Lattices

Each cell occupies a site on a lattice. A lattice provides a mapping between coordinates and indices of the underlying array.

N.B.: The structures are not true lattices, because they do not extend to infinity, but have finite size.

Currently, the following lattices are provided

They are subtypes of RealLattice. Populations without spatial structure "live" on a NoLattice.

Warning

Avoid manipulating the data field of a lattice directly. Doing so easily leads to an inconsistent state between lattice and metadata.

Use the getter and setter methods for Population instead.

Lattice Types

GrowthDynamics.Lattices.CubicLattice — Type

CubicLattice

Three dimensional primitive cubic lattice. Each site has six equidistant neighbors with a fourfold rotational symmetry in each of the three planes.

See https://en.wikipedia.org/wiki/Cubic_crystal_system

Fields

a: lattice constant
data: underlying array

Example

julia> using GrowthDynamics.Lattices

julia> l = CubicLattice(1/2, ones(16,16,16))
CubicLattice{Float64, Array{Float64, 3}}(0.5, [1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0;;; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0;;; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0;;; … ;;; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0;;; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0;;; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0])

julia> neighbors(l, (8,8,8))
6-element StaticArraysCore.MVector{6, CartesianIndex{3}} with indices SOneTo(6):
 CartesianIndex(7, 8, 8)
 CartesianIndex(9, 8, 8)
 CartesianIndex(8, 7, 8)
 CartesianIndex(8, 9, 8)
 CartesianIndex(8, 8, 7)
 CartesianIndex(8, 8, 9)

julia> coord.(Ref(l), ans)
6-element StaticArraysCore.MVector{6, GeometryBasics.Point{3, Float32}} with indices SOneTo(6):
 [3.0, 3.5, 3.5]
 [4.0, 3.5, 3.5]
 [3.5, 3.0, 3.5]
 [3.5, 4.0, 3.5]
 [3.5, 3.5, 3.0]
 [3.5, 3.5, 4.0]

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GrowthDynamics.Lattices.FCCLattice — Type

FCCLattice

Three dimensional face-centered cubic lattice. Each site has twelve neighbors.

See https://en.wikipedia.org/wiki/Cubic_crystal_system

Fields

a: lattice constant
data: underlying array

Example

julia> using GrowthDynamics.Lattices

julia> l = FCCLattice(1/2, ones(16,16,16))
FCCLattice{Float64, Array{Float64, 3}}(0.5, [1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0;;; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0;;; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0;;; … ;;; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0;;; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0;;; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0])

julia> neighbors(l, (8,8,8))
12-element StaticArraysCore.MVector{12, CartesianIndex{3}} with indices SOneTo(12):
 CartesianIndex(7, 8, 8)
 CartesianIndex(9, 8, 8)
 CartesianIndex(9, 7, 8)
 CartesianIndex(7, 7, 8)
 CartesianIndex(7, 8, 9)
 CartesianIndex(8, 7, 9)
 CartesianIndex(9, 8, 9)
 CartesianIndex(8, 8, 9)
 CartesianIndex(7, 8, 7)
 CartesianIndex(8, 7, 7)
 CartesianIndex(9, 8, 7)
 CartesianIndex(8, 8, 7)

julia> coord.(Ref(l), ans)
12-element StaticArraysCore.MVector{12, GeometryBasics.Point{3, Float32}} with indices SOneTo(12):
 [1.5, 3.75, 1.75]
 [2.0, 3.75, 1.75]
 [2.0, 3.25, 1.75]
 [1.5, 3.25, 1.75]
 [1.5, 3.5, 2.0]
 [1.75, 3.25, 2.0]
 [2.0, 3.5, 2.0]
 [1.75, 3.75, 2.0]
 [1.5, 3.5, 1.5]
 [1.75, 3.25, 1.5]
 [2.0, 3.5, 1.5]
 [1.75, 3.75, 1.5]

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GrowthDynamics.Lattices.HexagonalLattice — Type

HexagonalLattice

Each site has six equidistant neighbors with a sixfold rotational symmetry.

See https://en.wikipedia.org/wiki/Hexagonal_lattice

Fields

a: lattice constant
data: underlying array

Example

julia> using GrowthDynamics.Lattices

julia> l = HexagonalLattice(1/2, ones(32,32))
HexagonalLattice{Float64, Matrix{Float64}}(0.5, [1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0])

julia> neighbors(l, (16,16))
6-element StaticArraysCore.MVector{6, CartesianIndex{2}} with indices SOneTo(6):
 CartesianIndex(15, 16)
 CartesianIndex(15, 17)
 CartesianIndex(16, 17)
 CartesianIndex(17, 17)
 CartesianIndex(17, 16)
 CartesianIndex(16, 15)

julia> coord.(Ref(l), ans)
6-element StaticArraysCore.MVector{6, GeometryBasics.Point{2, Float32}} with indices SOneTo(6):
 [7.5, 6.0621777]
 [8.0, 6.0621777]
 [8.25, 6.4951906]
 [8.0, 6.928203]
 [7.5, 6.928203]
 [7.25, 6.4951906]

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GrowthDynamics.Lattices.LineLattice — Type

LineLattice

One dimensional lattice.

Fields

a: lattice spacing
data

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GrowthDynamics.Lattices.NoLattice — Type

NoLattice{T}

Dummy 'lattice' for systems without spatial structure."

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Each lattice type except NoLattice derives from RealLattice and implements the following methods

Common Methods

GrowthDynamics.Lattices.coord — Function

coord(L, I)

Coordinate of index I on the lattice L. I is either a tuple of integers, or an appropriate CartesianIndex.

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Dictionaries.index — Function

index(L, p)

Index of the coordinate p closest to the nearest site on lattice L. p is a GeometryBasics.Point.

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GrowthDynamics.Lattices.coordination — Function

Coordination number of the lattice.

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GrowthDynamics.Lattices.dimension — Function

Dimension of the lattice. Functionally equivalent to ndims(lattice.data).

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GrowthDynamics.Lattices.midpoint — Function

midpoint(L)

Return the index of the point nearest to the geometeric center of the lattice L.

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GrowthDynamics.Lattices.midpointcoord — Function

midpointcoord(L)

Return the coordinate of the point nearest to the geometeric center of the lattice L.

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GrowthDynamics.Lattices.neighbors — Function

neighbors(L::RealLattice, I)

Vector of neighbors of index I. ReturnsVector{CartesianIndex{dimension(L)}}``.

Does not check for bounds.

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GrowthDynamics.Lattices.neighbors! — Function

neighbors!(n, L, I)

In-place version of neighbors.

Use n = Neighbors(L) to allocate an appropriate vector.

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GrowthDynamics.Lattices.nneighbors — Function

nneighbors(L, I)

Boundary-aware number of nearest neighbors of site I on lattice L.

Example

julia> using GrowthDynamics.Lattices

julia> l = HexagonalLattice(1/2, ones(32,32))
HexagonalLattice{Float64, Matrix{Float64}}(0.5, [1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0; … ; 1.0 1.0 … 1.0 1.0; 1.0 1.0 … 1.0 1.0])

julia> nneighbors(l, (5,5))
6

julia> nneighbors(l, (1,5))
4

julia> nneighbors(l, (1,1))
2

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GrowthDynamics.Lattices.density — Function

density(L::RealLattice{T}, I)

Calculates (occupied sites)/(no. of neighbors)

"Occupied" is defined as not equal to zero(T).

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Radial geometries

Populations often grow with approximate radial symmetry. One might then be interested in for example grouping cells with respect to their distance from an origin.

GrowthDynamics.Lattices.shell — Function

shell(L::CubicLattice, r, o=midpointcoord(L))

Return indices of shell of radius r around o.

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GrowthDynamics.Lattices.isonshell — Function

isonshell(L::RealLattice, p, r, o)

Determine whether a lattice point p is on a shell with radius r wrt. the origin o. A shell is defined as the collection of points with |(p-o)|≤r+a/2 where a is the lattice spacing.

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GrowthDynamics.Lattices.conicsection — Function

conicsection(L, points, Ω; axis, o)

Filter those points that lie within a conic section of opening angle Ω around axis emanating from origin o.

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