Polytopes¶
This module provides access to polymake, which ‘has been developed since 1997 in the Discrete Geometry group at the Institute of Mathematics of Technische Universitat Berlin. Since 2004 the development is shared with Fachbereich Mathematik, Technische Universitat Darmstadt. The system offers access to a wide variety of algorithms and packages within a common framework. polymake is flexible and continuously expanding. The software supplies C++ and Perl interfaces which make it highly adaptable to individual needs.’
Note
If you have trouble with this module do:
sage: !polymake --reconfigure # not tested
at the command line.
AUTHORS:
- Ewgenij Gawrilow, Michael Joswig: main authors of polymake
- William Stein: Sage interface
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class
sage.geometry.polytope.
Polymake
¶ -
associahedron
(dimension)¶ Return the Associahedron.
INPUT:
dimension
– an integer
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birkhoff
(n)¶ Return the Birkhoff polytope.
INPUT:
n
– an integer
-
cell24
()¶ Return the 24-cell.
EXAMPLES:
sage: polymake.cell24() # optional - polymake The 24-cell
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convex_hull
(points=[])¶ EXAMPLES:
sage: R.<x,y,z> = PolynomialRing(QQ,3) sage: f = x^3 + y^3 + z^3 + x*y*z sage: e = f.exponents() sage: a = [[1] + list(v) for v in e] sage: a [[1, 3, 0, 0], [1, 0, 3, 0], [1, 1, 1, 1], [1, 0, 0, 3]] sage: n = polymake.convex_hull(a) # optional - polymake sage: n # optional - polymake Convex hull of points [[1, 0, 0, 3], [1, 0, 3, 0], [1, 1, 1, 1], [1, 3, 0, 0]] sage: n.facets() # optional - polymake [(0, 1, 0, 0), (3, -1, -1, 0), (0, 0, 1, 0)] sage: n.is_simple() # optional - polymake True sage: n.graph() # optional - polymake 'GRAPH\n{1 2}\n{0 2}\n{0 1}\n\n'
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cube
(dimension, scale=0)¶
-
from_data
(data)¶
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rand01
(d, n, seed=None)¶
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reconfigure
()¶ Reconfigure polymake.
Remember to run polymake.reconfigure() as soon as you have changed the customization file and/or installed missing software!
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class
sage.geometry.polytope.
Polytope
(datafile, desc)¶ Bases:
sage.structure.sage_object.SageObject
Create a polytope.
EXAMPLES:
sage: P = polymake.convex_hull([[1,0,0,0], [1,0,0,1], [1,0,1,0], [1,0,1,1], [1,1,0,0], [1,1,0,1], [1,1,1,0], [1,1,1,1]]) # optional - polymake
Note
If you have trouble with this module do:
sage: !polymake --reconfigure # not tested
at the command line.
-
cmd
(cmd)¶
-
data
()¶
-
facets
()¶ Return the facets.
EXAMPLES:
sage: P = polymake.convex_hull([[1,0,0,0], [1,0,0,1], [1,0,1,0], [1,0,1,1], [1,1,0,0], [1,1,0,1], [1,1,1,0], [1,1,1,1]]) # optional - polymake sage: P.facets() # optional - polymake [(0, 0, 0, 1), (0, 1, 0, 0), (0, 0, 1, 0), (1, 0, 0, -1), (1, 0, -1, 0), (1, -1, 0, 0)]
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graph
()¶
-
is_simple
()¶ Return
True
if this polytope is simple.A polytope is simple if the degree of each vertex equals the dimension of the polytope.
EXAMPLES:
sage: P = polymake.convex_hull([[1,0,0,0], [1,0,0,1], [1,0,1,0], [1,0,1,1], [1,1,0,0], [1,1,0,1], [1,1,1,0], [1,1,1,1]]) # optional - polymake sage: P.is_simple() # optional - polymake True
AUTHORS:
- Edwin O’Shea (2006-05-02): Definition of simple.
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n_facets
()¶ EXAMPLES:
sage: P = polymake.convex_hull([[1,0,0,0], [1,0,0,1], [1,0,1,0], [1,0,1,1], [1,1,0,0], [1,1,0,1], [1,1,1,0], [1,1,1,1]]) # optional - polymake sage: P.n_facets() # optional - polymake 6
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vertices
()¶ Return the vertices.
EXAMPLES:
sage: P = polymake.convex_hull([[1,0,0,0], [1,0,0,1], [1,0,1,0], [1,0,1,1], [1,1,0,0], [1,1,0,1], [1,1,1,0], [1,1,1,1]]) # optional - polymake sage: P.vertices() # optional - polymake [(1, 0, 0, 0), (1, 0, 0, 1), (1, 0, 1, 0), (1, 0, 1, 1), (1, 1, 0, 0), (1, 1, 0, 1), (1, 1, 1, 0), (1, 1, 1, 1)]
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visual
()¶
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write
(filename)¶
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