The $A_2$ lattice, also known as the **hexagonal lattice**, is the root lattice associated to the $A_2$ and $G_2$ root systems.

## Lattice Invariants

Dimension: | $2$ |

Determinant: | $3$ |

Level: | $3$ |

Label: | $2.3.3.1.1$ |

Density: | $0.906899682117108925297039128820\dots$ |

Group order: | $12$ |

Hermite number: | $1.15470053837925152901829756100\dots$ |

Minimal vector length: | $2$ |

Kissing number: | $6$ |

Normalized minimal vectors: |
$(1, 0)$, $(1, 1)$, $(0, 1)$ |

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## Theta Series

## Gram Matrix

$\left(\begin{array}{rr} 2 & -1 \\ -1 & 2 \end{array}\right)$

## Genus Structure

Class number: | $1$ |

Genus representatives: | $\left(\begin{array}{rr} 2 & 1 \\ 1 & 2 \end{array}\right)$ |

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## Comments

This integral lattice is the A2 lattice.

This is a root lattice.

# Additional information

This lattice is the unique solution of the sphere packing problem and the general kissing number problem in dimension 2.