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#
- class sage.interfaces.polymake.PolymakeAbstract(seed=None)#
Bases:
sage.interfaces.tab_completion.ExtraTabCompletion
,sage.interfaces.interface.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: 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]]) # optional - jupymake sage: q.H_VECTOR # optional - jupymake 1 5 5 1 sage: q.F_VECTOR # optional - jupymake 8 14 8 sage: q.VERY_AMPLE # optional - jupymake true
In the application ‘fan’, polymake can now compute the normal fan of \(q\) and its (primitive) rays:
sage: polymake.application('fan') # optional - jupymake sage: g = q.normal_fan() # optional - jupymake sage: g.RAYS # optional - jupymake -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() # optional - jupymake -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()
andpolymake_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 stringpager
, optional bool, defaultTrue
: 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: 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]]) # optional - jupymake sage: q.N_VERTICES # optional - jupymake 4 sage: q.BOUNDED # optional - jupymake true sage: q.VERTICES # optional - jupymake 1 2 0 4 1 3 0 8 1 2 1 8 1 3 1 8 sage: q.full_typename() # optional - jupymake 'Polytope<Rational>'
- set(var, value)#
Set the variable var to the given value.
Eventually,
var
is a reference tovalue
.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:
sage.interfaces.tab_completion.ExtraTabCompletion
,sage.interfaces.interface.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: c = polymake.cube(2) # optional - jupymake sage: c.contains # optional - jupymake Member function 'contains' of Polymake::polytope::Polytope__Rational object sage: V = polymake.new_object('Vector', [1,0,0]) # optional - jupymake sage: V # optional - jupymake 1 0 0 sage: c.contains(V) # optional - jupymake 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: p = polymake.rand_sphere(3, 13, seed=12) # optional - jupymake sage: p.typeof() # optional - jupymake ('Polymake::polytope::Polytope__Rational', 'ARRAY') sage: p.VERTICES.typeof() # optional - jupymake ('Polymake::common::Matrix_A_Rational_I_NonSymmetric_Z', 'ARRAY') sage: p.get_schedule('"F_VECTOR"').typeof() # optional - jupymake ('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:
sage.interfaces.interface.InterfaceFunctionElement
A callable (function or member function) bound to a polymake element.
EXAMPLES:
sage: c = polymake.cube(2) # optional - jupymake sage: V = polymake.new_object('Vector', [1,0,0]) # optional - jupymake sage: V # optional - jupymake 1 0 0 sage: c.contains # optional - jupymake Member function 'contains' of Polymake::polytope::Polytope__Rational object sage: c.contains(V) # optional - jupymake true
- class sage.interfaces.polymake.PolymakeJuPyMake(seed=None, verbose=False)#
Bases:
sage.interfaces.polymake.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. Typepolymake.Function?
for polymake’s help about a givenFunction
. Typepolymake(...)
to create a new polymake object, andpolymake.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: L = polymake.db_query({'"_id"': '"F.4D.0047"'}, # long time, optional - jupymake internet perl_mongodb ....: db='"LatticePolytopes"', ....: collection='"SmoothReflexive"'); L BigObjectArray sage: len(L) # long time, optional - jupymake internet perl_mongodb 1 sage: P = L[0] # long time, optional - jupymake internet perl_mongodb sage: sorted(P.list_properties(), key=str) # long time, optional - jupymake internet perl_mongodb [..., LATTICE_POINTS_GENERATORS, ..., POINTED, ...] sage: P.F_VECTOR # long time, optional - jupymake internet perl_mongodb 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 expect 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
ifself
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