The OSCAR Book Companion
The OSCAR Book Companion
The preprint of this chapter is available here.
julia> points = [1 0; 1 1; 0 1; -1 0; -1 -1];
julia> P = convex_hull(points)
Polyhedron in ambient dimension 2
julia> facets(P)
5-element SubObjectIterator{AffineHalfspace{QQFieldElem}} over the Halfspaces of R^2 described by:
-x_1 <= 1
-x_1 + x_2 <= 1
x_1 - 2*x_2 <= 1
x_1 <= 1
x_2 <= 1
julia> f_vector(P)
2-element Vector{ZZRingElem}:
5
5
julia> volume(P)
5//2
julia> lattice_points(P)
6-element SubObjectIterator{PointVector{ZZRingElem}}:
[-1, -1]
[-1, 0]
[0, 0]
[0, 1]
[1, 0]
[1, 1]
julia> P = polyhedron([1 5; 2 -1; -1 0; 0 -1], [20,10,0,0])
Polyhedron in ambient dimension 2
julia> LP = linear_program(P, [1,1], convention=:max)
Linear program
max{c*x + k | x in P}
where P is a Polyhedron{QQFieldElem} and
c=Polymake.LibPolymake.Rational[1 1]
k=0
julia> optimal_value(LP)
100/11
julia> optimal_vertex(LP)
2-element PointVector{QQFieldElem}:
70//11
30//11
julia> T = tetrahedron()
Polytope in ambient dimension 3
julia> vertices(T)
4-element SubObjectIterator{PointVector{QQFieldElem}}:
[1, -1, -1]
[-1, 1, -1]
[-1, -1, 1]
[1, 1, 1]
julia> [ f_vector(P) for P in [T, cube(3), cross_polytope(3), dodecahedron(), icosahedron()] ]
5-element Vector{Vector{ZZRingElem}}:
[4, 6, 4]
[8, 12, 6]
[6, 12, 8]
[20, 30, 12]
[12, 30, 20]
julia> D = dodecahedron()
Polytope in ambient dimension 3 with EmbeddedAbsSimpleNumFieldElem type coefficients
julia> vertices(D)[1]
3-element PointVector{EmbeddedAbsSimpleNumFieldElem}:
1//2 (0.50)
1//4*sqrt(5) + 3//4 (1.31)
0 (0.00)
orbit_counts = MSet{Int}();
for k in 1:92
J = johnson_solid(k)
Aut = automorphism_group(J)[:on_vertices]
push!(orbit_counts, length(orbits(Aut)))
end
julia> sort(collect(orbit_counts.dict))
14-element Vector{Pair{Int64, Int64}}:
2 => 26
3 => 24
4 => 17
5 => 8
7 => 5
8 => 2
9 => 2
12 => 1
14 => 1
15 => 1
17 => 1
20 => 1
27 => 1
29 => 2
julia> P3 = orbit_polytope([1,2,3,4], symmetric_group(4))
Polyhedron in ambient dimension 4
julia> dim(P3)
3
julia> describe(combinatorial_symmetries(P3))
"C2 x S4"
julia> Q = convex_hull([0 0 0], [0 1 0; 0 0 1], [1 0 0])
Polyhedron in ambient dimension 3
julia> vertices(Q)
0-element SubObjectIterator{PointVector{QQFieldElem}}
julia> minimal_faces(Q)
(base_points = PointVector{QQFieldElem}[[0, 0, 0]], lineality_basis = RayVector{QQFieldElem}[[1, 0, 0]])
julia> Rays = [1 0; 1 1; 0 1; -1 0; 0 -1];
julia> Cones = IncidenceMatrix([[1,2], [2,3], [3,4], [4,5], [5,1]]);
julia> PF = polyhedral_fan(Cones, Rays)
Polyhedral fan in ambient dimension 2
julia> P = rand_spherical_polytope(6,30)
Polytope in ambient dimension 6
julia> show(f_vector(P))
ZZRingElem[30, 336, 1468, 2874, 2568, 856]
julia> show(h_vector(P))
ZZRingElem[1, 24, 201, 404, 201, 24, 1]
julia> show(g_vector(P))
ZZRingElem[1, 23, 177, 203]
n_vertices = 30;
n_samples = 100;
g_vectors = Array{Int32}(undef,n_samples,2);
for i=1:n_samples
RS = rand_spherical_polytope(6,n_vertices)
g = g_vector(RS)
g_vectors[i,1] = g[3] # notice index shift as Julia counts from 1
g_vectors[i,2] = g[4]
end
using Plots
scatter(g_vectors[:,1], g_vectors[:,2],
xlabel="g_2", ylabel="g_3", legend=false);
julia> min_g2 = minimum(g_vectors[:,1])
159
julia> max_g2 = maximum(g_vectors[:,1])
192
julia> show(upper_bound_g_vector(6,30))
[1, 23, 276, 2300]
julia> ub = [ Int(Polymake.polytope.pseudopower(g2,2)) for g2 in min_g2:max_g2 ]
34-element Vector{Int64}:
990
997
1005
1014
1024
1035
1047
1060
1074
1089
1105
1122
1140
1141
1143
1146
1150
1155
1161
1168
1176
1185
1195
1206
1218
1231
1245
1260
1276
1293
1311
1330
1331
1333
julia> lambda = Partition([3,1,1])
[3, 1, 1]
julia> GT = gelfand_tsetlin_polytope(lambda)
Polyhedron in ambient dimension 6
julia> lattice_points(GT)
6-element SubObjectIterator{PointVector{ZZRingElem}}:
[3, 1, 1, 1, 1, 1]
[3, 1, 1, 2, 1, 1]
[3, 1, 1, 2, 1, 2]
[3, 1, 1, 3, 1, 1]
[3, 1, 1, 3, 1, 2]
[3, 1, 1, 3, 1, 3]
julia> ehrhart_polynomial(GT)
2*x^2 + 3*x + 1
julia> volume(project_full(GT))
2
julia> w0 = @perm (1,3,2)
(1,3,2)
julia> genGT = gelfand_tsetlin_polytope(lambda,w0)
Polyhedron in ambient dimension 6
julia> lattice_points(genGT)
3-element SubObjectIterator{PointVector{ZZRingElem}}:
[3, 1, 1, 3, 1, 1]
[3, 1, 1, 3, 1, 2]
[3, 1, 1, 3, 1, 3]
julia> dc = demazure_character(lambda,w0)
x1^3*x2*x3 + x1^2*x2^2*x3 + x1*x2^3*x3
julia> dc(1,1,1)
3
julia> C = cube(3, 0, 1);
julia> R, x, c = polynomial_ring(QQ, :x=>1:3, :c=>(0:1,0:1,0:1));
julia> f = sum(prod(x[i]^Int(p[i]) for i=1:3)
* c[(Vector{Int}(p)+[1,1,1])...] for p=lattice_points(C))
x[1]*x[2]*x[3]*c[1, 1, 1] + x[1]*x[2]*c[1, 1, 0] + x[1]*x[3]*c[1, 0, 1] + x[1]*c[1, 0, 0] + x[2]*x[3]*c[0, 1, 1] + x[2]*c[0, 1, 0] + x[3]*c[0, 0, 1] + c[0, 0, 0]
julia> I = ideal(R, vcat([derivative(f,t) for t = x], [f]));
julia> D222 = eliminate(I,x)[1]
c[0, 0, 0]^2*c[1, 1, 1]^2 - 2*c[0, 0, 0]*c[1, 0, 0]*c[0, 1, 1]*c[1, 1, 1] - 2*c[0, 0, 0]*c[0, 1, 0]*c[1, 0, 1]*c[1, 1, 1] - 2*c[0, 0, 0]*c[1, 1, 0]*c[0, 0, 1]*c[1, 1, 1] + 4*c[0, 0, 0]*c[1, 1, 0]*c[1, 0, 1]*c[0, 1, 1] + c[1, 0, 0]^2*c[0, 1, 1]^2 + 4*c[1, 0, 0]*c[0, 1, 0]*c[0, 0, 1]*c[1, 1, 1] - 2*c[1, 0, 0]*c[0, 1, 0]*c[1, 0, 1]*c[0, 1, 1] - 2*c[1, 0, 0]*c[1, 1, 0]*c[0, 0, 1]*c[0, 1, 1] + c[0, 1, 0]^2*c[1, 0, 1]^2 - 2*c[0, 1, 0]*c[1, 1, 0]*c[0, 0, 1]*c[1, 0, 1] + c[1, 1, 0]^2*c[0, 0, 1]^2
julia> CubeFacets = [lattice_points(F) for F in faces(C,2)];
julia> VariableIndices = [[c+[1,1,1] for c=Vector{Vector{Int}}(F)]
for F=CubeFacets];
julia> FacialDeterminants = [c[v[1]...]*c[v[4]...]-c[v[2]...]*c[v[3]...]
for v=VariableIndices]
6-element Vector{QQMPolyRingElem}:
c[0, 0, 0]*c[0, 1, 1] - c[0, 1, 0]*c[0, 0, 1]
c[1, 0, 0]*c[1, 1, 1] - c[1, 1, 0]*c[1, 0, 1]
c[0, 0, 0]*c[1, 0, 1] - c[1, 0, 0]*c[0, 0, 1]
c[0, 1, 0]*c[1, 1, 1] - c[1, 1, 0]*c[0, 1, 1]
c[0, 0, 0]*c[1, 1, 0] - c[1, 0, 0]*c[0, 1, 0]
c[0, 0, 1]*c[1, 1, 1] - c[1, 0, 1]*c[0, 1, 1]
julia> E222 = D222*prod(FacialDeterminants);
julia> NP = newton_polytope(E222);
julia> dim(NP)
4
julia> show(f_vector(NP))
ZZRingElem[74, 152, 100, 22]
julia> C = cube(3,0,1)
Polytope in ambient dimension 3
julia> SP = secondary_polytope(C)
Polytope in ambient dimension 8
julia> V = point_matrix(vertices(NP))[:, 4:11].+1;
julia> SP == convex_hull(V)
true
julia> AllTriangulations = all_triangulations(C);
julia> Tri_as_SOP = [subdivision_of_points(C,T)
for T in AllTriangulations];
julia> GKZ_Vectors = [gkz_vector(T) for T in Tri_as_SOP];
julia> SP_from_GKZ = convex_hull(GKZ_Vectors)
Polyhedron in ambient dimension 8
julia> SP_from_GKZ == SP
true
julia> T = AllTriangulations[1]
6-element Vector{Vector{Int64}}:
[1, 2, 3, 5]
[2, 3, 4, 5]
[2, 4, 5, 6]
[3, 4, 5, 7]
[4, 5, 6, 7]
[4, 6, 7, 8]
julia> S = subdivision_of_points(C,T)
Subdivision of points in ambient dimension 3
julia> is_regular(S)
true
julia> show(min_weights(S))
[3, 1, 1, 0, 0, 0, 0, 1]
julia> IM = IncidenceMatrix(T);
julia> V = matrix(QQ, vertices(C));
julia> PC = polyhedral_complex(IM,V);
julia> visualize(PC)
julia> G = automorphism_group(C)[:on_vertices]
Permutation group of degree 8
julia> OrbitRepresentatives=
unique([minimum(gset(G,permuted,[v]))
for v in GKZ_Vectors])
6-element Vector{Vector{QQFieldElem}}:
[1, 3, 3, 5, 5, 3, 3, 1]
[1, 3, 3, 5, 6, 2, 2, 2]
[1, 3, 4, 4, 6, 2, 1, 3]
[1, 4, 4, 3, 6, 1, 1, 4]
[1, 5, 5, 1, 5, 1, 1, 5]
[2, 2, 2, 6, 6, 2, 2, 2]
julia> OrbitSizes =
[length(
filter(x->minimum(gset(G,permuted,[x]))==u,
GKZ_Vectors)
) for u in OrbitRepresentatives];
julia> show(OrbitSizes)
[12, 24, 24, 8, 2, 4]
julia> P = rand_spherical_polytope(6, 500; seed=11)
Polytope in ambient dimension 6
julia> Polymake.prefer("ppl") do
@time n_facets(P)
end
205.427732 seconds (7.13 G allocations: 321.966 GiB, 0.68% gc time)
58163
julia> P = rand_spherical_polytope(6, 500; seed=11);
julia> Polymake.prefer("beneath_beyond") do
@time n_facets(P)
end
61.090983 seconds (335.63 M allocations: 5.309 GiB, 0.08% gc time)
58163
julia> P = rand_spherical_polytope(6, 500; seed=11);
julia> Polymake.prefer("libnormaliz") do
@time n_facets(P)
end
11.167154 seconds (19.12 M allocations: 811.301 MiB, 0.07% gc time)
58163