The Python backend, using number fields internally

class sage.geometry.polyhedron.backend_number_field.Polyhedron_number_field(parent, Vrep, Hrep, Vrep_minimal=None, Hrep_minimal=None, pref_rep=None, mutable=False, **kwds)[source]

Bases: Polyhedron_field, Polyhedron_base_number_field

Polyhedra whose data can be converted to number field elements.

All computations are done internally using a fixed real embedded number field, which is determined automatically.

INPUT:

  • Vrep – list [vertices, rays, lines] or None

  • Hrep – list [ieqs, eqns] or None

EXAMPLES:

sage: P = Polyhedron(vertices=[[1], [sqrt(2)]], backend='number_field'); P      # needs sage.rings.number_field sage.symbolic
A 1-dimensional polyhedron
 in (Symbolic Ring)^1 defined as the convex hull of 2 vertices
sage: P.vertices()                                                              # needs sage.rings.number_field sage.symbolic
(A vertex at (1), A vertex at (sqrt(2)))

sage: P = polytopes.icosahedron(exact=True, backend='number_field')             # needs sage.rings.number_field
sage: P                                                                         # needs sage.rings.number_field
A 3-dimensional polyhedron
 in (Number Field in sqrt5 with defining polynomial x^2 - 5
     with sqrt5 = 2.236067977499790?)^3
 defined as the convex hull of 12 vertices

sage: x = polygen(ZZ); P = Polyhedron(                                          # needs sage.rings.number_field sage.symbolic
....:     vertices=[[sqrt(2)], [AA.polynomial_root(x^3-2, RIF(0,3))]],
....:     backend='number_field')
sage: P                                                                         # needs sage.rings.number_field sage.symbolic
A 1-dimensional polyhedron
 in (Symbolic Ring)^1 defined as the convex hull of 2 vertices
sage: P.vertices()                                                              # needs sage.rings.number_field sage.symbolic
(A vertex at (sqrt(2)), A vertex at (2^(1/3)))

>>> from sage.all import *
>>> P = Polyhedron(vertices=[[Integer(1)], [sqrt(Integer(2))]], backend='number_field'); P      # needs sage.rings.number_field sage.symbolic
A 1-dimensional polyhedron
 in (Symbolic Ring)^1 defined as the convex hull of 2 vertices
>>> P.vertices()                                                              # needs sage.rings.number_field sage.symbolic
(A vertex at (1), A vertex at (sqrt(2)))

>>> P = polytopes.icosahedron(exact=True, backend='number_field')             # needs sage.rings.number_field
>>> P                                                                         # needs sage.rings.number_field
A 3-dimensional polyhedron
 in (Number Field in sqrt5 with defining polynomial x^2 - 5
     with sqrt5 = 2.236067977499790?)^3
 defined as the convex hull of 12 vertices

>>> x = polygen(ZZ); P = Polyhedron(                                          # needs sage.rings.number_field sage.symbolic
...     vertices=[[sqrt(Integer(2))], [AA.polynomial_root(x**Integer(3)-Integer(2), RIF(Integer(0),Integer(3)))]],
...     backend='number_field')
>>> P                                                                         # needs sage.rings.number_field sage.symbolic
A 1-dimensional polyhedron
 in (Symbolic Ring)^1 defined as the convex hull of 2 vertices
>>> P.vertices()                                                              # needs sage.rings.number_field sage.symbolic
(A vertex at (sqrt(2)), A vertex at (2^(1/3)))