Mesh Decomposition
We can decompose meshes into overlapping submeshes using Subobjects from Catlab. Given a mesh and a partition function on its vertices, we can create a cover of the mesh by creating submeshes corresponding to each part of the partition. The opens in the cover are by taking the non of the negation of the subobject defined by the vertices in each part. These submeshes can overlap along their boundaries.
We can then compute intersections of these submeshes using the meet operation. This allows us to analyze how the submeshes interact and overlap. The nerve of the cover can also be constructed, which encodes the combinatorial structure of the overlaps between the submeshes.
The idea is to use this nerve to build sheaves of vector spaces over the mesh decomposition, allowing for algebraic analysis of functions defined on the mesh.
using CombinatorialSpaces
using GeometryBasics: Point3d
using LinearAlgebra: norm
using CairoMakie
using Catlab
using Catlab.CategoricalAlgebra: ACSetCategory, ACSetCat
using Catlab.CategoricalAlgebra.Subobjects: Subobject, negate, non, meet, join
using Catlab.Theories: @withmodel
using Catlab.BasicSets: FinFunctionWe are going to draw our cover by drawing all the submeshes in orange and the total mesh in blue. The cover submeshes will be drawn on the top left triangle of a grid of plots. The diagonal entries are the individual submeshes, and the upper triangle entries are their pairwise intersections. A pairwise intersection will be empty if the two submeshes do not overlap, and the corresponding plot will be completely blue.
function draw(mesh; color=:blue)
f = Figure()
ax = CairoMakie.Axis(f[1,1], aspect=1)
draw!(ax, mesh, color=color)
return f
end
function draw!(ax, mesh::EmbeddedDeltaDualComplex1D; color=:blue)
ax = scatter!(ax, mesh[:point], color=color)
end
function draw!(ax, mesh::Union{EmbeddedDeltaSet2D,EmbeddedDeltaDualComplex2D}; color=:blue)
wireframe!(ax, mesh, color=color)
return ax
end
function draw!(ax, submesh::Subobject; color=:orange)
ϕ = hom(submesh)
@show nparts(dom(ϕ), :V)
@show nparts(codom(ϕ), :V)
draw!(ax, codom(ϕ),color=color)
draw!(ax, dom(ϕ), color=:orange)
end
function draw(cover::Vector{T}; color=:blue, cat) where T <: Subobject
f = Figure()
n = length(cover)
for i in 1:n
for j in i:n
ax = CairoMakie.Axis(f[i,j])
ui,uj = cover[i], cover[j]
@withmodel cat (meet,) begin
draw!(ax, meet(ui,uj), color=color)
end
end
end
f
enddraw (generic function with 2 methods)Example 1: Quadrants of a Rectangle
First we create a triangulated grid mesh and its dual complex. Our partition function will divide the mesh into four quadrants that overlap along their boundaries.
s = triangulated_grid(100,100,15,15,Point3d);
sd = EmbeddedDeltaDualComplex2D{Bool,Float64,Point3d}(s);
subdivide_duals!(sd, Barycenter());
# Create a category instance for working with Subobjects
# Pass the ACSet instance to ACSetCat constructor (Catlab 0.17 API)
const 𝒞 = ACSetCategory(ACSetCat(s))
f = draw(sd)
f
quadrants(x) = Int(x[1] > 50) + 2*Int(x[2] > 50)
function cover_mesh(partition_function, s, cat)
vertex_partition = map(partition_function, s[:point])
parts = map(unique(vertex_partition)) do p
vp = findall(i->i==p, vertex_partition)
subobj = Subobject(s; V=vp)
@withmodel cat (negate, non) begin
sp = non(negate(subobj))
end
end
return parts
end
quads = cover_mesh(quadrants, s, 𝒞)
q = quads[1]
draw(q)
We can look at the individual submeshes in the cover, their joins, and their intersections.
draw(quads[3])
@withmodel 𝒞 (join,) begin
q = join(quads[1], quads[3])
end
draw(q)
@withmodel 𝒞 (meet,) begin
draw(meet(quads[1], quads[2]))
end
draw(quads; cat=𝒞)
The nerve of the cover can be constructed by computing all pairwise intersections of the submeshes in the cover.
using Catlab.FreeDiagrams
function nerve(cover::Vector{T}, cat) where T <: Subobject
n = length(cover)
@withmodel cat (meet,) begin
map(1:n) do i
map(i:n) do j
ui,uj = cover[i], cover[j]
uij = meet(ui,uj)
(uij, i, j)
end
end |> Iterators.flatten
end
end
D = nerve(quads, 𝒞)Base.Iterators.Flatten{Vector{Vector{Tuple{Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}, Int64, Int64}}}}(Vector{Tuple{Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}, Int64, Int64}}[[(Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}(EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}:
V = 1:49
E = 1:120
Tri = 1:72
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 8, 8, 9, 9, 3, 9, 10, 10, 4, 10, 11, 11, 5, 11, 12, 12, 6, 12, 13, 13, 7, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49]
∂v1 : E → V = [1, 2, 1, 8, 2, 2, 3, 9, 3, 3, 4, 10, 4, 4, 5, 11, 5, 5, 6, 12, 6, 6, 7, 13, 7, 8, 15, 8, 9, 16, 9, 10, 17, 10, 11, 18, 11, 12, 19, 12, 13, 20, 13, 14, 16, 15, 22, 16, 17, 23, 17, 18, 24, 18, 19, 25, 19, 20, 26, 20, 21, 27, 21, 22, 29, 22, 23, 30, 23, 24, 31, 24, 25, 32, 25, 26, 33, 26, 27, 34, 27, 28, 30, 29, 36, 30, 31, 37, 31, 32, 38, 32, 33, 39, 33, 34, 40, 34, 35, 41, 35, 36, 43, 36, 37, 44, 37, 38, 45, 38, 39, 46, 39, 40, 47, 40, 41, 48, 41, 42]
∂e0 : Tri → E = [2, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120]
∂e1 : Tri → E = [3, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 48, 48, 51, 51, 54, 54, 57, 57, 60, 60, 63, 66, 66, 69, 69, 72, 72, 75, 75, 78, 78, 81, 81, 84, 86, 86, 89, 89, 92, 92, 95, 95, 98, 98, 101, 104, 104, 107, 107, 110, 110, 113, 113, 116, 116, 119, 119]
∂e2 : Tri → E = [1, 2, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 4, 29, 8, 32, 12, 35, 16, 38, 20, 41, 24, 27, 45, 30, 49, 33, 52, 36, 55, 39, 58, 42, 61, 64, 47, 67, 50, 70, 53, 73, 56, 76, 59, 79, 62, 65, 83, 68, 87, 71, 90, 74, 93, 77, 96, 80, 99, 102, 85, 105, 88, 108, 91, 111, 94, 114, 97, 117, 100]
edge_orientation : E → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
tri_orientation : Tri → Orientation = Bool[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
point : V → Point = GeometryBasics.Point{3, Float64}[[0.0, 0.0, 0.0], [13.953488372093023, 0.0, 0.0], [27.906976744186046, 0.0, 0.0], [41.860465116279066, 0.0, 0.0], [55.81395348837209, 0.0, 0.0], [69.76744186046511, 0.0, 0.0], [83.72093023255813, 0.0, 0.0], [6.976744186046512, 15.0, 0.0], [20.930232558139533, 15.0, 0.0], [34.883720930232556, 15.0, 0.0], [48.83720930232558, 15.0, 0.0], [62.7906976744186, 15.0, 0.0], [76.74418604651163, 15.0, 0.0], [90.69767441860465, 15.0, 0.0], [0.0, 30.0, 0.0], [13.953488372093023, 30.0, 0.0], [27.906976744186046, 30.0, 0.0], [41.860465116279066, 30.0, 0.0], [55.81395348837209, 30.0, 0.0], [69.76744186046511, 30.0, 0.0], [83.72093023255813, 30.0, 0.0], [6.976744186046512, 45.0, 0.0], [20.930232558139533, 45.0, 0.0], [34.883720930232556, 45.0, 0.0], [48.83720930232558, 45.0, 0.0], [62.7906976744186, 45.0, 0.0], [76.74418604651163, 45.0, 0.0], [90.69767441860465, 45.0, 0.0], [0.0, 60.0, 0.0], [13.953488372093023, 60.0, 0.0], [27.906976744186046, 60.0, 0.0], [41.860465116279066, 60.0, 0.0], [55.81395348837209, 60.0, 0.0], [69.76744186046511, 60.0, 0.0], [83.72093023255813, 60.0, 0.0], [6.976744186046512, 75.0, 0.0], [20.930232558139533, 75.0, 0.0], [34.883720930232556, 75.0, 0.0], [48.83720930232558, 75.0, 0.0], [62.7906976744186, 75.0, 0.0], [76.74418604651163, 75.0, 0.0], [90.69767441860465, 75.0, 0.0], [0.0, 90.0, 0.0], [13.953488372093023, 90.0, 0.0], [27.906976744186046, 90.0, 0.0], [41.860465116279066, 90.0, 0.0], [55.81395348837209, 90.0, 0.0], [69.76744186046511, 90.0, 0.0], [83.72093023255813, 90.0, 0.0]], (V = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(49), Bool[1, 1, 1, 1, 1, 0, 0, 1, 1, 1 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), E = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(120), Bool[1, 1, 1, 1, 1, 1, 1, 1, 1, 1 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Tri = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(72), Bool[1, 1, 1, 1, 1, 1, 1, 1, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Orientation = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]), Point = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]))), 1, 1), (Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}(EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}:
V = 1:49
E = 1:120
Tri = 1:72
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 8, 8, 9, 9, 3, 9, 10, 10, 4, 10, 11, 11, 5, 11, 12, 12, 6, 12, 13, 13, 7, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49]
∂v1 : E → V = [1, 2, 1, 8, 2, 2, 3, 9, 3, 3, 4, 10, 4, 4, 5, 11, 5, 5, 6, 12, 6, 6, 7, 13, 7, 8, 15, 8, 9, 16, 9, 10, 17, 10, 11, 18, 11, 12, 19, 12, 13, 20, 13, 14, 16, 15, 22, 16, 17, 23, 17, 18, 24, 18, 19, 25, 19, 20, 26, 20, 21, 27, 21, 22, 29, 22, 23, 30, 23, 24, 31, 24, 25, 32, 25, 26, 33, 26, 27, 34, 27, 28, 30, 29, 36, 30, 31, 37, 31, 32, 38, 32, 33, 39, 33, 34, 40, 34, 35, 41, 35, 36, 43, 36, 37, 44, 37, 38, 45, 38, 39, 46, 39, 40, 47, 40, 41, 48, 41, 42]
∂e0 : Tri → E = [2, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120]
∂e1 : Tri → E = [3, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 48, 48, 51, 51, 54, 54, 57, 57, 60, 60, 63, 66, 66, 69, 69, 72, 72, 75, 75, 78, 78, 81, 81, 84, 86, 86, 89, 89, 92, 92, 95, 95, 98, 98, 101, 104, 104, 107, 107, 110, 110, 113, 113, 116, 116, 119, 119]
∂e2 : Tri → E = [1, 2, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 4, 29, 8, 32, 12, 35, 16, 38, 20, 41, 24, 27, 45, 30, 49, 33, 52, 36, 55, 39, 58, 42, 61, 64, 47, 67, 50, 70, 53, 73, 56, 76, 59, 79, 62, 65, 83, 68, 87, 71, 90, 74, 93, 77, 96, 80, 99, 102, 85, 105, 88, 108, 91, 111, 94, 114, 97, 117, 100]
edge_orientation : E → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
tri_orientation : Tri → Orientation = Bool[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
point : V → Point = GeometryBasics.Point{3, Float64}[[0.0, 0.0, 0.0], [13.953488372093023, 0.0, 0.0], [27.906976744186046, 0.0, 0.0], [41.860465116279066, 0.0, 0.0], [55.81395348837209, 0.0, 0.0], [69.76744186046511, 0.0, 0.0], [83.72093023255813, 0.0, 0.0], [6.976744186046512, 15.0, 0.0], [20.930232558139533, 15.0, 0.0], [34.883720930232556, 15.0, 0.0], [48.83720930232558, 15.0, 0.0], [62.7906976744186, 15.0, 0.0], [76.74418604651163, 15.0, 0.0], [90.69767441860465, 15.0, 0.0], [0.0, 30.0, 0.0], [13.953488372093023, 30.0, 0.0], [27.906976744186046, 30.0, 0.0], [41.860465116279066, 30.0, 0.0], [55.81395348837209, 30.0, 0.0], [69.76744186046511, 30.0, 0.0], [83.72093023255813, 30.0, 0.0], [6.976744186046512, 45.0, 0.0], [20.930232558139533, 45.0, 0.0], [34.883720930232556, 45.0, 0.0], [48.83720930232558, 45.0, 0.0], [62.7906976744186, 45.0, 0.0], [76.74418604651163, 45.0, 0.0], [90.69767441860465, 45.0, 0.0], [0.0, 60.0, 0.0], [13.953488372093023, 60.0, 0.0], [27.906976744186046, 60.0, 0.0], [41.860465116279066, 60.0, 0.0], [55.81395348837209, 60.0, 0.0], [69.76744186046511, 60.0, 0.0], [83.72093023255813, 60.0, 0.0], [6.976744186046512, 75.0, 0.0], [20.930232558139533, 75.0, 0.0], [34.883720930232556, 75.0, 0.0], [48.83720930232558, 75.0, 0.0], [62.7906976744186, 75.0, 0.0], [76.74418604651163, 75.0, 0.0], [90.69767441860465, 75.0, 0.0], [0.0, 90.0, 0.0], [13.953488372093023, 90.0, 0.0], [27.906976744186046, 90.0, 0.0], [41.860465116279066, 90.0, 0.0], [55.81395348837209, 90.0, 0.0], [69.76744186046511, 90.0, 0.0], [83.72093023255813, 90.0, 0.0]], (V = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(49), Bool[0, 0, 0, 1, 1, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), E = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(120), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Tri = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(72), Bool[0, 0, 0, 0, 0, 0, 1, 1, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Orientation = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]), Point = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]))), 1, 2), (Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}(EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}:
V = 1:49
E = 1:120
Tri = 1:72
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 8, 8, 9, 9, 3, 9, 10, 10, 4, 10, 11, 11, 5, 11, 12, 12, 6, 12, 13, 13, 7, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49]
∂v1 : E → V = [1, 2, 1, 8, 2, 2, 3, 9, 3, 3, 4, 10, 4, 4, 5, 11, 5, 5, 6, 12, 6, 6, 7, 13, 7, 8, 15, 8, 9, 16, 9, 10, 17, 10, 11, 18, 11, 12, 19, 12, 13, 20, 13, 14, 16, 15, 22, 16, 17, 23, 17, 18, 24, 18, 19, 25, 19, 20, 26, 20, 21, 27, 21, 22, 29, 22, 23, 30, 23, 24, 31, 24, 25, 32, 25, 26, 33, 26, 27, 34, 27, 28, 30, 29, 36, 30, 31, 37, 31, 32, 38, 32, 33, 39, 33, 34, 40, 34, 35, 41, 35, 36, 43, 36, 37, 44, 37, 38, 45, 38, 39, 46, 39, 40, 47, 40, 41, 48, 41, 42]
∂e0 : Tri → E = [2, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120]
∂e1 : Tri → E = [3, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 48, 48, 51, 51, 54, 54, 57, 57, 60, 60, 63, 66, 66, 69, 69, 72, 72, 75, 75, 78, 78, 81, 81, 84, 86, 86, 89, 89, 92, 92, 95, 95, 98, 98, 101, 104, 104, 107, 107, 110, 110, 113, 113, 116, 116, 119, 119]
∂e2 : Tri → E = [1, 2, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 4, 29, 8, 32, 12, 35, 16, 38, 20, 41, 24, 27, 45, 30, 49, 33, 52, 36, 55, 39, 58, 42, 61, 64, 47, 67, 50, 70, 53, 73, 56, 76, 59, 79, 62, 65, 83, 68, 87, 71, 90, 74, 93, 77, 96, 80, 99, 102, 85, 105, 88, 108, 91, 111, 94, 114, 97, 117, 100]
edge_orientation : E → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
tri_orientation : Tri → Orientation = Bool[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
point : V → Point = GeometryBasics.Point{3, Float64}[[0.0, 0.0, 0.0], [13.953488372093023, 0.0, 0.0], [27.906976744186046, 0.0, 0.0], [41.860465116279066, 0.0, 0.0], [55.81395348837209, 0.0, 0.0], [69.76744186046511, 0.0, 0.0], [83.72093023255813, 0.0, 0.0], [6.976744186046512, 15.0, 0.0], [20.930232558139533, 15.0, 0.0], [34.883720930232556, 15.0, 0.0], [48.83720930232558, 15.0, 0.0], [62.7906976744186, 15.0, 0.0], [76.74418604651163, 15.0, 0.0], [90.69767441860465, 15.0, 0.0], [0.0, 30.0, 0.0], [13.953488372093023, 30.0, 0.0], [27.906976744186046, 30.0, 0.0], [41.860465116279066, 30.0, 0.0], [55.81395348837209, 30.0, 0.0], [69.76744186046511, 30.0, 0.0], [83.72093023255813, 30.0, 0.0], [6.976744186046512, 45.0, 0.0], [20.930232558139533, 45.0, 0.0], [34.883720930232556, 45.0, 0.0], [48.83720930232558, 45.0, 0.0], [62.7906976744186, 45.0, 0.0], [76.74418604651163, 45.0, 0.0], [90.69767441860465, 45.0, 0.0], [0.0, 60.0, 0.0], [13.953488372093023, 60.0, 0.0], [27.906976744186046, 60.0, 0.0], [41.860465116279066, 60.0, 0.0], [55.81395348837209, 60.0, 0.0], [69.76744186046511, 60.0, 0.0], [83.72093023255813, 60.0, 0.0], [6.976744186046512, 75.0, 0.0], [20.930232558139533, 75.0, 0.0], [34.883720930232556, 75.0, 0.0], [48.83720930232558, 75.0, 0.0], [62.7906976744186, 75.0, 0.0], [76.74418604651163, 75.0, 0.0], [90.69767441860465, 75.0, 0.0], [0.0, 90.0, 0.0], [13.953488372093023, 90.0, 0.0], [27.906976744186046, 90.0, 0.0], [41.860465116279066, 90.0, 0.0], [55.81395348837209, 90.0, 0.0], [69.76744186046511, 90.0, 0.0], [83.72093023255813, 90.0, 0.0]], (V = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(49), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), E = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(120), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Tri = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(72), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Orientation = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]), Point = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]))), 1, 3), (Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}(EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}:
V = 1:49
E = 1:120
Tri = 1:72
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 8, 8, 9, 9, 3, 9, 10, 10, 4, 10, 11, 11, 5, 11, 12, 12, 6, 12, 13, 13, 7, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49]
∂v1 : E → V = [1, 2, 1, 8, 2, 2, 3, 9, 3, 3, 4, 10, 4, 4, 5, 11, 5, 5, 6, 12, 6, 6, 7, 13, 7, 8, 15, 8, 9, 16, 9, 10, 17, 10, 11, 18, 11, 12, 19, 12, 13, 20, 13, 14, 16, 15, 22, 16, 17, 23, 17, 18, 24, 18, 19, 25, 19, 20, 26, 20, 21, 27, 21, 22, 29, 22, 23, 30, 23, 24, 31, 24, 25, 32, 25, 26, 33, 26, 27, 34, 27, 28, 30, 29, 36, 30, 31, 37, 31, 32, 38, 32, 33, 39, 33, 34, 40, 34, 35, 41, 35, 36, 43, 36, 37, 44, 37, 38, 45, 38, 39, 46, 39, 40, 47, 40, 41, 48, 41, 42]
∂e0 : Tri → E = [2, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120]
∂e1 : Tri → E = [3, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 48, 48, 51, 51, 54, 54, 57, 57, 60, 60, 63, 66, 66, 69, 69, 72, 72, 75, 75, 78, 78, 81, 81, 84, 86, 86, 89, 89, 92, 92, 95, 95, 98, 98, 101, 104, 104, 107, 107, 110, 110, 113, 113, 116, 116, 119, 119]
∂e2 : Tri → E = [1, 2, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 4, 29, 8, 32, 12, 35, 16, 38, 20, 41, 24, 27, 45, 30, 49, 33, 52, 36, 55, 39, 58, 42, 61, 64, 47, 67, 50, 70, 53, 73, 56, 76, 59, 79, 62, 65, 83, 68, 87, 71, 90, 74, 93, 77, 96, 80, 99, 102, 85, 105, 88, 108, 91, 111, 94, 114, 97, 117, 100]
edge_orientation : E → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
tri_orientation : Tri → Orientation = Bool[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
point : V → Point = GeometryBasics.Point{3, Float64}[[0.0, 0.0, 0.0], [13.953488372093023, 0.0, 0.0], [27.906976744186046, 0.0, 0.0], [41.860465116279066, 0.0, 0.0], [55.81395348837209, 0.0, 0.0], [69.76744186046511, 0.0, 0.0], [83.72093023255813, 0.0, 0.0], [6.976744186046512, 15.0, 0.0], [20.930232558139533, 15.0, 0.0], [34.883720930232556, 15.0, 0.0], [48.83720930232558, 15.0, 0.0], [62.7906976744186, 15.0, 0.0], [76.74418604651163, 15.0, 0.0], [90.69767441860465, 15.0, 0.0], [0.0, 30.0, 0.0], [13.953488372093023, 30.0, 0.0], [27.906976744186046, 30.0, 0.0], [41.860465116279066, 30.0, 0.0], [55.81395348837209, 30.0, 0.0], [69.76744186046511, 30.0, 0.0], [83.72093023255813, 30.0, 0.0], [6.976744186046512, 45.0, 0.0], [20.930232558139533, 45.0, 0.0], [34.883720930232556, 45.0, 0.0], [48.83720930232558, 45.0, 0.0], [62.7906976744186, 45.0, 0.0], [76.74418604651163, 45.0, 0.0], [90.69767441860465, 45.0, 0.0], [0.0, 60.0, 0.0], [13.953488372093023, 60.0, 0.0], [27.906976744186046, 60.0, 0.0], [41.860465116279066, 60.0, 0.0], [55.81395348837209, 60.0, 0.0], [69.76744186046511, 60.0, 0.0], [83.72093023255813, 60.0, 0.0], [6.976744186046512, 75.0, 0.0], [20.930232558139533, 75.0, 0.0], [34.883720930232556, 75.0, 0.0], [48.83720930232558, 75.0, 0.0], [62.7906976744186, 75.0, 0.0], [76.74418604651163, 75.0, 0.0], [90.69767441860465, 75.0, 0.0], [0.0, 90.0, 0.0], [13.953488372093023, 90.0, 0.0], [27.906976744186046, 90.0, 0.0], [41.860465116279066, 90.0, 0.0], [55.81395348837209, 90.0, 0.0], [69.76744186046511, 90.0, 0.0], [83.72093023255813, 90.0, 0.0]], (V = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(49), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), E = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(120), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Tri = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(72), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Orientation = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]), Point = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]))), 1, 4)], [(Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}(EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}:
V = 1:49
E = 1:120
Tri = 1:72
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 8, 8, 9, 9, 3, 9, 10, 10, 4, 10, 11, 11, 5, 11, 12, 12, 6, 12, 13, 13, 7, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49]
∂v1 : E → V = [1, 2, 1, 8, 2, 2, 3, 9, 3, 3, 4, 10, 4, 4, 5, 11, 5, 5, 6, 12, 6, 6, 7, 13, 7, 8, 15, 8, 9, 16, 9, 10, 17, 10, 11, 18, 11, 12, 19, 12, 13, 20, 13, 14, 16, 15, 22, 16, 17, 23, 17, 18, 24, 18, 19, 25, 19, 20, 26, 20, 21, 27, 21, 22, 29, 22, 23, 30, 23, 24, 31, 24, 25, 32, 25, 26, 33, 26, 27, 34, 27, 28, 30, 29, 36, 30, 31, 37, 31, 32, 38, 32, 33, 39, 33, 34, 40, 34, 35, 41, 35, 36, 43, 36, 37, 44, 37, 38, 45, 38, 39, 46, 39, 40, 47, 40, 41, 48, 41, 42]
∂e0 : Tri → E = [2, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120]
∂e1 : Tri → E = [3, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 48, 48, 51, 51, 54, 54, 57, 57, 60, 60, 63, 66, 66, 69, 69, 72, 72, 75, 75, 78, 78, 81, 81, 84, 86, 86, 89, 89, 92, 92, 95, 95, 98, 98, 101, 104, 104, 107, 107, 110, 110, 113, 113, 116, 116, 119, 119]
∂e2 : Tri → E = [1, 2, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 4, 29, 8, 32, 12, 35, 16, 38, 20, 41, 24, 27, 45, 30, 49, 33, 52, 36, 55, 39, 58, 42, 61, 64, 47, 67, 50, 70, 53, 73, 56, 76, 59, 79, 62, 65, 83, 68, 87, 71, 90, 74, 93, 77, 96, 80, 99, 102, 85, 105, 88, 108, 91, 111, 94, 114, 97, 117, 100]
edge_orientation : E → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
tri_orientation : Tri → Orientation = Bool[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
point : V → Point = GeometryBasics.Point{3, Float64}[[0.0, 0.0, 0.0], [13.953488372093023, 0.0, 0.0], [27.906976744186046, 0.0, 0.0], [41.860465116279066, 0.0, 0.0], [55.81395348837209, 0.0, 0.0], [69.76744186046511, 0.0, 0.0], [83.72093023255813, 0.0, 0.0], [6.976744186046512, 15.0, 0.0], [20.930232558139533, 15.0, 0.0], [34.883720930232556, 15.0, 0.0], [48.83720930232558, 15.0, 0.0], [62.7906976744186, 15.0, 0.0], [76.74418604651163, 15.0, 0.0], [90.69767441860465, 15.0, 0.0], [0.0, 30.0, 0.0], [13.953488372093023, 30.0, 0.0], [27.906976744186046, 30.0, 0.0], [41.860465116279066, 30.0, 0.0], [55.81395348837209, 30.0, 0.0], [69.76744186046511, 30.0, 0.0], [83.72093023255813, 30.0, 0.0], [6.976744186046512, 45.0, 0.0], [20.930232558139533, 45.0, 0.0], [34.883720930232556, 45.0, 0.0], [48.83720930232558, 45.0, 0.0], [62.7906976744186, 45.0, 0.0], [76.74418604651163, 45.0, 0.0], [90.69767441860465, 45.0, 0.0], [0.0, 60.0, 0.0], [13.953488372093023, 60.0, 0.0], [27.906976744186046, 60.0, 0.0], [41.860465116279066, 60.0, 0.0], [55.81395348837209, 60.0, 0.0], [69.76744186046511, 60.0, 0.0], [83.72093023255813, 60.0, 0.0], [6.976744186046512, 75.0, 0.0], [20.930232558139533, 75.0, 0.0], [34.883720930232556, 75.0, 0.0], [48.83720930232558, 75.0, 0.0], [62.7906976744186, 75.0, 0.0], [76.74418604651163, 75.0, 0.0], [90.69767441860465, 75.0, 0.0], [0.0, 90.0, 0.0], [13.953488372093023, 90.0, 0.0], [27.906976744186046, 90.0, 0.0], [41.860465116279066, 90.0, 0.0], [55.81395348837209, 90.0, 0.0], [69.76744186046511, 90.0, 0.0], [83.72093023255813, 90.0, 0.0]], (V = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(49), Bool[0, 0, 0, 1, 1, 1, 1, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), E = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(120), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Tri = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(72), Bool[0, 0, 0, 0, 0, 0, 1, 1, 1, 1 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Orientation = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]), Point = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]))), 2, 2), (Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}(EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}:
V = 1:49
E = 1:120
Tri = 1:72
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 8, 8, 9, 9, 3, 9, 10, 10, 4, 10, 11, 11, 5, 11, 12, 12, 6, 12, 13, 13, 7, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49]
∂v1 : E → V = [1, 2, 1, 8, 2, 2, 3, 9, 3, 3, 4, 10, 4, 4, 5, 11, 5, 5, 6, 12, 6, 6, 7, 13, 7, 8, 15, 8, 9, 16, 9, 10, 17, 10, 11, 18, 11, 12, 19, 12, 13, 20, 13, 14, 16, 15, 22, 16, 17, 23, 17, 18, 24, 18, 19, 25, 19, 20, 26, 20, 21, 27, 21, 22, 29, 22, 23, 30, 23, 24, 31, 24, 25, 32, 25, 26, 33, 26, 27, 34, 27, 28, 30, 29, 36, 30, 31, 37, 31, 32, 38, 32, 33, 39, 33, 34, 40, 34, 35, 41, 35, 36, 43, 36, 37, 44, 37, 38, 45, 38, 39, 46, 39, 40, 47, 40, 41, 48, 41, 42]
∂e0 : Tri → E = [2, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120]
∂e1 : Tri → E = [3, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 48, 48, 51, 51, 54, 54, 57, 57, 60, 60, 63, 66, 66, 69, 69, 72, 72, 75, 75, 78, 78, 81, 81, 84, 86, 86, 89, 89, 92, 92, 95, 95, 98, 98, 101, 104, 104, 107, 107, 110, 110, 113, 113, 116, 116, 119, 119]
∂e2 : Tri → E = [1, 2, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 4, 29, 8, 32, 12, 35, 16, 38, 20, 41, 24, 27, 45, 30, 49, 33, 52, 36, 55, 39, 58, 42, 61, 64, 47, 67, 50, 70, 53, 73, 56, 76, 59, 79, 62, 65, 83, 68, 87, 71, 90, 74, 93, 77, 96, 80, 99, 102, 85, 105, 88, 108, 91, 111, 94, 114, 97, 117, 100]
edge_orientation : E → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
tri_orientation : Tri → Orientation = Bool[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
point : V → Point = GeometryBasics.Point{3, Float64}[[0.0, 0.0, 0.0], [13.953488372093023, 0.0, 0.0], [27.906976744186046, 0.0, 0.0], [41.860465116279066, 0.0, 0.0], [55.81395348837209, 0.0, 0.0], [69.76744186046511, 0.0, 0.0], [83.72093023255813, 0.0, 0.0], [6.976744186046512, 15.0, 0.0], [20.930232558139533, 15.0, 0.0], [34.883720930232556, 15.0, 0.0], [48.83720930232558, 15.0, 0.0], [62.7906976744186, 15.0, 0.0], [76.74418604651163, 15.0, 0.0], [90.69767441860465, 15.0, 0.0], [0.0, 30.0, 0.0], [13.953488372093023, 30.0, 0.0], [27.906976744186046, 30.0, 0.0], [41.860465116279066, 30.0, 0.0], [55.81395348837209, 30.0, 0.0], [69.76744186046511, 30.0, 0.0], [83.72093023255813, 30.0, 0.0], [6.976744186046512, 45.0, 0.0], [20.930232558139533, 45.0, 0.0], [34.883720930232556, 45.0, 0.0], [48.83720930232558, 45.0, 0.0], [62.7906976744186, 45.0, 0.0], [76.74418604651163, 45.0, 0.0], [90.69767441860465, 45.0, 0.0], [0.0, 60.0, 0.0], [13.953488372093023, 60.0, 0.0], [27.906976744186046, 60.0, 0.0], [41.860465116279066, 60.0, 0.0], [55.81395348837209, 60.0, 0.0], [69.76744186046511, 60.0, 0.0], [83.72093023255813, 60.0, 0.0], [6.976744186046512, 75.0, 0.0], [20.930232558139533, 75.0, 0.0], [34.883720930232556, 75.0, 0.0], [48.83720930232558, 75.0, 0.0], [62.7906976744186, 75.0, 0.0], [76.74418604651163, 75.0, 0.0], [90.69767441860465, 75.0, 0.0], [0.0, 90.0, 0.0], [13.953488372093023, 90.0, 0.0], [27.906976744186046, 90.0, 0.0], [41.860465116279066, 90.0, 0.0], [55.81395348837209, 90.0, 0.0], [69.76744186046511, 90.0, 0.0], [83.72093023255813, 90.0, 0.0]], (V = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(49), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), E = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(120), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Tri = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(72), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Orientation = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]), Point = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]))), 2, 3), (Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}(EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}:
V = 1:49
E = 1:120
Tri = 1:72
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 8, 8, 9, 9, 3, 9, 10, 10, 4, 10, 11, 11, 5, 11, 12, 12, 6, 12, 13, 13, 7, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49]
∂v1 : E → V = [1, 2, 1, 8, 2, 2, 3, 9, 3, 3, 4, 10, 4, 4, 5, 11, 5, 5, 6, 12, 6, 6, 7, 13, 7, 8, 15, 8, 9, 16, 9, 10, 17, 10, 11, 18, 11, 12, 19, 12, 13, 20, 13, 14, 16, 15, 22, 16, 17, 23, 17, 18, 24, 18, 19, 25, 19, 20, 26, 20, 21, 27, 21, 22, 29, 22, 23, 30, 23, 24, 31, 24, 25, 32, 25, 26, 33, 26, 27, 34, 27, 28, 30, 29, 36, 30, 31, 37, 31, 32, 38, 32, 33, 39, 33, 34, 40, 34, 35, 41, 35, 36, 43, 36, 37, 44, 37, 38, 45, 38, 39, 46, 39, 40, 47, 40, 41, 48, 41, 42]
∂e0 : Tri → E = [2, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120]
∂e1 : Tri → E = [3, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 48, 48, 51, 51, 54, 54, 57, 57, 60, 60, 63, 66, 66, 69, 69, 72, 72, 75, 75, 78, 78, 81, 81, 84, 86, 86, 89, 89, 92, 92, 95, 95, 98, 98, 101, 104, 104, 107, 107, 110, 110, 113, 113, 116, 116, 119, 119]
∂e2 : Tri → E = [1, 2, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 4, 29, 8, 32, 12, 35, 16, 38, 20, 41, 24, 27, 45, 30, 49, 33, 52, 36, 55, 39, 58, 42, 61, 64, 47, 67, 50, 70, 53, 73, 56, 76, 59, 79, 62, 65, 83, 68, 87, 71, 90, 74, 93, 77, 96, 80, 99, 102, 85, 105, 88, 108, 91, 111, 94, 114, 97, 117, 100]
edge_orientation : E → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
tri_orientation : Tri → Orientation = Bool[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
point : V → Point = GeometryBasics.Point{3, Float64}[[0.0, 0.0, 0.0], [13.953488372093023, 0.0, 0.0], [27.906976744186046, 0.0, 0.0], [41.860465116279066, 0.0, 0.0], [55.81395348837209, 0.0, 0.0], [69.76744186046511, 0.0, 0.0], [83.72093023255813, 0.0, 0.0], [6.976744186046512, 15.0, 0.0], [20.930232558139533, 15.0, 0.0], [34.883720930232556, 15.0, 0.0], [48.83720930232558, 15.0, 0.0], [62.7906976744186, 15.0, 0.0], [76.74418604651163, 15.0, 0.0], [90.69767441860465, 15.0, 0.0], [0.0, 30.0, 0.0], [13.953488372093023, 30.0, 0.0], [27.906976744186046, 30.0, 0.0], [41.860465116279066, 30.0, 0.0], [55.81395348837209, 30.0, 0.0], [69.76744186046511, 30.0, 0.0], [83.72093023255813, 30.0, 0.0], [6.976744186046512, 45.0, 0.0], [20.930232558139533, 45.0, 0.0], [34.883720930232556, 45.0, 0.0], [48.83720930232558, 45.0, 0.0], [62.7906976744186, 45.0, 0.0], [76.74418604651163, 45.0, 0.0], [90.69767441860465, 45.0, 0.0], [0.0, 60.0, 0.0], [13.953488372093023, 60.0, 0.0], [27.906976744186046, 60.0, 0.0], [41.860465116279066, 60.0, 0.0], [55.81395348837209, 60.0, 0.0], [69.76744186046511, 60.0, 0.0], [83.72093023255813, 60.0, 0.0], [6.976744186046512, 75.0, 0.0], [20.930232558139533, 75.0, 0.0], [34.883720930232556, 75.0, 0.0], [48.83720930232558, 75.0, 0.0], [62.7906976744186, 75.0, 0.0], [76.74418604651163, 75.0, 0.0], [90.69767441860465, 75.0, 0.0], [0.0, 90.0, 0.0], [13.953488372093023, 90.0, 0.0], [27.906976744186046, 90.0, 0.0], [41.860465116279066, 90.0, 0.0], [55.81395348837209, 90.0, 0.0], [69.76744186046511, 90.0, 0.0], [83.72093023255813, 90.0, 0.0]], (V = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(49), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), E = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(120), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Tri = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(72), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]), Orientation = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]), Point = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]))), 2, 4)], [(Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}(EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}:
V = 1:49
E = 1:120
Tri = 1:72
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 8, 8, 9, 9, 3, 9, 10, 10, 4, 10, 11, 11, 5, 11, 12, 12, 6, 12, 13, 13, 7, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49]
∂v1 : E → V = [1, 2, 1, 8, 2, 2, 3, 9, 3, 3, 4, 10, 4, 4, 5, 11, 5, 5, 6, 12, 6, 6, 7, 13, 7, 8, 15, 8, 9, 16, 9, 10, 17, 10, 11, 18, 11, 12, 19, 12, 13, 20, 13, 14, 16, 15, 22, 16, 17, 23, 17, 18, 24, 18, 19, 25, 19, 20, 26, 20, 21, 27, 21, 22, 29, 22, 23, 30, 23, 24, 31, 24, 25, 32, 25, 26, 33, 26, 27, 34, 27, 28, 30, 29, 36, 30, 31, 37, 31, 32, 38, 32, 33, 39, 33, 34, 40, 34, 35, 41, 35, 36, 43, 36, 37, 44, 37, 38, 45, 38, 39, 46, 39, 40, 47, 40, 41, 48, 41, 42]
∂e0 : Tri → E = [2, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120]
∂e1 : Tri → E = [3, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 48, 48, 51, 51, 54, 54, 57, 57, 60, 60, 63, 66, 66, 69, 69, 72, 72, 75, 75, 78, 78, 81, 81, 84, 86, 86, 89, 89, 92, 92, 95, 95, 98, 98, 101, 104, 104, 107, 107, 110, 110, 113, 113, 116, 116, 119, 119]
∂e2 : Tri → E = [1, 2, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 4, 29, 8, 32, 12, 35, 16, 38, 20, 41, 24, 27, 45, 30, 49, 33, 52, 36, 55, 39, 58, 42, 61, 64, 47, 67, 50, 70, 53, 73, 56, 76, 59, 79, 62, 65, 83, 68, 87, 71, 90, 74, 93, 77, 96, 80, 99, 102, 85, 105, 88, 108, 91, 111, 94, 114, 97, 117, 100]
edge_orientation : E → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
tri_orientation : Tri → Orientation = Bool[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
point : V → Point = GeometryBasics.Point{3, Float64}[[0.0, 0.0, 0.0], [13.953488372093023, 0.0, 0.0], [27.906976744186046, 0.0, 0.0], [41.860465116279066, 0.0, 0.0], [55.81395348837209, 0.0, 0.0], [69.76744186046511, 0.0, 0.0], [83.72093023255813, 0.0, 0.0], [6.976744186046512, 15.0, 0.0], [20.930232558139533, 15.0, 0.0], [34.883720930232556, 15.0, 0.0], [48.83720930232558, 15.0, 0.0], [62.7906976744186, 15.0, 0.0], [76.74418604651163, 15.0, 0.0], [90.69767441860465, 15.0, 0.0], [0.0, 30.0, 0.0], [13.953488372093023, 30.0, 0.0], [27.906976744186046, 30.0, 0.0], [41.860465116279066, 30.0, 0.0], [55.81395348837209, 30.0, 0.0], [69.76744186046511, 30.0, 0.0], [83.72093023255813, 30.0, 0.0], [6.976744186046512, 45.0, 0.0], [20.930232558139533, 45.0, 0.0], [34.883720930232556, 45.0, 0.0], [48.83720930232558, 45.0, 0.0], [62.7906976744186, 45.0, 0.0], [76.74418604651163, 45.0, 0.0], [90.69767441860465, 45.0, 0.0], [0.0, 60.0, 0.0], [13.953488372093023, 60.0, 0.0], [27.906976744186046, 60.0, 0.0], [41.860465116279066, 60.0, 0.0], [55.81395348837209, 60.0, 0.0], [69.76744186046511, 60.0, 0.0], [83.72093023255813, 60.0, 0.0], [6.976744186046512, 75.0, 0.0], [20.930232558139533, 75.0, 0.0], [34.883720930232556, 75.0, 0.0], [48.83720930232558, 75.0, 0.0], [62.7906976744186, 75.0, 0.0], [76.74418604651163, 75.0, 0.0], [90.69767441860465, 75.0, 0.0], [0.0, 90.0, 0.0], [13.953488372093023, 90.0, 0.0], [27.906976744186046, 90.0, 0.0], [41.860465116279066, 90.0, 0.0], [55.81395348837209, 90.0, 0.0], [69.76744186046511, 90.0, 0.0], [83.72093023255813, 90.0, 0.0]], (V = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(49), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 1, 0, 0, 1, 1, 1, 1, 1, 0, 0]), E = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(120), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 1, 1, 1, 1, 0, 0, 0, 0, 0, 0]), Tri = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(72), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 1, 1, 1, 1, 1, 1, 0, 0, 0, 0]), Orientation = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]), Point = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]))), 3, 3), (Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}(EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}:
V = 1:49
E = 1:120
Tri = 1:72
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 8, 8, 9, 9, 3, 9, 10, 10, 4, 10, 11, 11, 5, 11, 12, 12, 6, 12, 13, 13, 7, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49]
∂v1 : E → V = [1, 2, 1, 8, 2, 2, 3, 9, 3, 3, 4, 10, 4, 4, 5, 11, 5, 5, 6, 12, 6, 6, 7, 13, 7, 8, 15, 8, 9, 16, 9, 10, 17, 10, 11, 18, 11, 12, 19, 12, 13, 20, 13, 14, 16, 15, 22, 16, 17, 23, 17, 18, 24, 18, 19, 25, 19, 20, 26, 20, 21, 27, 21, 22, 29, 22, 23, 30, 23, 24, 31, 24, 25, 32, 25, 26, 33, 26, 27, 34, 27, 28, 30, 29, 36, 30, 31, 37, 31, 32, 38, 32, 33, 39, 33, 34, 40, 34, 35, 41, 35, 36, 43, 36, 37, 44, 37, 38, 45, 38, 39, 46, 39, 40, 47, 40, 41, 48, 41, 42]
∂e0 : Tri → E = [2, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120]
∂e1 : Tri → E = [3, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 48, 48, 51, 51, 54, 54, 57, 57, 60, 60, 63, 66, 66, 69, 69, 72, 72, 75, 75, 78, 78, 81, 81, 84, 86, 86, 89, 89, 92, 92, 95, 95, 98, 98, 101, 104, 104, 107, 107, 110, 110, 113, 113, 116, 116, 119, 119]
∂e2 : Tri → E = [1, 2, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 4, 29, 8, 32, 12, 35, 16, 38, 20, 41, 24, 27, 45, 30, 49, 33, 52, 36, 55, 39, 58, 42, 61, 64, 47, 67, 50, 70, 53, 73, 56, 76, 59, 79, 62, 65, 83, 68, 87, 71, 90, 74, 93, 77, 96, 80, 99, 102, 85, 105, 88, 108, 91, 111, 94, 114, 97, 117, 100]
edge_orientation : E → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
tri_orientation : Tri → Orientation = Bool[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
point : V → Point = GeometryBasics.Point{3, Float64}[[0.0, 0.0, 0.0], [13.953488372093023, 0.0, 0.0], [27.906976744186046, 0.0, 0.0], [41.860465116279066, 0.0, 0.0], [55.81395348837209, 0.0, 0.0], [69.76744186046511, 0.0, 0.0], [83.72093023255813, 0.0, 0.0], [6.976744186046512, 15.0, 0.0], [20.930232558139533, 15.0, 0.0], [34.883720930232556, 15.0, 0.0], [48.83720930232558, 15.0, 0.0], [62.7906976744186, 15.0, 0.0], [76.74418604651163, 15.0, 0.0], [90.69767441860465, 15.0, 0.0], [0.0, 30.0, 0.0], [13.953488372093023, 30.0, 0.0], [27.906976744186046, 30.0, 0.0], [41.860465116279066, 30.0, 0.0], [55.81395348837209, 30.0, 0.0], [69.76744186046511, 30.0, 0.0], [83.72093023255813, 30.0, 0.0], [6.976744186046512, 45.0, 0.0], [20.930232558139533, 45.0, 0.0], [34.883720930232556, 45.0, 0.0], [48.83720930232558, 45.0, 0.0], [62.7906976744186, 45.0, 0.0], [76.74418604651163, 45.0, 0.0], [90.69767441860465, 45.0, 0.0], [0.0, 60.0, 0.0], [13.953488372093023, 60.0, 0.0], [27.906976744186046, 60.0, 0.0], [41.860465116279066, 60.0, 0.0], [55.81395348837209, 60.0, 0.0], [69.76744186046511, 60.0, 0.0], [83.72093023255813, 60.0, 0.0], [6.976744186046512, 75.0, 0.0], [20.930232558139533, 75.0, 0.0], [34.883720930232556, 75.0, 0.0], [48.83720930232558, 75.0, 0.0], [62.7906976744186, 75.0, 0.0], [76.74418604651163, 75.0, 0.0], [90.69767441860465, 75.0, 0.0], [0.0, 90.0, 0.0], [13.953488372093023, 90.0, 0.0], [27.906976744186046, 90.0, 0.0], [41.860465116279066, 90.0, 0.0], [55.81395348837209, 90.0, 0.0], [69.76744186046511, 90.0, 0.0], [83.72093023255813, 90.0, 0.0]], (V = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(49), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 1, 0, 0, 0, 0, 0, 1, 1, 0, 0]), E = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(120), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 1, 1, 1, 1, 0, 0, 0, 0, 0, 0]), Tri = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(72), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 1, 1, 0, 0, 0, 0]), Orientation = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]), Point = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]))), 3, 4)], [(Catlab.CategoricalAlgebra.Pointwise.SubCSets.SubACSetComponentwise{EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}, @NamedTuple{V::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, E::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Tri::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Orientation::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}, Point::Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}}}(EmbeddedDeltaSet2D{Bool, GeometryBasics.Point{3, Float64}}:
V = 1:49
E = 1:120
Tri = 1:72
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 8, 8, 9, 9, 3, 9, 10, 10, 4, 10, 11, 11, 5, 11, 12, 12, 6, 12, 13, 13, 7, 13, 14, 14, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 19, 19, 19, 20, 20, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 25, 25, 25, 26, 26, 26, 27, 27, 27, 28, 28, 29, 30, 30, 30, 31, 31, 31, 32, 32, 32, 33, 33, 33, 34, 34, 34, 35, 35, 35, 36, 36, 37, 37, 37, 38, 38, 38, 39, 39, 39, 40, 40, 40, 41, 41, 41, 42, 42, 43, 44, 44, 44, 45, 45, 45, 46, 46, 46, 47, 47, 47, 48, 48, 48, 49, 49, 49]
∂v1 : E → V = [1, 2, 1, 8, 2, 2, 3, 9, 3, 3, 4, 10, 4, 4, 5, 11, 5, 5, 6, 12, 6, 6, 7, 13, 7, 8, 15, 8, 9, 16, 9, 10, 17, 10, 11, 18, 11, 12, 19, 12, 13, 20, 13, 14, 16, 15, 22, 16, 17, 23, 17, 18, 24, 18, 19, 25, 19, 20, 26, 20, 21, 27, 21, 22, 29, 22, 23, 30, 23, 24, 31, 24, 25, 32, 25, 26, 33, 26, 27, 34, 27, 28, 30, 29, 36, 30, 31, 37, 31, 32, 38, 32, 33, 39, 33, 34, 40, 34, 35, 41, 35, 36, 43, 36, 37, 44, 37, 38, 45, 38, 39, 46, 39, 40, 47, 40, 41, 48, 41, 42]
∂e0 : Tri → E = [2, 4, 7, 8, 11, 12, 15, 16, 19, 20, 23, 24, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 49, 50, 52, 53, 55, 56, 58, 59, 61, 62, 65, 67, 68, 70, 71, 73, 74, 76, 77, 79, 80, 82, 83, 85, 87, 88, 90, 91, 93, 94, 96, 97, 99, 100, 103, 105, 106, 108, 109, 111, 112, 114, 115, 117, 118, 120]
∂e1 : Tri → E = [3, 5, 5, 9, 9, 13, 13, 17, 17, 21, 21, 25, 28, 28, 31, 31, 34, 34, 37, 37, 40, 40, 43, 43, 46, 48, 48, 51, 51, 54, 54, 57, 57, 60, 60, 63, 66, 66, 69, 69, 72, 72, 75, 75, 78, 78, 81, 81, 84, 86, 86, 89, 89, 92, 92, 95, 95, 98, 98, 101, 104, 104, 107, 107, 110, 110, 113, 113, 116, 116, 119, 119]
∂e2 : Tri → E = [1, 2, 6, 7, 10, 11, 14, 15, 18, 19, 22, 23, 26, 4, 29, 8, 32, 12, 35, 16, 38, 20, 41, 24, 27, 45, 30, 49, 33, 52, 36, 55, 39, 58, 42, 61, 64, 47, 67, 50, 70, 53, 73, 56, 76, 59, 79, 62, 65, 83, 68, 87, 71, 90, 74, 93, 77, 96, 80, 99, 102, 85, 105, 88, 108, 91, 111, 94, 114, 97, 117, 100]
edge_orientation : E → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]
tri_orientation : Tri → Orientation = Bool[1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1]
point : V → Point = GeometryBasics.Point{3, Float64}[[0.0, 0.0, 0.0], [13.953488372093023, 0.0, 0.0], [27.906976744186046, 0.0, 0.0], [41.860465116279066, 0.0, 0.0], [55.81395348837209, 0.0, 0.0], [69.76744186046511, 0.0, 0.0], [83.72093023255813, 0.0, 0.0], [6.976744186046512, 15.0, 0.0], [20.930232558139533, 15.0, 0.0], [34.883720930232556, 15.0, 0.0], [48.83720930232558, 15.0, 0.0], [62.7906976744186, 15.0, 0.0], [76.74418604651163, 15.0, 0.0], [90.69767441860465, 15.0, 0.0], [0.0, 30.0, 0.0], [13.953488372093023, 30.0, 0.0], [27.906976744186046, 30.0, 0.0], [41.860465116279066, 30.0, 0.0], [55.81395348837209, 30.0, 0.0], [69.76744186046511, 30.0, 0.0], [83.72093023255813, 30.0, 0.0], [6.976744186046512, 45.0, 0.0], [20.930232558139533, 45.0, 0.0], [34.883720930232556, 45.0, 0.0], [48.83720930232558, 45.0, 0.0], [62.7906976744186, 45.0, 0.0], [76.74418604651163, 45.0, 0.0], [90.69767441860465, 45.0, 0.0], [0.0, 60.0, 0.0], [13.953488372093023, 60.0, 0.0], [27.906976744186046, 60.0, 0.0], [41.860465116279066, 60.0, 0.0], [55.81395348837209, 60.0, 0.0], [69.76744186046511, 60.0, 0.0], [83.72093023255813, 60.0, 0.0], [6.976744186046512, 75.0, 0.0], [20.930232558139533, 75.0, 0.0], [34.883720930232556, 75.0, 0.0], [48.83720930232558, 75.0, 0.0], [62.7906976744186, 75.0, 0.0], [76.74418604651163, 75.0, 0.0], [90.69767441860465, 75.0, 0.0], [0.0, 90.0, 0.0], [13.953488372093023, 90.0, 0.0], [27.906976744186046, 90.0, 0.0], [41.860465116279066, 90.0, 0.0], [55.81395348837209, 90.0, 0.0], [69.76744186046511, 90.0, 0.0], [83.72093023255813, 90.0, 0.0]], (V = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(49), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 1, 1, 1, 0, 0, 0, 1, 1, 1, 1]), E = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(120), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]), Tri = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(72), Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0 … 0, 0, 0, 0, 1, 1, 1, 1, 1, 1]), Orientation = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]), Point = Catlab.CategoricalAlgebra.SetCats.Subsets.SubFinSetVector{Catlab.BasicSets.FinSets.FinSet}(FinSet(0), Bool[]))), 4, 4)]])Example 2: Arcs of a Circle
We can also create a circular mesh and decompose it into overlapping arcs. Each arc will overlap with its neighbors along their endpoints.
function circle(n, c)
mesh = EmbeddedDeltaSet1D{Bool, Point2D}()
map(range(0, 2pi - (pi/(2^(n-1))); step=pi/(2^(n-1)))) do t
add_vertex!(mesh, point=Point2D(cos(t),sin(t))*(c/2pi))
end
add_edges!(mesh, 1:(nv(mesh)-1), 2:nv(mesh))
add_edge!(mesh, nv(mesh), 1)
dualmesh = EmbeddedDeltaDualComplex1D{Bool, Float64, Point2D}(mesh)
subdivide_duals!(dualmesh, Circumcenter())
mesh,dualmesh
end
mesh,dualmesh = circle(9, 100)(EmbeddedDeltaSet1D{Bool, GeometryBasics.Point{2, Float64}}:
V = 1:512
E = 1:512
Orientation = 1:0
Point = 1:0
∂v0 : E → V = [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 1]
∂v1 : E → V = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512]
edge_orientation : E → Orientation = [nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing, nothing]
point : V → Point = GeometryBasics.Point{2, Float64}[[15.915494309189533, 0.0], [15.914295901738946, 0.19530759775137696], [15.910700859862938, 0.39058578289693036], [15.904709724961592, 0.5858051472602669], [15.896323399277785, 0.7809362915231868], [15.88554314576132, 0.9759498296531126], [15.872370587878725, 1.1708163933285185], [15.856807709368761, 1.3655066363616901], [15.83885685394369, 1.5599912391181536], [15.81852072493631, 1.7542409129321048], [15.795802384892845, 1.9482264045171738], [15.770705255111743, 2.141918500371862], [15.743233115128437, 2.3352880311789903], [15.713390102146152, 2.528305876198487], [15.681180710412875, 2.720942967652868], [15.646609790544526, 2.913170295104736], [15.609682548794474, 3.1049589098256427], [15.5704045462695, 3.2962799291556673], [15.528781698092308, 3.4871045408530303], [15.48482027251073, 3.67740400743312], [15.438526889953756, 3.867149670496245], [15.38990852203452, 4.056312955043488], [15.3389724905004, 4.2448653737799935], [15.285726466130384, 4.432778531405049], [15.230178467579892, 4.620024128888316], [15.17233686017318, 4.806573967731557], [15.112210354643567, 4.992399954215227], [15.049808005821621, 5.17747410362928], [14.985139211271546, 5.361768544487575], [14.918213709875927, 5.545255522725204], [14.849041580369104, 5.727907405878162], [14.777633239819341, 5.909696687244696], [14.70399944206007, 6.090595990027704], [14.628151276070387, 6.270578071457586], [14.550100164305096, 6.449615826894905], [14.469857860974535, 6.627682293912245], [14.387436450274432, 6.804750656354654], [14.302848344566067, 6.980794248378055], [14.216106282507022, 7.15578655846503], [14.127223327132779, 7.329701233417352], [14.036212863889483, 7.502512082324682], [13.943088598618132, 7.674193080508821], [13.847864555490542, 7.844718373442936], [13.75055507489734, 8.014062280645154], [13.65117481128838, 8.182199299545942], [13.549738730965805, 8.349104109328715], [13.4462621098302, 8.51475157474304], [13.340760531080099, 8.679116749889928], [13.233249882865186, 8.84217488197858], [13.123746355893635, 9.003901415054084], [13.012266440993818, 9.164271993695435], [12.898826926630866, 9.323262466683374], [12.783444896378377, 9.480848890637485], [12.6661377263457, 9.63700753362197], [12.546923082561152, 9.79171487871959], [12.425818918311577, 9.944947627573235], [12.302843471438662, 10.096682703894558], [12.178015261592362, 10.246897256939192], [12.051353087441926, 10.395568664947982], [11.922876023844882, 10.542674538553738], [11.792603418974434, 10.68819272415299], [11.660554891405688, 10.832101307242251], [11.526750327161192, 10.974378615718239], [11.391209876716143, 11.115003223141638], [11.253953951963826, 11.253953951963824], [11.115003223141638, 11.391209876716143], [10.97437861571824, 11.52675032716119], [10.832101307242253, 11.660554891405688], [10.688192724152993, 11.792603418974432], [10.542674538553738, 11.92287602384488], [10.395568664947984, 12.051353087441926], [10.246897256939194, 12.178015261592362], [10.096682703894558, 12.30284347143866], [9.944947627573237, 12.425818918311577], [9.79171487871959, 12.54692308256115], [9.63700753362197, 12.666137726345697], [9.480848890637487, 12.783444896378375], [9.323262466683374, 12.898826926630866], [9.164271993695435, 13.012266440993818], [9.003901415054086, 13.123746355893635], [8.842174881978583, 13.233249882865186], [8.679116749889928, 13.340760531080097], [8.514751574743041, 13.446262109830199], [8.349104109328714, 13.549738730965805], [8.182199299545942, 13.65117481128838], [8.014062280645154, 13.75055507489734], [7.844718373442937, 13.84786455549054], [7.6741930805088225, 13.943088598618132], [7.502512082324684, 14.036212863889482], [7.329701233417352, 14.127223327132779], [7.155786558465031, 14.216106282507022], [6.980794248378056, 14.302848344566067], [6.804750656354655, 14.387436450274432], [6.6276822939122475, 14.469857860974534], [6.449615826894905, 14.550100164305096], [6.270578071457586, 14.628151276070387], 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[15.528781698092306, -3.487104540853036], [15.5704045462695, -3.296279929155667], [15.609682548794472, -3.1049589098256503], [15.646609790544526, -2.9131702951047367], [15.681180710412873, -2.720942967652877], [15.71339010214615, -2.5283058761984893], [15.743233115128435, -2.335288031179], [15.770705255111743, -2.1419185003718657], [15.795802384892845, -1.948226404517171], [15.81852072493631, -1.7542409129321097], [15.83885685394369, -1.559991239118152], [15.856807709368761, -1.3655066363616961], [15.872370587878725, -1.1708163933285183], [15.88554314576132, -0.9759498296531203], [15.896323399277785, -0.780936291523188], [15.90470972496159, -0.5858051472602759], [15.910700859862938, -0.3905857828969329], [15.914295901738946, -0.19530759775138726]], EmbeddedDeltaDualComplex1D{Bool, Float64, GeometryBasics.Point{2, Float64}}:
V = 1:512
E = 1:512
DualV = 1:1024
DualE = 1:1024
Orientation = 1:0
Real = 1:0
Point = 1:0
∂v0 : E → V = [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 1]
∂v1 : E → V = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512]
D_∂v0 : DualE → DualV = [513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 888, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998, 999, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 888, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998, 999, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024]
D_∂v1 : DualE → DualV = [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 1, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512]
vertex_center : V → DualV = [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512]
edge_center : E → DualV = [513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 888, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998, 999, 1000, 1001, 1002, 1003, 1004, 1005, 1006, 1007, 1008, 1009, 1010, 1011, 1012, 1013, 1014, 1015, 1016, 1017, 1018, 1019, 1020, 1021, 1022, 1023, 1024]
edge_orientation : E → Orientation = Bool[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
D_edge_orientation : DualE → Orientation = Bool[0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]
point : V → Point = GeometryBasics.Point{2, Float64}[[15.915494309189533, 0.0], [15.914295901738946, 0.19530759775137696], [15.910700859862938, 0.39058578289693036], [15.904709724961592, 0.5858051472602669], [15.896323399277785, 0.7809362915231868], [15.88554314576132, 0.9759498296531126], [15.872370587878725, 1.1708163933285185], [15.856807709368761, 1.3655066363616901], [15.83885685394369, 1.5599912391181536], [15.81852072493631, 1.7542409129321048], [15.795802384892845, 1.9482264045171738], [15.770705255111743, 2.141918500371862], [15.743233115128437, 2.3352880311789903], [15.713390102146152, 2.528305876198487], [15.681180710412875, 2.720942967652868], [15.646609790544526, 2.913170295104736], [15.609682548794474, 3.1049589098256427], [15.5704045462695, 3.2962799291556673], [15.528781698092308, 3.4871045408530303], [15.48482027251073, 3.67740400743312], [15.438526889953756, 3.867149670496245], [15.38990852203452, 4.056312955043488], [15.3389724905004, 4.2448653737799935], 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dual_point : DualV → Point = GeometryBasics.Point{2, Float64}[[15.915494309189533, 0.0], [15.914295901738946, 0.19530759775137696], [15.910700859862938, 0.39058578289693036], [15.904709724961592, 0.5858051472602669], [15.896323399277785, 0.7809362915231868], [15.88554314576132, 0.9759498296531126], [15.872370587878725, 1.1708163933285185], [15.856807709368761, 1.3655066363616901], [15.83885685394369, 1.5599912391181536], [15.81852072493631, 1.7542409129321048], [15.795802384892845, 1.9482264045171738], [15.770705255111743, 2.141918500371862], [15.743233115128437, 2.3352880311789903], [15.713390102146152, 2.528305876198487], [15.681180710412875, 2.720942967652868], [15.646609790544526, 2.913170295104736], [15.609682548794474, 3.1049589098256427], [15.5704045462695, 3.2962799291556673], [15.528781698092308, 3.4871045408530303], [15.48482027251073, 3.67740400743312], [15.438526889953756, 3.867149670496245], [15.38990852203452, 4.056312955043488], [15.3389724905004, 4.2448653737799935], 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dual_length : DualE → Real = [0.09765563721546147, 0.09765563721546144, 0.09765563721546146, 0.09765563721546143, 0.09765563721546147, 0.09765563721546161, 0.09765563721546142, 0.09765563721546133, 0.09765563721546167, 0.09765563721546143, 0.09765563721546133, 0.09765563721546165, 0.09765563721546143, 0.09765563721546143, 0.09765563721546168, 0.09765563721546121, 0.09765563721546144, 0.09765563721546133, 0.0976556372154615, 0.09765563721546187, 0.09765563721546175, 0.09765563721546132, 0.09765563721546078, 0.09765563721546118, 0.0976556372154617, 0.0976556372154617, 0.09765563721546222, 0.09765563721546194, 0.09765563721546176, 0.09765563721546111, 0.09765563721546194, 0.09765563721546179, 0.09765563721546144, 0.0976556372154613, 0.0976556372154612, 0.09765563721546153, 0.09765563721546165, 0.09765563721546112, 0.09765563721546283, 0.09765563721546114, 0.09765563721546075, 0.09765563721546128, 0.09765563721546242, 0.09765563721545968, 0.0976556372154625, 0.09765563721546149, 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0.09765563721545742, 0.09765563721546473, 0.0976556372154588, 0.09765563721546482, 0.09765563721545889, 0.09765563721546532, 0.09765563721545671, 0.09765563721546473, 0.09765563721546473, 0.09765563721545671, 0.09765563721546532, 0.09765563721545889, 0.09765563721546362, 0.09765563721545763, 0.09765563721546473, 0.09765563721545742, 0.09765563721546561, 0.09765563721545747, 0.0976556372154638, 0.09765563721546339, 0.09765563721545678, 0.09765563721546446, 0.09765563721545777, 0.09765563721546447, 0.09765563721545797, 0.09765563721546551, 0.09765563721545753, 0.09765563721546343, 0.09765563721545725, 0.09765563721546483, 0.09765563721546448, 0.09765563721545756, 0.0976556372154635, 0.09765563721545725, 0.0976556372154651, 0.0976556372154569, 0.09765563721546469, 0.09765563721545722, 0.09765563721546455, 0.09765563721545747, 0.09765563721546343, 0.09765563721546525, 0.09765563721545716, 0.09765563721546487, 0.09765563721545717, 0.0976556372154656, 0.0976556372154569, 0.09765563721546476, 0.09765563721545777, 0.09765563721546419, 0.09765563721546475, 0.09765563721545788, 0.09765563721546489, 0.09765563721545764, 0.09765563721546437, 0.09765563721545753, 0.0976556372154646, 0.09765563721545731, 0.09765563721546477, 0.09765563721545764, 0.09765563721546429, 0.09765563721546441, 0.09765563721545789, 0.09765563721546476, 0.09765563721545757, 0.09765563721546473, 0.09765563721545763, 0.09765563721546469, 0.09765563721545754, 0.09765563721546468, 0.09765563721545757, 0.09765563721546662])draw(dualmesh)
The circle can be decomposed into four overlapping arcs using a partition function based on the quadrants of the Cartesian plane. Notice how we supply a partition function rather than a list of vertex indices. The cover_mesh function will compute the vertex indices for us.
function pizza_slices(x)
(x[1] > 0) + 2*(x[2] > 0)
end
# Create a category instance for the circle mesh
# Pass the ACSet instance to ACSetCat constructor (Catlab 0.17 API)
const 𝒞₁ = ACSetCategory(ACSetCat(dualmesh))
circ_quads = cover_mesh(pizza_slices, dualmesh, 𝒞₁)
draw(circ_quads[1])
draw(circ_quads; cat=𝒞₁)
Diagram Interpretation in Vect
In order to build sheaves over the mesh decomposition, we first need to create a diagram representing the cover. Each object in the diagram is a morphism in FinSet representing the inclusion of one submesh into another.
function finsetdiagram(cover; object=:V)
n = length(cover)
u1 = cover[1]
f = hom(u1)
X = codom(f)
ObT = FinSet
HomT = FinFunction
homs = [(hom(cover[i]).components[object], i+1, 1) for i in 1:n]
opens = dom.(hom.(cover))
obs = [X]
append!(obs, opens)
obs = FinSet.(nparts.(obs, object))
# @show obs
# edge_homs = map(homs) do (ui, s, t)
# println("$s --> $t = $ui")
# end
diag = FreeGraph(obs, homs)
end
diag = finsetdiagram(quads)
Catlab.CategoricalAlgebra.Cats.FreeDiagrams.FreeGraphs.FreeGraph{Catlab.BasicSets.FinSets.FinSet, Catlab.BasicSets.FinFunctions.FinFunction} {V:5, E:4, Ob:0, Hom:0}
| V | ob |
|---|---|
| 1 | FinSet(49) |
| 2 | FinSet(25) |
| 3 | FinSet(19) |
| 4 | FinSet(19) |
| 5 | FinSet(16) |
| E | src | tgt | hom |
|---|---|---|---|
| 1 | 2 | 1 | FinFunction([1, 2, 3, 4, 5, 8, 9, 10, 11, 12, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 29, 30, 31, 32, 33], FinSet(49)) |
| 2 | 3 | 1 | FinFunction([4, 5, 6, 7, 11, 12, 13, 14, 18, 19, 20, 21, 25, 26, 27, 28, 33, 34, 35], FinSet(49)) |
| 3 | 4 | 1 | FinFunction([22, 23, 24, 25, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 43, 44, 45, 46, 47], FinSet(49)) |
| 4 | 5 | 1 | FinFunction([25, 26, 27, 28, 32, 33, 34, 35, 39, 40, 41, 42, 46, 47, 48, 49], FinSet(49)) |
The free vector space sheaf over the diagram can be constructed by composing the diagram with the free vector space functor and the appropriate pushforward or pullback operations. This creates a diagram in Vect representing the sheaf of vector spaces over the mesh decomposition. This is just the vertex component; similar constructions can be done for edges and faces.
using Catlab.Sheaves: FVect
import Catlab.Sheaves: pullback_matrix, FMatPullback, FMatPushforward
import Catlab.CategoricalAlgebra.Matrices: MatrixDom
MatrixDom(n::Int64) = MatrixDom{Matrix}(n)
Vdiag = force(compose(FinDomFunctor(diag), FMatPushforward))# compose(FinDomFunctor(diag), op(FMatPullback))
Catlab.dom(m::Matrix) = FinSet(size(m, 2))
Catlab.codom(m::Matrix) = FinSet(size(m, 1))
function vectdiagram(diag)
obs = diag[:ob]
homs = map(enumerate(diag[edges(diag), :hom])) do (e, f)
(pullback_matrix(f), diag[e, :src], diag[e,:tgt])
end
# FreeDiagram(obs, homs)
return obs, homs
end
vectdiagram(diag)(Catlab.BasicSets.FinSets.FinSet[FinSet(49), FinSet(25), FinSet(19), FinSet(19), FinSet(16)], Tuple{SparseArrays.SparseMatrixCSC{Int64, Int64}, Int64, Int64}[(sparse([1, 2, 3, 4, 5, 6, 7, 8, 9, 10 … 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], [1, 2, 3, 4, 5, 8, 9, 10, 11, 12 … 22, 23, 24, 25, 26, 29, 30, 31, 32, 33], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1 … 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 25, 49), 2, 1), (sparse([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [4, 5, 6, 7, 11, 12, 13, 14, 18, 19, 20, 21, 25, 26, 27, 28, 33, 34, 35], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 19, 49), 3, 1), (sparse([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19], [22, 23, 24, 25, 29, 30, 31, 32, 33, 36, 37, 38, 39, 40, 43, 44, 45, 46, 47], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 19, 49), 4, 1), (sparse([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], [25, 26, 27, 28, 32, 33, 34, 35, 39, 40, 41, 42, 46, 47, 48, 49], [1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1], 16, 49), 5, 1)])Nerve Cover Type
This data is getting rather involved, so we will encapsulate it in a NerveCover type for easier use and display.
import Catlab.Sheaves: AbstractCover
struct NerveCover{T, X, C} <: AbstractCover
vertices::Dict{T, Int}
basis::Vector{X}
cat::C
end
function NerveCover(subobjects::Vector{X}, cat) where X <: Subobject
lookup = enumerate(subobjects)
vertices = Dict{Int, Int}(i=>i for (i, _) in lookup)
return NerveCover{Int, Subobject, typeof(cat)}(vertices, subobjects, cat)
end
function NerveCover(subobjects::Dict{T,Subobject}, cat) where T
lookup = enumerate(keys(subobjects))
vertices = Dict{T, Int}(k=>i for (i, k) in lookup)
opens = collect(values(subobjects))
return NerveCover{T, Subobject, typeof(cat)}(vertices, opens, cat)
end
Base.length(K::NerveCover) = length(K.basis)
Base.show(io::IO, K::NerveCover) = begin
print(io, "$(typeof(K)) with $(length(K)) generating opens:\n\tNV, NE, NT")
for (i, ui) in enumerate(K.basis)
print(io, "\n ")
V = nv(dom(hom(ui)))
E = ne(dom(hom(ui)))
T = ntriangles(dom(hom(ui)))
print(io, "K[$i]: $V, $E, $T")
end
end
import Catlab.CategoricalAlgebra.CSets: SubACSetComponentwise
function Base.show(io::IO, U::SubACSetComponentwise{X}) where X <: HasDeltaSet
print(io, "Subdelta-set")
V = nv(dom(hom(U)))
E = ne(dom(hom(U)))
T = ntriangles(dom(hom(U)))
print(io, "with size $V, $E, $T")
V = nv(codom(hom(U)))
E = ne(codom(hom(U)))
T = ntriangles(codom(hom(U)))
print(io, " of object with size $V, $E, $T")
end
function Base.getindex(K::NerveCover, I::Vararg{Int})
@withmodel K.cat (meet,) begin
map(I) do i
K.basis[i]
end |> x->foldl(meet, x)
end
end
using Combinatorics: powerset
function resolve(K::NerveCover, dim=2)
map(powerset(K.vertices, 1, dim)) do S
S => K[S...]
end |> Dict
endK = NerveCover(quads, 𝒞)
K[1,2]resolve(K)resolve(K, 3)