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No 7-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=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.

[
2D lattices |
GA with Cinderella
]

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