def platonic_icosahedron():
r"""Produce a triple consisting of a polyhedral version of the platonic icosahedron,
the associated cone surface, and a ConeSurfaceToPolyhedronMap from the surface
to the polyhedron.
EXAMPLES::
sage: from flatsurf.geometry.polyhedra import platonic_icosahedron
sage: polyhedron,surface,surface_to_polyhedron = platonic_icosahedron()
sage: TestSuite(surface).run()
r"""
vertices=[]
phi=AA(1+sqrt(5))/2
F=NumberField(phi.minpoly(),"phi",embedding=phi)
phi=F.gen()
for i in xrange(3):
for s1 in xrange(-1,3,2):
for s2 in xrange(-1,3,2):
p=3*[None]
p[i]=s1*phi
p[(i+1)%3]=s2
p[(i+2)%3]=0
vertices.append(vector(F,p))
p=Polyhedron(vertices=vertices)
s,m = polyhedron_to_cone_surface(p)
return p,s,m
def platonic_icosahedron():
r"""Produce a triple consisting of a polyhedral version of the platonic icosahedron,
the associated cone surface, and a ConeSurfaceToPolyhedronMap from the surface
to the polyhedron.
EXAMPLES::
sage: from flatsurf.geometry.polyhedra import platonic_icosahedron
sage: polyhedron,surface,surface_to_polyhedron = platonic_icosahedron()
sage: TestSuite(surface).run()
r"""
vertices = []
phi = AA(1 + sqrt(5)) / 2
F = NumberField(phi.minpoly(), "phi", embedding=phi)
phi = F.gen()
for i in range(3):
for s1 in range(-1, 3, 2):
for s2 in range(-1, 3, 2):
p = 3 * [None]
p[i] = s1 * phi
p[(i + 1) % 3] = s2
p[(i + 2) % 3] = 0
vertices.append(vector(F, p))
p = Polyhedron(vertices=vertices)
s, m = polyhedron_to_cone_surface(p)
return p, s, m
def platonic_dodecahedron():
r"""Produce a triple consisting of a polyhedral version of the platonic dodecahedron,
the associated cone surface, and a ConeSurfaceToPolyhedronMap from the surface
to the polyhedron.
EXAMPLES::
sage: from flatsurf.geometry.polyhedra import platonic_dodecahedron
sage: polyhedron,surface,surface_to_polyhedron = platonic_dodecahedron()
sage: TestSuite(surface).run()
r"""
vertices=[]
phi=AA(1+sqrt(5))/2
F=NumberField(phi.minpoly(),"phi",embedding=phi)
phi=F.gen()
for x in xrange(-1,3,2):
for y in xrange(-1,3,2):
for z in xrange(-1,3,2):
vertices.append(vector(F,(x,y,z)))
for x in xrange(-1,3,2):
for y in xrange(-1,3,2):
vertices.append(vector(F,(0,x*phi,y/phi)))
vertices.append(vector(F,(y/phi,0,x*phi)))
vertices.append(vector(F,(x*phi,y/phi,0)))
scale=AA(2/sqrt(1+(phi-1)**2+(1/phi-1)**2))
p=Polyhedron(vertices=vertices)
s,m = polyhedron_to_cone_surface(p,scaling_factor=scale)
return p,s,m
def platonic_dodecahedron():
r"""Produce a triple consisting of a polyhedral version of the platonic dodecahedron,
the associated cone surface, and a ConeSurfaceToPolyhedronMap from the surface
to the polyhedron.
EXAMPLES::
sage: from flatsurf.geometry.polyhedra import platonic_dodecahedron
sage: polyhedron,surface,surface_to_polyhedron = platonic_dodecahedron()
sage: TestSuite(surface).run()
r"""
vertices = []
phi = AA(1 + sqrt(5)) / 2
F = NumberField(phi.minpoly(), "phi", embedding=phi)
phi = F.gen()
for x in range(-1, 3, 2):
for y in range(-1, 3, 2):
for z in range(-1, 3, 2):
vertices.append(vector(F, (x, y, z)))
for x in range(-1, 3, 2):
for y in range(-1, 3, 2):
vertices.append(vector(F, (0, x * phi, y / phi)))
vertices.append(vector(F, (y / phi, 0, x * phi)))
vertices.append(vector(F, (x * phi, y / phi, 0)))
scale = AA(2 / sqrt(1 + (phi - 1)**2 + (1 / phi - 1)**2))
p = Polyhedron(vertices=vertices)
s, m = polyhedron_to_cone_surface(p, scaling_factor=scale)
return p, s, m