(obj, name, memberfunction=False)#

Bases: InterfaceFunctionElement

A callable (function or member function) bound to a polymake element.

EXAMPLES:

sage: # optional - jupymake
sage: c = polymake.cube(2)
sage: V = polymake.new_object('Vector', [1,0,0])
sage: V
1 0 0
sage: c.contains
Member function 'contains' of Polymake::polytope::Polytope__Rational object
sage: c.contains(V)
true

class sage.interfaces.polymake.PolymakeJuPyMake(seed=None, verbose=False)#

Bases: PolymakeAbstract

Interface to the polymake interpreter using JuPyMake.

In order to use this interface, you need to either install the optional polymake package for Sage, or install polymake system-wide on your computer; it is available from https://polymake.org. Also install the jupymake Python package.

Type polymake.[tab] for a list of most functions available from your polymake install. Type polymake.Function? for polymake’s help about a given Function. Type polymake(...) to create a new polymake object, and polymake.eval(...) to run a string using polymake and get the result back as a string.

EXAMPLES:

sage: from sage.interfaces.polymake import polymake_jupymake as polymake
sage: type(polymake)
<...sage.interfaces.polymake.PolymakeJuPyMake...
sage: p = polymake.rand_sphere(4, 20, seed=5)       # optional - jupymake
sage: p                                             # optional - jupymake
Random spherical polytope of dimension 4; seed=5...
sage: set_verbose(3)
sage: p.H_VECTOR;                                   # optional - jupymake # random
used package ppl
  The Parma Polyhedra Library ...
sage: p.H_VECTOR                                    # optional - jupymake
1 16 40 16 1
sage: set_verbose(0)
sage: p.F_VECTOR                                    # optional - jupymake
20 94 148 74
sage: print(p.F_VECTOR._sage_doc_())                # optional - jupymake # random
property_types/Algebraic Types/Vector:
 A type for vectors with entries of type Element.

 You can perform algebraic operations such as addition or scalar multiplication.

 You can create a new Vector by entering its elements, e.g.:
    $v = new Vector<Int>(1,2,3);
 or
    $v = new Vector<Int>([1,2,3]);

Python strings are translated to polymake (Perl) identifiers. To obtain Perl strings, use strings containing double-quote characters. Python dicts are translated to Perl hashes.

sage: # long time, optional - internet jupymake perl_mongodb
sage: L = polymake.db_query({'"_id"': '"F.4D.0047"'},
....:                       db='"LatticePolytopes"',
....:                       collection='"SmoothReflexive"'); L
BigObjectArray
sage: len(L)
1
sage: P = L[0]
sage: sorted(P.list_properties(), key=str)
[..., LATTICE_POINTS_GENERATORS, ..., POINTED, ...]
sage: P.F_VECTOR
20 40 29 9

eval(code, **kwds)#

Evaluate a command.

INPUT:

code – a command (string) to be evaluated

Different reaction types of polymake, including warnings, comments, errors, request for user interaction, and yielding a continuation prompt, are taken into account.

EXAMPLES:

sage: from sage.interfaces.polymake import polymake_jupymake as polymake
sage: p = polymake.cube(3)              # optional - jupymake  # indirect doctest

Here we see that remarks printed by polymake are displayed if the verbosity is positive:

sage: set_verbose(1)
sage: p.N_LATTICE_POINTS                # optional - jupymake # random
used package latte
  LattE (Lattice point Enumeration) is a computer software dedicated to the
  problems of counting lattice points and integration inside convex polytopes.
  Copyright by Matthias Koeppe, Jesus A. De Loera and others.
  http://www.math.ucdavis.edu/~latte/
27
sage: set_verbose(0)

If polymake raises an error, the polymake interface raises a PolymakeError:

sage: polymake.eval('FOOBAR(3);')       # optional - jupymake
Traceback (most recent call last):
...
PolymakeError: Undefined subroutine &Polymake::User::FOOBAR called...

If a command is incomplete, then polymake returns a continuation prompt. In that case, we raise an error:

sage: polymake.eval('print 3')          # optional - jupymake
Traceback (most recent call last):
...
SyntaxError: Incomplete polymake command 'print 3'
sage: polymake.eval('print 3;')         # optional - jupymake
'3'

However, if the command contains line breaks but eventually is complete, no error is raised:

sage: print(polymake.eval('$tmp="abc";\nprint $tmp;'))  # optional - jupymake
abc

When requesting help, polymake sometimes expects the user to choose from a list. In that situation, we abort with a warning, and show the list from which the user can choose; we could demonstrate this using the help() method, but here we use an explicit code evaluation:

sage: print(polymake.eval('help "TRIANGULATION";'))     # optional - jupymake # random
doctest:warning
...
UserWarning: Polymake expects user interaction. We abort and return
the options that Polymake provides.
There are 5 help topics matching 'TRIANGULATION':
1: objects/Cone/properties/Triangulation and volume/TRIANGULATION
2: objects/Polytope/properties/Triangulation and volume/TRIANGULATION
3: objects/Visualization/Visual::PointConfiguration/methods/TRIANGULATION
4: objects/Visualization/Visual::Polytope/methods/TRIANGULATION
5: objects/PointConfiguration/properties/Triangulation and volume/TRIANGULATION

By default, we just wait until polymake returns a result. However, it is possible to explicitly set a timeout. The following usually does work in an interactive session and often in doc tests, too. However, sometimes it hangs, and therefore we remove it from the tests, for now:

sage: c = polymake.cube(15)             # optional - jupymake
sage: polymake.eval('print {}->F_VECTOR;'.format(c.name()), timeout=1) # not tested # optional - jupymake
Traceback (most recent call last):
...
RuntimeError: Polymake fails to respond timely

We verify that after the timeout, polymake is still able to give answers:

sage: c                                 # optional - jupymake
cube of dimension 15
sage: c.N_VERTICES                      # optional - jupymake
32768

Note, however, that the recovery after a timeout is not perfect. It may happen that in some situation the interface collapses and thus polymake would automatically be restarted, thereby losing all data that have been computed before.

is_running()#: Return True if self is currently running.

sage.interfaces.polymake.polymake_console(command='')#

Spawn a new polymake command-line session.

EXAMPLES:

sage: from sage.interfaces.polymake import polymake_console
sage: polymake_console()        # not tested
Welcome to polymake version ...
...
Ewgenij Gawrilow, Michael Joswig (TU Berlin)
http://www.polymake.org

This is free software licensed under GPL; see the source for copying conditions.
There is NO warranty; not even for MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.

Press F1 or enter 'help;' for basic instructions.

Application polytope currently uses following third-party software packages:
4ti2, bliss, cdd, latte, libnormaliz, lrs, permlib, ppl, sketch, sympol, threejs, tikz, topcom, tosimplex
For more details:  show_credits;
polytope >

sage.interfaces.polymake.reduce_load_Polymake()#

Return the polymake interface object defined in sage.interfaces.polymake.

EXAMPLES:

sage: from sage.interfaces.polymake import reduce_load_Polymake
sage: reduce_load_Polymake()
Polymake