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
-
class
sage.geometry.polytope.
Polymake
¶ -
associahedron
(dimension)¶ Return the Associahedron.
INPUT:
dimension
– an integer
-
birkhoff
(n)¶ Return the Birkhoff polytope.
INPUT:
n
– an integer
-
cell24
()¶ Return the 24-cell.
EXAMPLES:
sage: polymake.cell24() # not tested The 24-cell
-
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) # not tested sage: n # not tested Convex hull of points [[1, 0, 0, 3], [1, 0, 3, 0], [1, 1, 1, 1], [1, 3, 0, 0]] sage: n.facets() # not tested [(0, 1, 0, 0), (3, -1, -1, 0), (0, 0, 1, 0)] sage: n.is_simple() # not tested True sage: n.graph() # not tested 'GRAPH\n{1 2}\n{0 2}\n{0 1}\n\n'
-
cube
(dimension, scale=0)¶
-
from_data
(data)¶
-
rand01
(d, n, seed=None)¶
-
reconfigure
()¶ Reconfigure polymake.
Remember to run polymake.reconfigure() as soon as you have changed the customization file and/or installed missing software!
-
-
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]]) # not tested
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]]) # not tested sage: P.facets() # not tested [(0, 0, 0, 1), (0, 1, 0, 0), (0, 0, 1, 0), (1, 0, 0, -1), (1, 0, -1, 0), (1, -1, 0, 0)]
-
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]]) # not tested sage: P.is_simple() # not tested True
AUTHORS:
- Edwin O’Shea (2006-05-02): Definition of simple.
-
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]]) # not tested sage: P.n_facets() # not tested 6
-
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]]) # not tested sage: P.vertices() # not tested [(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)]
-
visual
()¶
-
write
(filename)¶
-