LatticeData

LatticeData[lattice, "property"]
gives the specified property for a lattice.

LatticeData[n]
gives a list of named lattices of dimension n.

DetailsDetails

  • Lattices can be specified by standard names such as and .
  • LatticeData[] gives a list of classical named lattices.
  • LatticeData[patt] gives a list of all named lattices that match the string pattern patt.
  • LatticeData[{"type", id}, ...] gives data for the lattice of the specified type with identifier id.
  • Type-related lattices include:
  • {"BarnesWall",n} Barnes-Wall lattice
    {"Bravais",{"system","centering"}}Bravais lattice for the specified crystal system
    {"CoxeterBarnes", {r, n}}Coxeter-Barnes lattice
    {"DualRootLattice",group}dual root lattice for the specified Lie group
    {"IntegerLattice",n}integer lattice
    {"KappaLattice",{m,n}}-lattice
    {"LaminatedLattice",{m,n}}laminated lattice
    {"MordellWeil",n}Mordell-Weil lattice
    {"Niemeier",n}^(th) Niemeier lattice
    {"PerfectLattice",n}perfect lattice
    {"Quebbemann",n}Quebbemann lattice
    {"RootLattice",group}root lattice for the specified Lie group
  • Crystal systems are specified by standard names such as , , etc. Centering is , , or .
  • Groups can be specified either for example as or .
  • LatticeData[group, ...] is equivalent to .
  • LatticeData["Properties"] gives a list of possible properties for lattices.
  • Lattice points properties include:
  • "Basis"basis vectors
    "Determinant"determinant of Gram matrix
    "Dimension"dimension of lattice
    "Dual"dual lattice
    "GeneratorMatrix"matrix of generators
    "Genus"genus of lattice
    "GlueVectors"glue vectors (when applicable)
    "GramMatrix"Gram matrix
    "Image"configuration of points (when applicable)
    "KissingNumber"kissing number
    "MinimalNorm"minimal norm
    "MinimalVectors"minimal vectors
    "ModularNumber"modular number
    "RadialFunction"lattice points as a function of radius ( series coefficients)
    "ThetaSeriesFunction"pure function for the theta series
  • Lattice-packing-related properties include:
  • "CenterDensity"center density
    "CoveringRadius"covering radius
    "CoxeterNumber"Coxeter number
    "Density"average sphere packing density
    "HermiteInvariant"Hermite invariant
    "PackingRadius"packing radius
    "Thickness"thickness
    "Volume"volume of the fundamental region
  • Other properties include:
  • "QuadraticForm"quadratic form
    "AutomorphismGroupOrder"order of the automorphism group
  • Naming-related properties include:
  • "AlternateNames"alternate English names
    "Name"English name
    "Notation"standard notation for display
    "StandardName"Mathematica name
  • LatticeData[lattice, "Classes"] gives a list of the classes in which lattice occurs.
  • LatticeData["class"] gives a list of named lattices in the specified class.
  • LatticeData[lattice, "class"] gives True or False depending on whether lattice is in the specified class.
  • Basic classes of lattices include:
  • "Even"even
    "Extremal"extremal
    "Integral"integral
    "Odd"odd lattice
    "Unimodular"unimodular
  • Negative classes of lattices include:
  • "Nonextremal"not extremal
    "Nonintegral"not integral
    "Nonunimodular"not unimodular
  • Using LatticeData may require internet connectivity.
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