def cross_polytope(self, dim_n):
"""
Return a cross-polytope in dimension ``dim_n``. These are
the generalization of the octahedron.
INPUT:
- ``dim_n`` -- integer. The dimension of the cross-polytope.
OUTPUT:
A Polyhedron object of the ``dim_n``-dimensional cross-polytope,
with exact coordinates.
EXAMPLES::
sage: four_cross = polytopes.cross_polytope(4)
sage: four_cross.is_simple()
False
sage: four_cross.n_vertices()
8
"""
verts = permutations([0 for i in range(dim_n-1)] + [1])
verts = verts + permutations([0 for i in range(dim_n-1)] + [-1])
return Polyhedron(vertices=verts)
def six_hundred_cell(self):
"""
Return the standard 600-cell polytope.
OUTPUT:
A Polyhedron object of the 4-dimensional 600-cell, a regular
polytope. In many ways this is an analogue of the
icosahedron. The coordinates of this polytope are rational
approximations of the true coordinates of the 600-cell, some
of which involve the (irrational) golden ratio.
EXAMPLES::
sage: p600 = polytopes.six_hundred_cell() # not tested - very long time
sage: len(list(p600.bounded_edges())) # not tested - very long time
120
"""
verts = []
q12 = QQ(1) / 2
base = [q12, q12, q12, q12]
for i in range(2):
for j in range(2):
for k in range(2):
for l in range(2):
verts.append([x for x in base])
base[3] = base[3] * (-1)
base[2] = base[2] * (-1)
base[1] = base[1] * (-1)
base[0] = base[0] * (-1)
for x in permutations([0, 0, 0, 1]):
verts.append(x)
for x in permutations([0, 0, 0, -1]):
verts.append(x)
g = QQ(1618033) / 1000000 # Golden ratio approximation
verts = verts + [
i([q12, g / 2, 1 / (g * 2), 0]) for i in AlternatingGroup(4)
]
verts = verts + [
i([q12, g / 2, -1 / (g * 2), 0]) for i in AlternatingGroup(4)
]
verts = verts + [
i([q12, -g / 2, 1 / (g * 2), 0]) for i in AlternatingGroup(4)
]
verts = verts + [
i([q12, -g / 2, -1 / (g * 2), 0]) for i in AlternatingGroup(4)
]
verts = verts + [
i([-q12, g / 2, 1 / (g * 2), 0]) for i in AlternatingGroup(4)
]
verts = verts + [
i([-q12, g / 2, -1 / (g * 2), 0]) for i in AlternatingGroup(4)
]
verts = verts + [
i([-q12, -g / 2, 1 / (g * 2), 0]) for i in AlternatingGroup(4)
]
verts = verts + [
i([-q12, -g / 2, -1 / (g * 2), 0]) for i in AlternatingGroup(4)
]
return Polyhedron(vertices=verts)
def cross_polytope(self, dim_n):
"""
Return a cross-polytope in dimension ``dim_n``. These are
the generalization of the octahedron.
INPUT:
- ``dim_n`` -- integer. The dimension of the cross-polytope.
OUTPUT:
A Polyhedron object of the ``dim_n``-dimensional cross-polytope,
with exact coordinates.
EXAMPLES::
sage: four_cross = polytopes.cross_polytope(4)
sage: four_cross.is_simple()
False
sage: four_cross.n_vertices()
8
"""
verts = permutations([0 for i in range(dim_n - 1)] + [1])
verts = verts + permutations([0 for i in range(dim_n - 1)] + [-1])
return Polyhedron(vertices=verts)
def permutahedron(self, n, project = True):
"""
The standard permutahedron of (1,...,n) projected into n-1
dimensions.
INPUT:
- ``n`` -- the numbers ``(1,...,n)`` are permuted
- ``project`` -- If ``False`` the polyhedron is left in dimension ``n``.
OUTPUT:
A Polyhedron object representing the permutahedron.
EXAMPLES::
sage: perm4 = polytopes.permutahedron(4)
sage: perm4
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 24 vertices
sage: polytopes.permutahedron(5).show() # long time
"""
verts = range(1,n+1)
verts = permutations(verts)
if project:
verts = [Polytopes.project_1(x) for x in verts]
p = Polyhedron(vertices=verts)
return p
def hypersimplex(self, dim_n, k, project = True):
"""
The hypersimplex in dimension dim_n with d choose k vertices,
projected into (dim_n - 1) dimensions.
INPUT:
- ``n`` -- the numbers ``(1,...,n)`` are permuted
- ``project`` -- If ``False``, the polyhedron is left in
dimension ``n``.
OUTPUT:
A Polyhedron object representing the hypersimplex.
EXAMPLES::
sage: h_4_2 = polytopes.hypersimplex(4,2) # combinatorially equivalent to octahedron
sage: h_4_2.n_vertices()
6
sage: h_4_2.n_inequalities()
8
"""
vert0 = [0]*(dim_n-k) + [1]*k
verts = permutations(vert0)
if project:
verts = [Polytopes.project_1(x) for x in verts]
return Polyhedron(vertices=verts)
def n_simplex(self, dim_n=3, project = True):
"""
Return a rational approximation to a regular simplex in
dimension ``dim_n``.
INPUT:
- ``dim_n`` -- The dimension of the cross-polytope, a positive
integer.
- ``project`` -- Optional argument, whether to project
orthogonally. Default is True.
OUTPUT:
A Polyhedron object of the ``dim_n``-dimensional simplex.
EXAMPLES::
sage: s5 = polytopes.n_simplex(5)
sage: s5.dim()
5
"""
verts = permutations([0 for i in range(dim_n)] + [1])
if project: verts = [Polytopes.project_1(x) for x in verts]
return Polyhedron(vertices=verts)
def permutahedron(self, n, project=True):
"""
The standard permutahedron of (1,...,n) projected into n-1
dimensions.
INPUT:
- ``n`` -- the numbers ``(1,...,n)`` are permuted
- ``project`` -- If ``False`` the polyhedron is left in dimension ``n``.
OUTPUT:
A Polyhedron object representing the permutahedron.
EXAMPLES::
sage: perm4 = polytopes.permutahedron(4)
sage: perm4
A 3-dimensional polyhedron in QQ^3 defined as the convex hull of 24 vertices
sage: polytopes.permutahedron(5).show() # long time
"""
verts = range(1, n + 1)
verts = permutations(verts)
if project:
verts = [Polytopes.project_1(x) for x in verts]
p = Polyhedron(vertices=verts)
return p
def hypersimplex(self, dim_n, k, project=True):
"""
The hypersimplex in dimension dim_n with d choose k vertices,
projected into (dim_n - 1) dimensions.
INPUT:
- ``n`` -- the numbers ``(1,...,n)`` are permuted
- ``project`` -- If ``False``, the polyhedron is left in
dimension ``n``.
OUTPUT:
A Polyhedron object representing the hypersimplex.
EXAMPLES::
sage: h_4_2 = polytopes.hypersimplex(4,2) # combinatorially equivalent to octahedron
sage: h_4_2.n_vertices()
6
sage: h_4_2.n_inequalities()
8
"""
vert0 = [0] * (dim_n - k) + [1] * k
verts = permutations(vert0)
if project:
verts = [Polytopes.project_1(x) for x in verts]
return Polyhedron(vertices=verts)
def n_simplex(self, dim_n=3, project=True):
"""
Return a rational approximation to a regular simplex in
dimension ``dim_n``.
INPUT:
- ``dim_n`` -- The dimension of the cross-polytope, a positive
integer.
- ``project`` -- Optional argument, whether to project
orthogonally. Default is True.
OUTPUT:
A Polyhedron object of the ``dim_n``-dimensional simplex.
EXAMPLES::
sage: s5 = polytopes.n_simplex(5)
sage: s5.dim()
5
"""
verts = permutations([0 for i in range(dim_n)] + [1])
if project: verts = [Polytopes.project_1(x) for x in verts]
return Polyhedron(vertices=verts)
def six_hundred_cell(self):
"""
Return the standard 600-cell polytope.
OUTPUT:
A Polyhedron object of the 4-dimensional 600-cell, a regular
polytope. In many ways this is an analogue of the
icosahedron. The coordinates of this polytope are rational
approximations of the true coordinates of the 600-cell, some
of which involve the (irrational) golden ratio.
EXAMPLES::
sage: p600 = polytopes.six_hundred_cell() # not tested - very long time
sage: len(list(p600.bounded_edges())) # not tested - very long time
120
"""
verts = []
q12 = QQ(1)/2
base = [q12,q12,q12,q12]
for i in range(2):
for j in range(2):
for k in range(2):
for l in range(2):
verts.append([x for x in base])
base[3] = base[3]*(-1)
base[2] = base[2]*(-1)
base[1] = base[1]*(-1)
base[0] = base[0]*(-1)
for x in permutations([0,0,0,1]):
verts.append(x)
for x in permutations([0,0,0,-1]):
verts.append(x)
g = QQ(1618033)/1000000 # Golden ratio approximation
verts = verts + [i([q12,g/2,1/(g*2),0]) for i in AlternatingGroup(4)]
verts = verts + [i([q12,g/2,-1/(g*2),0]) for i in AlternatingGroup(4)]
verts = verts + [i([q12,-g/2,1/(g*2),0]) for i in AlternatingGroup(4)]
verts = verts + [i([q12,-g/2,-1/(g*2),0]) for i in AlternatingGroup(4)]
verts = verts + [i([-q12,g/2,1/(g*2),0]) for i in AlternatingGroup(4)]
verts = verts + [i([-q12,g/2,-1/(g*2),0]) for i in AlternatingGroup(4)]
verts = verts + [i([-q12,-g/2,1/(g*2),0]) for i in AlternatingGroup(4)]
verts = verts + [i([-q12,-g/2,-1/(g*2),0]) for i in AlternatingGroup(4)]
return Polyhedron(vertices=verts)
def twenty_four_cell(self):
"""
Return the standard 24-cell polytope.
OUTPUT:
A Polyhedron object of the 4-dimensional 24-cell, a regular
polytope. The coordinates of this polytope are exact.
EXAMPLES::
sage: p24 = polytopes.twenty_four_cell()
sage: v = p24.vertex_generator().next()
sage: for adj in v.neighbors(): print adj
A vertex at (-1/2, -1/2, -1/2, 1/2)
A vertex at (-1/2, -1/2, 1/2, -1/2)
A vertex at (-1, 0, 0, 0)
A vertex at (-1/2, 1/2, -1/2, -1/2)
A vertex at (0, -1, 0, 0)
A vertex at (0, 0, -1, 0)
A vertex at (0, 0, 0, -1)
A vertex at (1/2, -1/2, -1/2, -1/2)
"""
verts = []
q12 = QQ(1)/2
base = [q12,q12,q12,q12]
for i in range(2):
for j in range(2):
for k in range(2):
for l in range(2):
verts.append([x for x in base])
base[3] = base[3]*(-1)
base[2] = base[2]*(-1)
base[1] = base[1]*(-1)
base[0] = base[0]*(-1)
verts = verts + permutations([0,0,0,1])
verts = verts + permutations([0,0,0,-1])
return Polyhedron(vertices=verts)
def twenty_four_cell(self):
"""
Return the standard 24-cell polytope.
OUTPUT:
A Polyhedron object of the 4-dimensional 24-cell, a regular
polytope. The coordinates of this polytope are exact.
EXAMPLES::
sage: p24 = polytopes.twenty_four_cell()
sage: v = p24.vertex_generator().next()
sage: for adj in v.neighbors(): print adj
A vertex at (-1/2, -1/2, -1/2, 1/2)
A vertex at (-1/2, -1/2, 1/2, -1/2)
A vertex at (-1, 0, 0, 0)
A vertex at (-1/2, 1/2, -1/2, -1/2)
A vertex at (0, -1, 0, 0)
A vertex at (0, 0, -1, 0)
A vertex at (0, 0, 0, -1)
A vertex at (1/2, -1/2, -1/2, -1/2)
"""
verts = []
q12 = QQ(1) / 2
base = [q12, q12, q12, q12]
for i in range(2):
for j in range(2):
for k in range(2):
for l in range(2):
verts.append([x for x in base])
base[3] = base[3] * (-1)
base[2] = base[2] * (-1)
base[1] = base[1] * (-1)
base[0] = base[0] * (-1)
verts = verts + permutations([0, 0, 0, 1])
verts = verts + permutations([0, 0, 0, -1])
return Polyhedron(vertices=verts)
def Birkhoff_polytope(self, n):
"""
Return the Birkhoff polytope with n! vertices. Each vertex
is a (flattened) n by n permutation matrix.
INPUT:
- ``n`` -- a positive integer giving the size of the permutation matrices.
EXAMPLES::
sage: b3 = polytopes.Birkhoff_polytope(3)
sage: b3.n_vertices()
6
"""
perms = permutations(range(1, n + 1))
verts = []
for p in perms:
verts += [[Polytopes._pfunc(i, j, p) for j in range(1, n + 1) for i in range(1, n + 1)]]
return Polyhedron(vertices=verts)
def Birkhoff_polytope(self, n):
"""
Return the Birkhoff polytope with n! vertices. Each vertex
is a (flattened) n by n permutation matrix.
INPUT:
- ``n`` -- a positive integer giving the size of the permutation matrices.
EXAMPLES::
sage: b3 = polytopes.Birkhoff_polytope(3)
sage: b3.n_vertices()
6
"""
perms = permutations(range(1,n+1))
verts = []
for p in perms:
verts += [ [Polytopes._pfunc(i,j,p) for j in range(1,n+1)
for i in range(1,n+1) ] ]
return Polyhedron(vertices=verts)
