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, 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. The applet below shows why p=7 does not work. Trying to combine eight septagons to a lattice inevitably creates overlaps, which cannot be avoided no matter how one may try to arrange the septagons. Compare also the applets for p=6 and p=5.
Soli Deo Gloria. Created with Cinderella v2.0beta and v1.2 by Eckhard Hitzer (Fukui).