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    Cubic Lattice

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    Common names

    primitive cubic | 3-dimensional integer lattice

    Description of lattice

    basis | (1 | 0 | 0) | (0 | 1 | 0) | (0 | 0 | 1) Gram matrix | (1 | 0 | 0 0 | 1 | 0 0 | 0 | 1)

    Lattice invariants

    dimension | 3 determinant | 1 minimal squared norm | 1 smallest vectors | (1 | 0 | 0) | (0 | 1 | 0) | (0 | 0 | 1) | (-1 | 0 | 0) | (0 | -1 | 0) | (0 | 0 | -1) kissing number | 6

    Lattice-packing invariants

    packing radius | 1/2 = 0.5 covering radius | sqrt(3)/2≈0.866025 density | π/6≈0.523599 center density | 1/8 = 0.125 Hermite invariant | 1 thickness | (sqrt(3) π)/2≈2.7207 volume | 1

    Quadratic form and theta series

    quadratic form | x^2 + y^2 + z^2 theta series (closed series) | ϑ_3(0, e^(i π x))^3

    More properties

    dual | integer lattice | 3 modular number | 1 number of symmetries | 48

    Common properties

    integral | odd | unimodular

    Crystallographic properties

    lattice system | cubic crystal system | cubic crystal family | cubic required point group symmetry | 4 3-fold rotation axes point groups | 5 space groups | 36

    Point groups

    crystal class | Schönflies | Hermann-Mauguin tetartoidal | T | 23 diploidal | T_h | m3^_ gyroidal | O | 432 tetrahedral | T_d | 4^_3m hexoctahedral | O_h | m3^_m

    Space groups

    crystal class | IUCr number | Hermann-Mauguin tetartoidal | 198 | 199 | P213 | P23 diploidal | 204 | 205 | 206 | Pa3^_ | Pm3^_ | Pn3^_ gyroidal | 211 | 212 | 213 | 214 | P4132 | P4232 | P432 | P4332 tetrahedral | 219 | 220 | P43m^_ | P43n^_ hexoctahedral | 227 | 228 | 229 | 230 | Pm3m^_ | Pm3n^_ | Pn3m^_ | Pn3n^_

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