def small_rhombicuboctahedron(self, exact=True, base_ring=None):
"""
Return the (small) rhombicuboctahedron.
The rhombicuboctahedron is an Archimedean solid with 24 vertices and 26
faces. See the :wikipedia:`Rhombicuboctahedron` for more information.
INPUT:
- ``exact`` -- (boolean, default ``True``) If ``False`` use an
approximate ring for the coordinates.
- ``base_ring`` -- the ring in which the coordinates will belong to. If
it is not provided and ``exact=True`` it will be a the number field
`\QQ[\phi]` where `\phi` is the golden ratio and if ``exact=False`` it
will be the real double field.
EXAMPLES::
sage: sr = polytopes.small_rhombicuboctahedron()
sage: sr.f_vector()
(1, 24, 48, 26, 1)
sage: sr.volume()
80/3*sqrt2 + 32
The faces are `8` equilateral triangles and `18` squares::
sage: sum(1 for f in sr.faces(2) if len(f.vertices()) == 3)
8
sage: sum(1 for f in sr.faces(2) if len(f.vertices()) == 4)
18
Its non exact version::
sage: sr = polytopes.small_rhombicuboctahedron(False)
sage: sr
A 3-dimensional polyhedron in RDF^3 defined as the convex hull of 24
vertices
sage: sr.f_vector()
(1, 24, 48, 26, 1)
"""
if base_ring is None and exact:
from sage.rings.number_field.number_field import QuadraticField
K = QuadraticField(2, 'sqrt2')
sqrt2 = K.gen()
base_ring = K
else:
if base_ring is None:
base_ring = RDF
sqrt2 = base_ring(2).sqrt()
one = base_ring.one()
a = sqrt2 + one
verts = []
verts.extend([s1*one, s2*one, s3*a] for s1,s2,s3 in itertools.product([1,-1], repeat=3))
verts.extend([s1*one, s3*a, s2*one] for s1,s2,s3 in itertools.product([1,-1], repeat=3))
verts.extend([s1*a, s2*one, s3*one] for s1,s2,s3 in itertools.product([1,-1], repeat=3))
return Polyhedron(vertices=verts)
def great_rhombicuboctahedron(self, exact=True, base_ring=None):
"""
Return the great rhombicuboctahedron.
The great rohombicuboctahedron (or truncated cuboctahedron) is an
Archimedean solid with 48 vertices and 26 faces. For more information
see the :wikipedia:`Truncated_cuboctahedron`.
INPUT:
- ``exact`` -- (boolean, default ``True``) If ``False`` use an
approximate ring for the coordinates.
- ``base_ring`` -- the ring in which the coordinates will belong to. If
it is not provided and ``exact=True`` it will be a the number field
`\QQ[\phi]` where `\phi` is the golden ratio and if ``exact=False`` it
will be the real double field.
EXAMPLES::
sage: gr = polytopes.great_rhombicuboctahedron() # long time ~ 3sec
sage: gr.f_vector() # long time
(1, 48, 72, 26, 1)
A faster implementation is obtained by setting ``exact=False``::
sage: gr = polytopes.great_rhombicuboctahedron(exact=False)
sage: gr.f_vector()
(1, 48, 72, 26, 1)
Its faces are 4 squares, 8 regular hexagons and 6 regular octagons::
sage: sum(1 for f in gr.faces(2) if len(f.vertices()) == 4)
12
sage: sum(1 for f in gr.faces(2) if len(f.vertices()) == 6)
8
sage: sum(1 for f in gr.faces(2) if len(f.vertices()) == 8)
6
"""
if base_ring is None and exact:
from sage.rings.number_field.number_field import QuadraticField
K = QuadraticField(2, 'sqrt2')
sqrt2 = K.gen()
base_ring = K
else:
if base_ring is None:
base_ring = RDF
sqrt2 = base_ring(2).sqrt()
one = base_ring.one()
v1 = sqrt2 + 1
v2 = 2*sqrt2 + 1
verts = [ [s1*z1, s2*z2, s3*z3]
for z1,z2,z3 in itertools.permutations([one,v1,v2])
for s1,s2,s3 in itertools.product([1,-1], repeat=3)]
return Polyhedron(vertices=verts, base_ring=base_ring)
def great_rhombicuboctahedron(self, exact=True, base_ring=None):
"""
Return the great rhombicuboctahedron.
The great rohombicuboctahedron (or truncated cuboctahedron) is an
Archimedean solid with 48 vertices and 26 faces. For more information
see the :wikipedia:`Truncated_cuboctahedron`.
INPUT:
- ``exact`` -- (boolean, default ``True``) If ``False`` use an
approximate ring for the coordinates.
- ``base_ring`` -- the ring in which the coordinates will belong to. If
it is not provided and ``exact=True`` it will be a the number field
`\QQ[\phi]` where `\phi` is the golden ratio and if ``exact=False`` it
will be the real double field.
EXAMPLES::
sage: gr = polytopes.great_rhombicuboctahedron() # long time ~ 3sec
sage: gr.f_vector() # long time
(1, 48, 72, 26, 1)
A faster implementation is obtained by setting ``exact=False``::
sage: gr = polytopes.great_rhombicuboctahedron(exact=False)
sage: gr.f_vector()
(1, 48, 72, 26, 1)
Its faces are 4 squares, 8 regular hexagons and 6 regular octagons::
sage: sum(1 for f in gr.faces(2) if len(f.vertices()) == 4)
12
sage: sum(1 for f in gr.faces(2) if len(f.vertices()) == 6)
8
sage: sum(1 for f in gr.faces(2) if len(f.vertices()) == 8)
6
"""
if base_ring is None and exact:
from sage.rings.number_field.number_field import QuadraticField
K = QuadraticField(2, 'sqrt2')
sqrt2 = K.gen()
base_ring = K
else:
if base_ring is None:
base_ring = RDF
sqrt2 = base_ring(2).sqrt()
one = base_ring.one()
v1 = sqrt2 + 1
v2 = 2 * sqrt2 + 1
verts = [[s1 * z1, s2 * z2, s3 * z3]
for z1, z2, z3 in itertools.permutations([one, v1, v2])
for s1, s2, s3 in itertools.product([1, -1], repeat=3)]
return Polyhedron(vertices=verts, base_ring=base_ring)
def six_hundred_cell(self, exact=False):
"""
Return the standard 600-cell polytope.
The 600-cell is a 4-dimensional regular polytope. In many ways this is
an analogue of the icosahedron.
.. WARNING::
The coordinates are not exact by default. The computation with exact
coordinates takes a huge amount of time.
INPUT:
- ``exact`` - (boolean, default ``False``) if ``True`` use exact
coordinates instead of floating point approximations
EXAMPLES::
sage: p600 = polytopes.six_hundred_cell()
sage: p600
A 4-dimensional polyhedron in RDF^4 defined as the convex hull of 120 vertices
sage: p600.f_vector()
(1, 120, 720, 1200, 600, 1)
Computation with exact coordinates is currently too long to be useful::
sage: p600 = polytopes.six_hundred_cell(exact=True) # not tested - very long time
sage: len(list(p600.bounded_edges())) # not tested - very long time
120
"""
if exact:
from sage.rings.number_field.number_field import QuadraticField
K = QuadraticField(5, 'sqrt5')
sqrt5 = K.gen()
g = (1 + sqrt5) / 2
base_ring = K
else:
g = (1 + RDF(5).sqrt()) / 2
base_ring = RDF
q12 = base_ring(1) / base_ring(2)
z = base_ring.zero()
verts = [[s1 * q12, s2 * q12, s3 * q12, s4 * q12]
for s1, s2, s3, s4 in itertools.product([1, -1], repeat=4)]
V = (base_ring)**4
verts.extend(V.basis())
verts.extend(-v for v in V.basis())
pts = [[s1 * q12, s2 * g / 2, s3 / (2 * g), z]
for (s1, s2, s3) in itertools.product([1, -1], repeat=3)]
for p in AlternatingGroup(4):
verts.extend(p(x) for x in pts)
return Polyhedron(vertices=verts, base_ring=base_ring)
def six_hundred_cell(self, exact=False):
"""
Return the standard 600-cell polytope.
The 600-cell is a 4-dimensional regular polytope. In many ways this is
an analogue of the icosahedron.
.. WARNING::
The coordinates are not exact by default. The computation with exact
coordinates takes a huge amount of time.
INPUT:
- ``exact`` - (boolean, default ``False``) if ``True`` use exact
coordinates instead of floating point approximations
EXAMPLES::
sage: p600 = polytopes.six_hundred_cell()
sage: p600
A 4-dimensional polyhedron in RDF^4 defined as the convex hull of 120 vertices
sage: p600.f_vector()
(1, 120, 720, 1200, 600, 1)
Computation with exact coordinates is currently too long to be useful::
sage: p600 = polytopes.six_hundred_cell(exact=True) # not tested - very long time
sage: len(list(p600.bounded_edges())) # not tested - very long time
120
"""
if exact:
from sage.rings.number_field.number_field import QuadraticField
K = QuadraticField(5, 'sqrt5')
sqrt5 = K.gen()
g = (1 + sqrt5) / 2
base_ring = K
else:
g = (1 + RDF(5).sqrt()) / 2
base_ring = RDF
q12 = base_ring(1) / base_ring(2)
z = base_ring.zero()
verts = [[s1*q12, s2*q12, s3*q12, s4*q12] for s1,s2,s3,s4 in itertools.product([1,-1], repeat=4)]
V = (base_ring)**4
verts.extend(V.basis())
verts.extend(-v for v in V.basis())
pts = [[s1 * q12, s2*g/2, s3/(2*g), z] for (s1,s2,s3) in itertools.product([1,-1], repeat=3)]
for p in AlternatingGroup(4):
verts.extend(p(x) for x in pts)
return Polyhedron(vertices=verts, base_ring=base_ring)
def is_quasigeometric(self):
"""
Decide whether the binary recurrence sequence is degenerate and similar to a geometric sequence,
i.e. the union of multiple geometric sequences, or geometric after term ``u0``.
If `\\alpha/\\beta` is a `k` th root of unity, where `k>1`, then necessarily `k = 2, 3, 4, 6`.
Then `F = [[0,1],[c,b]` is diagonalizable, and `F^k = [[\\alpha^k, 0], [0,\\beta^k]]` is scaler
matrix. Thus for all values of `j` mod `k`, the `j` mod `k` terms of `u_n` form a geometric
series.
If `\\alpha` or `\\beta` is zero, this implies that `c=0`. This is the case when `F` is
singular. In this case, `u_1, u_2, u_3, ...` is geometric.
EXAMPLES::
sage: S = BinaryRecurrenceSequence(0,1)
sage: [S(i) for i in range(10)]
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
sage: S.is_quasigeometric()
True
sage: R = BinaryRecurrenceSequence(3,0)
sage: [R(i) for i in range(10)]
[0, 1, 3, 9, 27, 81, 243, 729, 2187, 6561]
sage: R.is_quasigeometric()
True
"""
#First test if F is singular... i.e. beta = 0
if self.c == 0:
return True
#Otherwise test if alpha/beta is a root of unity that is not 1
else:
if (self.b**2 + 4 * self.c) != 0: #thus alpha/beta != 1
if (self.b**2 + 4 * self.c).is_square():
A = sqrt((self.b**2 + 4 * self.c))
else:
K = QuadraticField((self.b**2 + 4 * self.c), 'x')
A = K.gen()
if ((self.b + A) / (self.b - A))**(6) == 1:
return True
return False
def is_quasigeometric(self):
"""
Decide whether the binary recurrence sequence is degenerate and similar to a geometric sequence,
i.e. the union of multiple geometric sequences, or geometric after term ``u0``.
If `\\alpha/\\beta` is a `k` th root of unity, where `k>1`, then necessarily `k = 2, 3, 4, 6`.
Then `F = [[0,1],[c,b]` is diagonalizable, and `F^k = [[\\alpha^k, 0], [0,\\beta^k]]` is scaler
matrix. Thus for all values of `j` mod `k`, the `j` mod `k` terms of `u_n` form a geometric
series.
If `\\alpha` or `\\beta` is zero, this implies that `c=0`. This is the case when `F` is
singular. In this case, `u_1, u_2, u_3, ...` is geometric.
EXAMPLES::
sage: S = BinaryRecurrenceSequence(0,1)
sage: [S(i) for i in range(10)]
[0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
sage: S.is_quasigeometric()
True
sage: R = BinaryRecurrenceSequence(3,0)
sage: [R(i) for i in range(10)]
[0, 1, 3, 9, 27, 81, 243, 729, 2187, 6561]
sage: R.is_quasigeometric()
True
"""
#First test if F is singular... i.e. beta = 0
if self.c == 0:
return True
#Otherwise test if alpha/beta is a root of unity that is not 1
else:
if (self.b**2+4*self.c) != 0: #thus alpha/beta != 1
if (self.b**2+4*self.c).is_square():
A = sqrt((self.b**2+4*self.c))
else:
K = QuadraticField((self.b**2+4*self.c), 'x')
A = K.gen()
if ((self.b+A)/(self.b-A))**(6) == 1:
return True
return False
def icosidodecahedron(self, exact=True):
"""
Return the Icosidodecahedron
The Icosidodecahedron is a polyhedron with twenty triangular faces and
twelve pentagonal faces. For more information see the
:wikipedia:`Icosidodecahedron`.
INPUT:
- ``exact`` -- (boolean, default ``True``) If ``False`` use an
approximate ring for the coordinates.
EXAMPLES::
sage: gr = polytopes.icosidodecahedron()
sage: gr.f_vector()
(1, 30, 60, 32, 1)
TESTS::
sage: polytopes.icosidodecahedron(exact=False)
A 3-dimensional polyhedron in RDF^3 defined as the convex hull of 30 vertices
"""
from sage.rings.number_field.number_field import QuadraticField
from itertools import product
K = QuadraticField(5, 'sqrt5')
one = K.one()
phi = (one + K.gen()) / 2
gens = [((-1)**a * one / 2, (-1)**b * phi / 2,
(-1)**c * (one + phi) / 2)
for a, b, c in product([0, 1], repeat=3)]
gens.extend([(0, 0, phi), (0, 0, -phi)])
verts = []
for p in AlternatingGroup(3):
verts.extend(p(x) for x in gens)
if exact:
return Polyhedron(vertices=verts, base_ring=K)
else:
verts = [(RR(x), RR(y), RR(z)) for x, y, z in verts]
return Polyhedron(vertices=verts)
def icosidodecahedron(self, exact=True):
"""
Return the Icosidodecahedron
The Icosidodecahedron is a polyhedron with twenty triangular faces and
twelve pentagonal faces. For more information see the
:wikipedia:`Icosidodecahedron`.
INPUT:
- ``exact`` -- (boolean, default ``True``) If ``False`` use an
approximate ring for the coordinates.
EXAMPLES::
sage: gr = polytopes.icosidodecahedron()
sage: gr.f_vector()
(1, 30, 60, 32, 1)
TESTS::
sage: polytopes.icosidodecahedron(exact=False)
A 3-dimensional polyhedron in RDF^3 defined as the convex hull of 30 vertices
"""
from sage.rings.number_field.number_field import QuadraticField
from itertools import product
K = QuadraticField(5, 'sqrt5')
one = K.one()
phi = (one+K.gen())/2
gens = [((-1)**a*one/2, (-1)**b*phi/2, (-1)**c*(one+phi)/2)
for a,b,c in product([0,1],repeat=3)]
gens.extend([(0,0,phi), (0,0,-phi)])
verts = []
for p in AlternatingGroup(3):
verts.extend(p(x) for x in gens)
if exact:
return Polyhedron(vertices=verts,base_ring=K)
else:
verts = [(RR(x),RR(y),RR(z)) for x,y,z in verts]
return Polyhedron(vertices=verts)
def is_degenerate(self):
"""
Decide whether the binary recurrence sequence is degenerate.
Let `\\alpha` and `\\beta` denote the roots of the characteristic polynomial
`p(x) = x^2-bx -c`. Let `a = u_1-u_0\\beta/(\\beta - \\alpha)` and
`b = u_1-u_0\\alpha/(\\beta - \\alpha)`. The sequence is, thus, given by
`u_n = a \\alpha^n - b\\beta^n`. Then we say that the sequence is nondegenerate
if and only if `a*b*\\alpha*\\beta \\neq 0` and `\\alpha/\\beta` is not a
root of unity.
More concretely, there are 4 classes of degeneracy, that can all be formulated
in terms of the matrix `F = [[0,1], [c, b]]`.
- `F` is singular -- this corresponds to ``c`` = 0, and thus `\\alpha*\\beta = 0`. This sequence is geometric after term ``u0`` and so we call it ``quasigeometric``.
- `v = [[u_0], [u_1]]` is an eigenvector of `F` -- this corresponds to a ``geometric`` sequence with `a*b = 0`.
- `F` is nondiagonalizable -- this corresponds to `\\alpha = \\beta`. This sequence will be the point-wise product of an arithmetic and geometric sequence.
- `F^k` is scaler, for some `k>1` -- this corresponds to `\\alpha/\\beta` a `k` th root of unity. This sequence is a union of several geometric sequences, and so we again call it ``quasigeometric``.
EXAMPLES::
sage: S = BinaryRecurrenceSequence(0,1)
sage: S.is_degenerate()
True
sage: S.is_geometric()
False
sage: S.is_quasigeometric()
True
sage: R = BinaryRecurrenceSequence(3,-2)
sage: R.is_degenerate()
False
sage: T = BinaryRecurrenceSequence(2,-1)
sage: T.is_degenerate()
True
sage: T.is_arithmetic()
True
"""
if (self.b**2 + 4 * self.c) != 0:
if (self.b**2 + 4 * self.c).is_square():
A = sqrt((self.b**2 + 4 * self.c))
else:
K = QuadraticField((self.b**2 + 4 * self.c), 'x')
A = K.gen()
aa = (self.u1 - self.u0 *
(self.b + A) / 2) / (A) #called `a` in Docstring
bb = (self.u1 - self.u0 *
(self.b - A) / 2) / (A) #called `b` in Docstring
#(b+A)/2 is called alpha in Docstring, (b-A)/2 is called beta in Docstring
if (self.b - A) != 0:
if ((self.b + A) / (self.b - A))**(6) == 1:
return True
else:
return True
if aa * bb * (self.b + A) * (self.b - A) == 0:
return True
return False
return True
def icosahedron(self, exact=True, base_ring=None):
"""
Return an icosahedron with edge length 1.
The icosahedron is one of the Platonic sold. It has 20 faces and is dual
to the :meth:`dodecahedron`.
INPUT:
- ``exact`` -- (boolean, default ``True``) If ``False`` use an
approximate ring for the coordinates.
- ``base_ring`` -- (optional) the ring in which the coordinates will
belong to. Note that this ring must contain `\sqrt(5)`. If it is not
provided and ``exact=True`` it will be the number field
`\QQ[\sqrt(5)]` and if ``exact=False`` it will be the real double
field.
EXAMPLES::
sage: ico = polytopes.icosahedron()
sage: ico.f_vector()
(1, 12, 30, 20, 1)
sage: ico.volume()
5/12*sqrt5 + 5/4
Its non exact version::
sage: ico = polytopes.icosahedron(exact=False)
sage: ico.base_ring()
Real Double Field
sage: ico.volume()
2.1816949907715726
A version using `AA <sage.rings.qqbar.AlgebraicRealField>`::
sage: ico = polytopes.icosahedron(base_ring=AA) # long time
sage: ico.base_ring() # long time
Algebraic Real Field
sage: ico.volume() # long time
2.181694990624913?
Note that if base ring is provided it must contain the square root of
`5`. Otherwise you will get an error::
sage: polytopes.icosahedron(base_ring=QQ)
Traceback (most recent call last):
...
TypeError: unable to convert 1/4*sqrt(5) + 1/4 to a rational
"""
if base_ring is None and exact:
from sage.rings.number_field.number_field import QuadraticField
K = QuadraticField(5, 'sqrt5')
sqrt5 = K.gen()
g = (1 + sqrt5) / 2
base_ring = K
else:
if base_ring is None:
base_ring = RDF
g = (1 + base_ring(5).sqrt()) / 2
r12 = base_ring.one() / 2
z = base_ring.zero()
pts = [[z, s1*r12, s2*g/2] for s1,s2 in itertools.product([1,-1],repeat=2)]
verts = [p(v) for p in AlternatingGroup(3) for v in pts]
return Polyhedron(vertices=verts, base_ring=base_ring)
def is_degenerate(self):
"""
Decide whether the binary recurrence sequence is degenerate.
Let `\\alpha` and `\\beta` denote the roots of the characteristic polynomial
`p(x) = x^2-bx -c`. Let `a = u_1-u_0\\beta/(\\beta - \\alpha)` and
`b = u_1-u_0\\alpha/(\\beta - \\alpha)`. The sequence is, thus, given by
`u_n = a \\alpha^n - b\\beta^n`. Then we say that the sequence is nondegenerate
if and only if `a*b*\\alpha*\\beta \\neq 0` and `\\alpha/\\beta` is not a
root of unity.
More concretely, there are 4 classes of degeneracy, that can all be formulated
in terms of the matrix `F = [[0,1], [c, b]]`.
- `F` is singular -- this corresponds to ``c`` = 0, and thus `\\alpha*\\beta = 0`. This sequence is geometric after term ``u0`` and so we call it ``quasigeometric``.
- `v = [[u_0], [u_1]]` is an eigenvector of `F` -- this corresponds to a ``geometric`` sequence with `a*b = 0`.
- `F` is nondiagonalizable -- this corresponds to `\\alpha = \\beta`. This sequence will be the point-wise product of an arithmetic and geometric sequence.
- `F^k` is scaler, for some `k>1` -- this corresponds to `\\alpha/\\beta` a `k` th root of unity. This sequence is a union of several geometric sequences, and so we again call it ``quasigeometric``.
EXAMPLES::
sage: S = BinaryRecurrenceSequence(0,1)
sage: S.is_degenerate()
True
sage: S.is_geometric()
False
sage: S.is_quasigeometric()
True
sage: R = BinaryRecurrenceSequence(3,-2)
sage: R.is_degenerate()
False
sage: T = BinaryRecurrenceSequence(2,-1)
sage: T.is_degenerate()
True
sage: T.is_arithmetic()
True
"""
if (self.b**2+4*self.c) != 0:
if (self.b**2+4*self.c).is_square():
A = sqrt((self.b**2+4*self.c))
else:
K = QuadraticField((self.b**2+4*self.c), 'x')
A = K.gen()
aa = (self.u1 - self.u0*(self.b + A)/2)/(A) #called `a` in Docstring
bb = (self.u1 - self.u0*(self.b - A)/2)/(A) #called `b` in Docstring
#(b+A)/2 is called alpha in Docstring, (b-A)/2 is called beta in Docstring
if (self.b - A) != 0:
if ((self.b+A)/(self.b-A))**(6) == 1:
return True
else:
return True
if aa*bb*(self.b + A)*(self.b - A) == 0:
return True
return False
return True
def small_rhombicuboctahedron(self, exact=True, base_ring=None):
"""
Return the (small) rhombicuboctahedron.
The rhombicuboctahedron is an Archimedean solid with 24 vertices and 26
faces. See the :wikipedia:`Rhombicuboctahedron` for more information.
INPUT:
- ``exact`` -- (boolean, default ``True``) If ``False`` use an
approximate ring for the coordinates.
- ``base_ring`` -- the ring in which the coordinates will belong to. If
it is not provided and ``exact=True`` it will be a the number field
`\QQ[\phi]` where `\phi` is the golden ratio and if ``exact=False`` it
will be the real double field.
EXAMPLES::
sage: sr = polytopes.small_rhombicuboctahedron()
sage: sr.f_vector()
(1, 24, 48, 26, 1)
sage: sr.volume()
80/3*sqrt2 + 32
The faces are `8` equilateral triangles and `18` squares::
sage: sum(1 for f in sr.faces(2) if len(f.vertices()) == 3)
8
sage: sum(1 for f in sr.faces(2) if len(f.vertices()) == 4)
18
Its non exact version::
sage: sr = polytopes.small_rhombicuboctahedron(False)
sage: sr
A 3-dimensional polyhedron in RDF^3 defined as the convex hull of 24
vertices
sage: sr.f_vector()
(1, 24, 48, 26, 1)
"""
if base_ring is None and exact:
from sage.rings.number_field.number_field import QuadraticField
K = QuadraticField(2, 'sqrt2')
sqrt2 = K.gen()
base_ring = K
else:
if base_ring is None:
base_ring = RDF
sqrt2 = base_ring(2).sqrt()
one = base_ring.one()
a = sqrt2 + one
verts = []
verts.extend([s1 * one, s2 * one, s3 * a]
for s1, s2, s3 in itertools.product([1, -1], repeat=3))
verts.extend([s1 * one, s3 * a, s2 * one]
for s1, s2, s3 in itertools.product([1, -1], repeat=3))
verts.extend([s1 * a, s2 * one, s3 * one]
for s1, s2, s3 in itertools.product([1, -1], repeat=3))
return Polyhedron(vertices=verts)
def icosahedron(self, exact=True, base_ring=None):
"""
Return an icosahedron with edge length 1.
The icosahedron is one of the Platonic sold. It has 20 faces and is dual
to the :meth:`dodecahedron`.
INPUT:
- ``exact`` -- (boolean, default ``True``) If ``False`` use an
approximate ring for the coordinates.
- ``base_ring`` -- (optional) the ring in which the coordinates will
belong to. Note that this ring must contain `\sqrt(5)`. If it is not
provided and ``exact=True`` it will be the number field
`\QQ[\sqrt(5)]` and if ``exact=False`` it will be the real double
field.
EXAMPLES::
sage: ico = polytopes.icosahedron()
sage: ico.f_vector()
(1, 12, 30, 20, 1)
sage: ico.volume()
5/12*sqrt5 + 5/4
Its non exact version::
sage: ico = polytopes.icosahedron(exact=False)
sage: ico.base_ring()
Real Double Field
sage: ico.volume()
2.1816949907715726
A version using `AA <sage.rings.qqbar.AlgebraicRealField>`::
sage: ico = polytopes.icosahedron(base_ring=AA) # long time
sage: ico.base_ring() # long time
Algebraic Real Field
sage: ico.volume() # long time
2.181694990624913?
Note that if base ring is provided it must contain the square root of
`5`. Otherwise you will get an error::
sage: polytopes.icosahedron(base_ring=QQ)
Traceback (most recent call last):
...
TypeError: unable to convert 1/4*sqrt(5) + 1/4 to a rational
"""
if base_ring is None and exact:
from sage.rings.number_field.number_field import QuadraticField
K = QuadraticField(5, 'sqrt5')
sqrt5 = K.gen()
g = (1 + sqrt5) / 2
base_ring = K
else:
if base_ring is None:
base_ring = RDF
g = (1 + base_ring(5).sqrt()) / 2
r12 = base_ring.one() / 2
z = base_ring.zero()
pts = [[z, s1 * r12, s2 * g / 2]
for s1, s2 in itertools.product([1, -1], repeat=2)]
verts = [p(v) for p in AlternatingGroup(3) for v in pts]
return Polyhedron(vertices=verts, base_ring=base_ring)
