Hexagonal lattice

The hexagonal lattice or triangular lattice is one of the five two-dimensional Bravais lattice types. The symmetry category of the lattice is wallpaper group p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° and are of equal lengths,

Honeycomb lattice

The honeycomb lattice is a special case of the hexagonal lattice with a two-atom basis. The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb lattice can be seen as the union of two offset triangular lattices.

In nature, carbon atoms of the two-dimensional material graphene are arranged in a honeycomb lattice.

<!--A triangular lattice itself can be divided into 3 offset triangular lattices, shown above in red, green and blue. A triangular lattice is also called an A2 lattice, A2, and the union of three triangular lattices is A*2.

In addition to these points, or instead of them, the sides of the hexagons may be shown; depending on application they may be called lattice bonds.

With respect to a hexagonal lattice we can distinguish two sets or three directions:

A pattern with 3- or 6-fold rotational symmetry has a lattice of 3-fold rotocenters (including possible 6-fold rotocenters) that is this finer lattice relative to the lattice of translational symmetry.

In the case of 6-fold rotational symmetry the 6-fold centers form a lattice as coarse as the lattice of translational symmetry, i.e. there is one 6-fold center and there are two 3-fold centers per primitive cell.

For reflection axes, there are two possible sets of directions, mentioned above. In the case of 3-fold symmetry either none (p3) or one of the two applies: In the 6-fold case either none (p6) or both (p6m) apply.

If there are reflection axes in the main translation directions, one of the three sets of rotocenters play a different role than the other two: these reflection axes pass through them. With p6 one set is special because of being 6-fold.

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Crystal classes

The hexagonal lattice class names, Schönflies notation, Hermann-Mauguin notation, orbifold notation, Coxeter notation, and wallpaper groups are listed in the table below.
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