Any fixed-point symmetry of a lattice transforms lattice points into new lattice points.
Its matrix elements with respect to a basis of lattice point vectors must therefore all be integers.
As a consequence also the trace
trace = 2 cos(theta) + n-2
of any lattice rotation must be integer, with n the dimension of the space. 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.
Below two applets show graphically why no two-dimensional lattices exist with five (p=5) and seven (p=7) fold rotation symmetries. In contrast to this the hexagonal (p=6) honeycomb lattice is well known to exist.
Soli Deo Gloria. Created with Cinderella v2.0beta and v1.2 by Eckhard Hitzer (Fukui).