--- title: "Lattice" language: "en" type: "Entity" summary: "Notable mathematical lattices." canonical_url: "https://reference.wolfram.com/language/ref/entity/Lattice.html" source: "Wolfram Language Documentation" related_guides: - title: "Entity Types" link: "https://reference.wolfram.com/language/guide/EntityTypes.en.md" related_functions: - title: "LatticeSystem" link: "https://reference.wolfram.com/language/ref/entity/LatticeSystem.en.md" - title: "Graph" link: "https://reference.wolfram.com/language/ref/entity/Graph.en.md" - title: "Knot" link: "https://reference.wolfram.com/language/ref/entity/Knot.en.md" - title: "FiniteGroup" link: "https://reference.wolfram.com/language/ref/entity/FiniteGroup.en.md" - title: "IntegerSequence" link: "https://reference.wolfram.com/language/ref/entity/IntegerSequence.en.md" --- # Lattice Notable mathematical lattices. [Related Interpreter](https://reference.wolfram.com/language/ref/interpreter/Lattice.en.md) Entity["Lattice", name] or Entity[...] represents an entity of type "Lattice". Entity[...][prop] gives the value of a specified property. Entity[...][{propi, …}] gives the value of a list of properties. EntityClass["Lattice", {propi -> speci, …}] represents a class of entities with values of propi defined by speci. ## Sample Entities * Entity["Lattice", "BaseCenteredMonoclinic"]Entity["Lattice", "BodyCenteredCubic"]Entity["Lattice", "CoxeterTodd"]Entity["Lattice", "FaceCenteredCubic"]Entity["Lattice", "HexagonalLattice"]Entity["Lattice", "Leech"]Entity["Lattice", "SimpleCubic"]Entity["Lattice", "SimpleMonoclinic"]Entity["Lattice", "SquareLattice"]Entity["Lattice", "TetrahedralPacking"]**…** ## Sample Entity Classes * EntityClass["Lattice", "Even"]EntityClass["Lattice", "Extremal"]EntityClass["Lattice", "Integral"]EntityClass["Lattice", "Nonextremal"]EntityClass["Lattice", "Nonintegral"]EntityClass["Lattice", "Nonunimodular"]EntityClass["Lattice", "Odd"]EntityClass["Lattice", "Unimodular"]**…** ## Properties | | | | ---------------------- | -------------------- | | AlternateNames | alternate names | | AutomorphismGroupOrder | number of symmetries | | Basis | basis | | CenterDensity | center density | | Classes | classes | | CoveringRadius | covering radius | | CoxeterNumber | Coxeter number | | CrystalFamily | crystal family | | CrystalSystems | crystal system | | Density | density | | Determinant | determinant | | Dimension | dimension | | Dual | dual | | EntityClasses | entity classes | | EntityTypeList | entity type list | | Even | even | | Extremal | extremal | | GeneratorMatrix | generator matrix | | GramMatrix | Gram matrix | | Graphics3D | 3D graphic | | HermiteInvariant | Hermite invariant | | Image | image | | Integral | integral | | KissingNumber | kissing number | | LatticeSystem | lattice system | | MinimalNorm | minimal squared norm | | MinimalVectors | smallest vectors | | ModularNumber | modular number | | Name | name | | Nonextremal | nonextremal | | Nonintegral | nonintegral | | Nonunimodular | nonunimodular | | Notation | notation | | Odd | odd | | PackingRadius | packing radius | | QuadraticForm | quadratic form | | RadialFunction | radial function | | ThetaSeriesFunction | theta series | | Thickness | thickness | | Unimodular | unimodular | | Volume | volume | ## Details * ``"Lattice"`` entities include particular named mathematical lattices as well as members of parametrized families. * ``"Lattice"`` entity classes include classifications based on various mathematical properties. * Some properties are available for the ``"Lattice"`` entity type as a whole and can be given using the form ``EntityValue["Lattice", property]``. Such properties include: | | | | ----------------------------- | ------------------------------------------------------------------ | | "Properties" | the list of available properties | | "PropertyCanonicalNames" | the standard names of available properties | | "SampleEntities" | a sample list of available entities (typically of length 10) | | "SampleEntityClasses" | a sample list of available entity classes (typically of length 10) | | "EntityCount" | number of entities available | | "Entities" | the list of available entities | | "EntityCanonicalNames" | the standard names of available entities | | "EntityClasses" | the list of available entity classes | | "EntityClassCanonicalNames" | the standard names of available entity classes | | "PropertyClasses" | the list of available property classes | | "PropertyClassCanonicalNames" | the standard names of available property classes | | "PropertyCount" | number of properties available | * The following annotations can be used in the third argument of ``EntityValue["Lattice", property, annotation]`` : | | | | --- | --- | | "EntityAssociation" | an association of entities and entity-property values | | "PropertyAssociation" | an association of properties and entity-property values | | "EntityPropertyAssociation" | an association in which the specified entities are keys, and values are a nested association of properties and entity-property values | | "PropertyEntityAssociation" | an association in which the specified properties are keys, and values are a nested association of entities and entity-property values | | "Dataset" | a dataset in which the specified entities are keys, and values are an association of property names and entity-property values | * The following annotations can be used in the second argument of ``EntityValue[property, annotation]`` : "Description" a brief textual description of the property --- ## Examples (4) ### Basic Examples (4) Use ctrl+= for entity discovery: ```wl In[1]:= \[FreeformPrompt]["coxeter-todd lattice"] Out[1]= Entity["Lattice", "CoxeterTodd"] ``` --- Find a property value for an entity: ```wl In[1]:= Entity["Lattice", "SquareLattice"]["ThetaSeriesFunction"][x] Out[1]= EllipticTheta[3, 0, E^I π x]^2 ``` --- Retrieve a dataset of all available properties for an entity: ```wl In[1]:= Entity["Lattice", "HexagonalLattice"]["Dataset"]//DeleteMissing Out[1]= Dataset[Association[EntityProperty["Lattice", "AlternateNames"] -> {}, EntityProperty["Lattice", "Basis"] -> SparseArray[Automatic, {2, 2}, 0, {1, {{0, 2, 3}, {{2}, {1}, {2}}}, {1, -1, -1}}], EntityProperty["Lattice", "CenterDensity"] -> ... {EntityClass["Lattice", "Integral"], EntityClass["Lattice", "Nonunimodular"], EntityClass["Lattice", "Odd"]}, EntityProperty["Thing", "EntityTypeList"] -> {EntityType["Lattice"]}, EntityProperty["Thing", "Name"] -> "hexagonal lattice"]] ``` --- Return lattices in three dimensions: ```wl In[1]:= EntityClass["Lattice", "Dimension" -> 3]//EntityList Out[1]= {Entity["Lattice", "BaseCenteredMonoclinic"], Entity["Lattice", "BaseCenteredOrthorhombic"], Entity["Lattice", "BodyCenteredCubic"], Entity["Lattice", "BodyCenteredOrthorhombic"], Entity["Lattice", "CenteredTetragonal"], Entity["Lattice", "FaceCent ... Hexagonal"], Entity["Lattice", "SimpleMonoclinic"], Entity["Lattice", "SimpleOrthorhombic"], Entity["Lattice", "SimpleTetragonal"], Entity["Lattice", "SimpleTriclinic"], Entity["Lattice", "SimpleTrigonal"], Entity["Lattice", "TetrahedralPacking"]} ``` ## See Also * [`LatticeSystem`](https://reference.wolfram.com/language/ref/entity/LatticeSystem.en.md) * [`Graph`](https://reference.wolfram.com/language/ref/entity/Graph.en.md) * [`Knot`](https://reference.wolfram.com/language/ref/entity/Knot.en.md) * [`FiniteGroup`](https://reference.wolfram.com/language/ref/entity/FiniteGroup.en.md) * [`IntegerSequence`](https://reference.wolfram.com/language/ref/entity/IntegerSequence.en.md) ## Related Guides * [Entity Types](https://reference.wolfram.com/language/guide/EntityTypes.en.md) ## History * [Introduced in 2019 (12.0)](https://reference.wolfram.com/language/guide/SummaryOfNewFeaturesIn120.en.md)