File: /home/vbraun/Sage/sage/local/lib/python2.6/site-packages/sage/geometry/polyhedra.py
Type: <type ‘instancemethod’>
Definition: cell24.f_vector()
Docstring:
Return the f-vector.
OUTPUT:
Returns a vector whose i-th entry is the number of i-dimensional faces of the polytope.
EXAMPLES:
sage: p = Polyhedron(vertices = [[1, 2, 3], [1, 3, 2], [2, 1, 3], [2, 3, 1], [3, 1, 2], [3, 2, 1], [0, 0, 0]]) sage: p.f_vector() (1, 7, 12, 7, 1)
Here is a picture of the 24-cell projected into 3 dimensions:
{{{id=1| cell24.plot() /// }}}The "round" 24-cell can be $GL(4,\mathbb{Q})$-squished into a lattice polytope:
{{{id=13| cell24 = Polyhedron(vertices=[ (1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1),(1,-1,-1,1),(0,0,-1,1), (0,-1,0,1),(-1,0,0,1),(1,0,0,-1),(0,1,0,-1),(0,0,1,-1),(-1,1,1,-1), (1,-1,-1,0),(0,0,-1,0),(0,-1,0,0),(-1,0,0,0),(1,-1,0,0),(1,0,-1,0), (0,1,1,-1),(-1,1,1,0),(-1,1,0,0),(-1,0,1,0),(0,-1,-1,1),(0,0,0,-1)]) cell24.f_vector() /// (1, 24, 96, 96, 24, 1) }}} {{{id=67| cell24.lattice_polytope().is_reflexive() /// True }}}Here is the symmetry group of the 24-cell
{{{id=27| Aut = cell24.restricted_automorphism_group() Aut.cardinality() /// 1152 }}} {{{id=34| Aut.gens() /// [(3,9)(4,18)(7,13)(8,14)(10,20)(22,24), (2,3)(6,7)(10,11)(14,15)(17,18)(21,22), (2,4)(3,19)(5,18)(7,10)(8,21)(9,17)(12,22)(14,23)(15,24), (2,18)(3,17)(4,5)(8,23)(9,19)(12,24)(13,20)(14,21)(15,22), (2,19)(3,4)(5,17)(6,11)(8,22)(9,18)(12,21)(14,24)(15,23), (1,2,19,10,12,24,16,15,23,7,5,4)(3,18,20,9,21,11,14,22,13,8,17,6), (1,16)(2,21)(3,22)(4,8)(5,23)(9,24)(12,19)(14,18)(15,17)] }}}Pick the following $G$-permutation action on the vertices of $\nabla$
{{{id=36| G = PermutationGroup([ '(1,14,22)(2,24,7)(3,18,16)(4,10,15)(5,21,11)(6,12,17)(8,19,13)(9,23,20)', '(1,10,16,7)(2,12,15,5)(3,9,14,8)(4,19,24,23)(6,20,11,13)(17,18,21,22)']) G.is_subgroup(Aut) and G.is_isomorphic( SL(2,3).as_matrix_group().as_permutation_group() ) /// True }}} {{{id=85| G.orbits() /// [[1, 10, 14, 16, 15, 8, 22, 7, 3, 5, 4, 19, 17, 2, 9, 18, 21, 24, 13, 6, 12, 23, 11, 20]] }}}Representatives for the 7 conjugacy classes:
{{{id=44| G.conjugacy_classes_representatives() /// [(), (1,2,21,16,15,17)(3,19,10,14,23,7)(4,20,12,24,13,5)(6,8,22,11,9,18), (1,4,8,16,24,9)(2,6,23,15,11,19)(3,20,21,14,13,17)(5,7,22,12,10,18), (1,6,16,11)(2,14,15,3)(4,21,24,17)(5,8,12,9)(7,20,10,13)(18,23,22,19), (1,8,24)(2,23,11)(3,21,13)(4,16,9)(5,22,10)(6,15,19)(7,12,18)(14,17,20), (1,16)(2,15)(3,14)(4,24)(5,12)(6,11)(7,10)(8,9)(13,20)(17,21)(18,22)(19,23), (1,21,15)(2,16,17)(3,10,23)(4,12,13)(5,20,24)(6,22,9)(7,19,14)(8,11,18)] }}}Here is the orbit of $1$ under the permutation $g_3$:
{{{id=52| (G.1).orbit(1) /// [1, 14, 22] }}}The polynomial $P=P_0+P_\infty$ is
{{{id=57| anticanonical_bundle = -Pnabla.K() P = sum(anticanonical_bundle.sections_monomials()) P([1]*24) # g_3 and g_4^2-fixed point z_i=1 /// 25 }}}One of the maximal cones is $\langle p_1,p_2,p_3,p_4,p_{19},p_{20}\rangle$:
{{{id=69| cone = fan.generating_cone(8) cone.ambient_ray_indices() /// (0, 1, 2, 3, 18, 19) }}}The singularity of $\mathbb{P}_\nabla$ is where the corresponding homogeneous variables vanish:
{{{id=63| P(0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,1) /// 1 }}}The patches are non-complete intersections: 9 equations in $\mathbb{C}^8$ cutting out a $4$-dimensional affine toric variety
{{{id=71| ambient_patch = Pnabla.affine_algebraic_patch(cone, names='x+') ambient_patch /// Closed subscheme of Affine Space of dimension 8 over Finite Field of size 101 defined by: -x1*x4 + x0*x7, -x1*x5 + x0*x6, -x4*x6 + x5*x7, x0*x4 - x3*x5, -x2*x4 + x3*x7, -x1*x4 + x3*x6, x0*x2 - x1*x3, x2*x6 - x1*x7, -x1*x4 + x2*x5 }}}Of course there is a singularity at $0\in \mathbb{C}^8$:
{{{id=76| ambient_patch.is_smooth() /// False }}}Now we throw in the equation $P=0$ and go to the same patch again:
{{{id=70| Xtilde = Pnabla.subscheme(P) patch = Xtilde.affine_algebraic_patch(cone, names='x+') patch /// Closed subscheme of Affine Space of dimension 8 over Finite Field of size 101 defined by: -x1*x4 + x0*x7, -x1*x5 + x0*x6, -x4*x6 + x5*x7, x0*x4 - x3*x5, -x2*x4 + x3*x7, -x1*x4 + x3*x6, x0*x2 - x1*x3, x2*x6 - x1*x7, -x1*x4 + x2*x5, x2^2*x5^2 + x0*x2*x5 + x2^2*x5 + x2*x3*x5 + x2*x4*x5 + x2*x5^2 + x0*x2*x6 + x2*x5*x6 + x2*x5*x7 + x0*x2 + x0*x4 + x2*x4 + x1*x5 + x2*x5 + x2*x6 + x5*x7 + x0 + x1 + x2 + x3 + x4 + x5 + x6 + x7 + 1 }}} {{{id=73| patch.is_smooth() /// True }}} {{{id=81| /// }}}