No 5-fold rotation symmetric 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, 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=5 does not work. Trying to combine six pentagons to a lattice leaves gaps, which cannot be closed no matter how one may try to arrange the pentagons. Compare also the applets for p=6 and p=7.

Please enable Java for an interactive construction (with Cinderella).
[ 2D lattices | GA with Cinderella ]

Soli Deo Gloria. Created with Cinderella v2.0beta and v1.2 by Eckhard Hitzer (Fukui).