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].

class
sage.interfaces.polymake.
Polymake
(script_subdirectory=None, logfile=None, server=None, server_tmpdir=None, seed=None, command=None)¶ Bases:
sage.interfaces.tab_completion.ExtraTabCompletion
,sage.interfaces.expect.Expect
Interface to the polymake interpreter.
In order to use this interface, you need to either install the optional polymake package for Sage, or install polymake systemwide on your computer; it is available from https://polymake.org.
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: p = polymake.rand_sphere(4, 20, seed=5) # optional  polymake sage: p # optional  polymake Random spherical polytope of dimension 4; seed=5... sage: set_verbose(3) sage: p.H_VECTOR # optional  polymake used package ppl The Parma Polyhedra Library (PPL): A C++ library for convex polyhedra and other numerical abstractions. http://www.cs.unipr.it/ppl/ 1 16 47 16 1 sage: set_verbose(0) sage: p.F_VECTOR # optional  polymake 20 101 162 81 sage: print(p.F_VECTOR._sage_doc_()) # optional  polymake # 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]);

_eval_line
(line, allow_use_file=True, wait_for_prompt=True, restart_if_needed=True, **kwds)¶ Evaluate a command.
INPUT:
line
, a command (string) to be evaluatedallow_use_file
(optional bool, defaultTrue
), whether or not to use a file if the line is very long.wait_for_prompt
(optional, defaultTrue
), whether or not to wait before polymake returns a prompt. If it is a string, it is considered as alternative prompt to be waited for.restart_if_needed
(optional bool, defaultTrue
), whether or not to restart polymake in case something goes wrong further optional arguments (e.g., timeout) that will be passed to
pexpect.pty_spawn.spawn.expect()
. Note that they are ignored if the line is too long and thus is evaluated via a file. So, if a timeout is defined, it should be accompanied byallow_use_file=False
.
Different reaction types of polymake, including warnings, comments, errors, request for user interaction, and yielding a continuation prompt, are taken into account.
Usually, this method is indirectly called via
eval()
.EXAMPLES:
sage: p = polymake.cube(3) # optional  polymake # 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  polymake 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  polymake 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  polymake Traceback (most recent call last): ... SyntaxError: Incomplete polymake command 'print 3' sage: polymake.eval('print 3;') # optional  polymake '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  polymake 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  polymake # 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  polymake sage: polymake.eval('print {}>F_VECTOR;'.format(c.name()), timeout=1) # optional  polymake # not tested 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  polymake cube of dimension 15 sage: c.N_VERTICES # optional  polymake 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.

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 hvector 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  polymake sage: q.H_VECTOR # optional  polymake 1 5 5 1 sage: q.F_VECTOR # optional  polymake 8 14 8 sage: q.VERY_AMPLE # optional  polymake 1
In the application ‘fan’, polymake can now compute the normal fan of \(q\) and its (primitive) rays:
sage: polymake.application('fan') # optional  polymake sage: g = q.normal_fan() # optional  polymake sage: g.RAYS # optional  polymake 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  polymake 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  polymake # 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  polymake '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  polymake 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  polymake # 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  polymake # 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  polymake 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  polymake sage: q.N_VERTICES # optional  polymake 4 sage: q.BOUNDED # optional  polymake 1 sage: q.VERTICES # optional  polymake 1 2 0 4 1 3 0 8 1 2 1 8 1 3 1 8 sage: q.full_typename() # optional  polymake '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  polymake # indirect doctest sage: d = polymake.cube(3) # optional  polymake
Equality is, for “big” objects such as polytopes, comparison by identity:
sage: c == d # optional  polymake False
However, the list of vertices is equal:
sage: c.VERTICES == d.VERTICES # optional  polymake True

version
()¶ Version of the polymake installation.
EXAMPLES:
sage: polymake.version() # optional  polymake '3...'


class
sage.interfaces.polymake.
PolymakeElement
(parent, value, is_name=False, name=None)¶ Bases:
sage.interfaces.tab_completion.ExtraTabCompletion
,sage.interfaces.expect.ExpectElement
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  polymake sage: p.typename() # optional  polymake 'Polytope' sage: p # optional  polymake 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  polymake 3319173990813887/4503599627370496

bool
()¶ Return whether this polymake element is equal to
True
.EXAMPLES:
sage: from sage.interfaces.polymake import polymake sage: polymake(0).bool() # optional polymake False sage: polymake(1).bool() # optional polymake True

full_typename
()¶ The name of the specialised type of this element.
EXAMPLES:
sage: c = polymake.cube(4) # optional  polymake sage: c.full_typename() # optional  polymake 'Polytope<Rational>' sage: c.VERTICES.full_typename() # optional  polymake '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  polymake
Normally, a property would be accessed as follows:
sage: p.F_VECTOR # optional  polymake 20 101 162 81
However, explicit access is possible as well:
sage: p.get_member('F_VECTOR') # optional  polymake 20 101 162 81
In some cases, the explicit access works better:
sage: p.type # optional  polymake Member function 'type' of Polymake::polytope::Polytope__Rational object sage: p.get_member('type') # optional  polymake Polytope<Rational>[SAGE...] sage: p.get_member('type').get_member('name') # optional  polymake Polytope
Note that in the last example calling the erroneously constructed member function
type
still works:sage: p.type() # optional  polymake 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  polymake sage: c.contains # optional  polymake Member function 'contains' of Polymake::polytope::Polytope__Rational object sage: V = polymake.new_object('Vector', [1,0,0]) # optional  polymake sage: V # optional  polymake 1 0 0 sage: c.contains(V) # optional  polymake 1
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  polymake Member function 'foo' of Polymake::polytope::Polytope__Rational object sage: c.get_member_function('foo')() # optional  polymake Traceback (most recent call last): ... TypeError: Can't locate object method "foo" via package "Polymake::polytope::Polytope__Rational" at input line 1.

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  polymake sage: c.known_properties() # optional  polymake ['AFFINE_HULL', 'BOUNDED', 'CONE_AMBIENT_DIM', 'CONE_DIM', ... 'VERTICES_IN_FACETS'] sage: c.list_properties() # optional  polymake 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  polymake 16 32 24 8 sage: c.known_properties() # optional  polymake ['AFFINE_HULL', 'BOUNDED', 'COMBINATORIAL_DIM', 'CONE_AMBIENT_DIM', 'CONE_DIM', 'DUAL_H_VECTOR', 'FACETS', 'FAR_FACE', 'FEASIBLE', 'FULL_DIM', 'F_VECTOR', 'GRAPH', 'LINEALITY_DIM', 'LINEALITY_SPACE', 'N_FACETS', 'N_VERTICES', 'POINTED', 'SIMPLE', 'SIMPLICIAL', 'VERTICES', 'VERTICES_IN_FACETS']

qualified_typename
()¶ The qualified name of the type of this element.
EXAMPLES:
sage: c = polymake.cube(4) # optional  polymake sage: c.qualified_typename() # optional  polymake 'polytope::Polytope<Rational>' sage: c.VERTICES.qualified_typename() # optional  polymake 'common::Matrix<Rational, NonSymmetric>'

typename
()¶ The name of the underlying base type of this element in polymake.
EXAMPLES:
sage: c = polymake.cube(4) # optional  polymake sage: c.typename() # optional  polymake 'Polytope' sage: c.VERTICES.typename() # optional  polymake 'Matrix'

typeof
()¶ Returns 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  polymake sage: p.typeof() # optional  polymake ('Polymake::polytope::Polytope__Rational', 'ARRAY') sage: p.VERTICES.typeof() # optional  polymake ('Polymake::common::Matrix_A_Rational_I_NonSymmetric_Z', 'ARRAY') sage: p.get_schedule("F_VECTOR").typeof() # optional  polymake ('Polymake::Core::Scheduler::RuleChain', 'ARRAY')
On “small” objects, it just returns empty strings:
sage: p.N_VERTICES.typeof() # optional  polymake ('', '') sage: p.list_properties().typeof() # optional  polymake ('', '')


exception
sage.interfaces.polymake.
PolymakeError
¶ Bases:
exceptions.RuntimeError
Raised if polymake yields an error message.

class
sage.interfaces.polymake.
PolymakeFunctionElement
(obj, name, memberfunction=False)¶ Bases:
sage.interfaces.expect.FunctionElement
A callable (function or member function) bound to a polymake element.
EXAMPLES:
sage: c = polymake.cube(2) # optional  polymake sage: V = polymake.new_object('Vector', [1,0,0]) # optional  polymake sage: V # optional  polymake 1 0 0 sage: c.contains # optional  polymake Member function 'contains' of Polymake::polytope::Polytope__Rational object sage: c.contains(V) # optional  polymake 1

sage.interfaces.polymake.
polymake_console
(command='')¶ Spawn a new polymake commandline 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 thirdparty 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
()¶ Returns the polymake interface object defined in
sage.interfaces.polymake
.EXAMPLES:
sage: from sage.interfaces.polymake import reduce_load_Polymake sage: reduce_load_Polymake() Polymake