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2D hexagonal lattice

Any fixed-point symmetry of a lattice transforms lattice points into new lattice points.
Its matrix elements must therefore all be integers. As a consequence also the trace

trace = 2 cos(theta) + n-2

of any lattice rotation must be integer. n is the dimension of the space concerned.
Together with the cyclic condition theta = 2Pi/p for a
rotation generating a rotation subgroup we get:

p = 1,2,3,4,6

but not 5,7,8, etc. The applet below shows a two-dimensional lattice with a six fold rotation
symmetry (p=6). You can drag the red line interactively to rotate the green lattice.
The six corners of the central hexagon correspond to six symmetry
rotations, which are elements of 2C_6.
You can even drag the center of the green lattice to different positions. That
way you can verify, that
it doesn't matter around which center of one of the
black hexagons you rotate. The rotation symmetry
is for rotations around any
hexagon center the same.
Compare also applets for p=5 and p=7
illustrating
why 5 and 7 fold two dimensional lattices don't exist.

[
2D lattices |
GA with Cinderella
]

Soli Deo Gloria. Created with Cinderella1.2
by Eckhard Hitzer (Fukui).