def cross(d):
r"""Produce a d-dimensional cross polytope.
Regular polytope corresponding to the Coxeter group of type B<sub>d-1</sub>
= C<sub>d-1</sub>.
"""
return call_polymake_function("polytope", "cross", d)
def cube(d, x_up=1, x_low=-1):
r"""Produce a d-dimensional cube
Regular polytope corresponding to the Coxeter group of type
`B^{d-1} = C^{d-1}`. The bounding hyperplanes are `x_i <= x_{up}` and
`x_i >= x_{low}`.
Keyword arguments:
d -- dimension
x_up -- upper bound in each dimension
x_low -- lower bound in each dimension
>>> import polymake
>>> polymake.cube(2).VERTICES
[ 1 -1 -1]
[ 1 1 -1]
[ 1 -1 1 ]
[ 1 1 1 ]
>>> polymake.cube(2,2,0).VERTICES
[ 1 0 0]
[ 1 2 0]
[ 1 0 2]
[ 1 2 2]
"""
return call_polymake_function("polytope", "cube", d, x_up, x_low)
def cyclic_caratheodory(d, n):
r"""Produce a d-dimensional cyclic polytope with n points.
Prototypical example of a neighborly polytope. Combinatorics completely known
due to Gale's evenness criterion. Coordinates are chosen on the trigonometric
moment curve. For cyclic polytopes from other curves, see :func:`cyclic`.
"""
return call_polymake_function("polytope", 'cyclic_caratheodory', d, n)
def associahedron(d):
r"""Produce a d-dimensional associahedron (or Stasheff polytope)
>>> import polymake
>>> a = polymake.associahedron(3)
>>> a.N_VERTICES
14
"""
return call_polymake_function("polytope", "associahedron", d)
def icosahedron():
r"""
Create exact regular icosahedron in Q(sqrt{5}). A Platonic solid.
>>> import polymake
>>> ico = polymake.icosahedron()
>>> ico
Polytope<QuadraticExtension<Rational>><...>
"""
return call_polymake_function("polytope", "icosahedron")
def dwarfed_cube(d):
r"""Produce a d-dimensional dwarfed cube.
Keyword arguments:
d -- the dimension
>>> import polymake
>>> polymake.dwarfed_cube(2).N_VERTICES
5
"""
return call_polymake_function("polytope", "dwarfed_cube", d)
def cycle_graph(n):
r"""Constructs a cycle graph on n nodes.
>>> import polymake
>>> g = polymake.cycle_graph(7)
>>> g
Graph<Undirected><...>
>>> g.N_NODES
7
>>> g.N_EDGES
7
"""
return call_polymake_function("polytope", "cycle_graph", n)
def neighborly_cubical(d, n):
r""" Produce the combinatorial description of a neighborly cubical polytope.
Keyword arguments:
d -- dimension of the polytope
n -- dimension of the equivalent cube
The facets are labelled in oriented matroid notation as in the cubical Gale
evenness criterion. See Joswig and Ziegler, Discr. Comput. Geom.
24:315-344 (2000).
"""
return call_polymake_function("polytope", "neighborly_cubical", d, n)
def rand_sphere(dim, npoints):
r"""Produce a d-dimensional polytope with n random vertices uniformly
distributed on the unit sphere.
Keyword arguments:
d -- the dimension
n -- the number of random vertices
>>> import polymake
>>> p = polymake.rand_sphere(3, 10)
>>> p.N_VERTICES
10
"""
return call_polymake_function("polytope", "rand_sphere", dim, npoints)
def kneser_graph(n, k):
r"""Create the Kneser graph on parameters (n,k).
It has one node for each set in \({[n]}\choose{k}\),
and an edge between two nodes iff the corresponding subsets are
disjoint.
>>> import polymake
>>> g = polymake.kneser_graph(6, 3)
>>> g
Graph<Undirected><...>
>>> g.N_NODES
20
>>> g.N_EDGES
10
"""
return call_polymake_function("polytope", "kneser_graph", n, k)
def johnson_graph(n, k):
r"""Create the Johnson graph on parameters (n,k).
It has one node for each set in \({[n]}\choose{k}\),
and an edge between two nodes iff the intersection of the corresponding
subsets is of size k-1.
>>> import polymake
>>> g = polymake.johnson_graph(5, 3)
>>> g
Graph<Undirected><...>
>>> g.N_NODES
10
>>> g.N_EDGES
30
"""
return call_polymake_function("polytope", "johnson_graph", n, k)
def birkhoff(n, even=False):
"""Constructs the Birkhoff polytope of dimension `n^2`.
It is the polytope of `n \\times n` stochastic matrices (encoded as `n^2`
row vectors), thus matrices with non-negative entries whose row and column
entries sum up to one. Its vertices are the permutation matrices.
Keyword arguments:
n -- integer
even -- boolean (default to False)
>>> import polymake
>>> b = polymake.birkhoff(3)
>>> b.N_VERTICES
6
"""
return call_polymake_function("polytope", "birkhoff", n, even)
def cuboctahedron():
r"""Create cuboctahedron. An Archimedean solid.
>>> import polymake
>>> polymake.cuboctahedron().VERTICES
[ 1 1 1 0 ]
[ 1 1 0 1 ]
[ 1 0 1 1 ]
[ 1 1 0 -1]
[ 1 0 1 -1]
[ 1 1 -1 0 ]
[ 1 0 -1 1 ]
[ 1 -1 1 0 ]
[ 1 -1 0 1 ]
[ 1 0 -1 -1]
[ 1 -1 0 -1]
[ 1 -1 -1 0 ]
"""
return call_polymake_function("polytope","cuboctahedron")
def delpezzo(d, scale=1):
r"""
Produce a d-dimensional del-Pezzo polytope
It is the convex hull of the cross polytope together with the all-ones and
minus all-ones vector. All coordinates are +/- scale or 0.
Keyword arguments:
d -- the dimension
scale -- the absolute value of each non-zero vertex coordinate.
>>> import polymake
>>> polymake.delpezzo(2).VERTICES
[ 1 1 0 ]
[ 1 0 1 ]
[ 1 -1 0 ]
[ 1 0 -1]
[ 1 1 1 ]
[ 1 -1 -1]
"""
return call_polymake_function("polytope", "delpezzo", d, scale)
def cyclic(d, n):
r"""Produce a d-dimensional cyclic polytope with n points.
Prototypical example of a neighborly polytope. Combinatorics completely
known due to Gale's evenness criterion. Coordinates are chosen on the
(spherical) moment curve at integer steps from start, or 0 if unspecified.
If spherical is true the vertices lie on the sphere with center
(1/2,0,...,0) and radius 1/2. In this case (the necessarily positive)
parameter start defaults to 1.
>>> import polymake
>>> polymake.cyclic(2, 4).VERTICES
[ 1 0 0]
[ 1 1 1]
[ 1 2 4]
[ 1 3 9]
>>> polymake.cyclic(3, 3)
Traceback (most recent call last):
...
ValueError: cyclic: d >= 2 and n > d required
<BLANKLINE>
"""
return call_polymake_function("polytope",'cyclic', d, n)
