18.701
2010
Plane
Crystallographic
Groups
with
Point
Group
D
1
.
This
note
describes
discrete
subgroups
G
of
isometries
of
the
plane
P
whose
translation
lattice
L
contains
two
independent
vectors,
and
whose
point
group
G
is
the
dihedral
group
D
1
,
which
consists
of
the
identity
and
a
reﬂection
about
the
origin.
Among
the
ten
possible
point
groups
C
n
or
D
n
with
n
= 1
,
2
,
3
,
4
,
6,
the
analysis
of
D
1
is
among
the
most
complicated.
There
are
three
different
types
of
group
with
this
point
group.
Let
G
be
a
group
of
the
type
that
we
are
considering.
We
choose
coordinates
so
that
the
reﬂection
in
G
is
about
the
horizontal
axis.
As
in
the
text,
we
put
bars
over
symbols
that
represent
elements
of
the
point
group
G
to
avoid
confusing
them
with
the
elements
of
G
.
So
we
denote
the
reﬂection
in
G
by
r
.
The
lattice
L
consists
of
the
vectors
v
such
that
t
v
is
in
G
,
and
we
know
that
elements
of
G
map
L
to
L
.
If
v
is
in
L
,
rv
is
also
in
L
.
I.
The
shape
of
the
lattice
Proposition
1.
There
are
horizontal
and
vertical
vectors
a
= (
a ,
t
1
0)
t
and
b
=
(0
, b
2
)
respectively,
such
that,
with
c
=
1
2
(
a
+
b
)
,
L
is
one
of
the
two
lattices
L
1
or
L
2
,
where
L
1
=
Z
a
+
Z
b,
is
a
‘rectangular’
lattice,
and
L
2
=
Z
a
+
Z
c,
is
a
‘triangular’
lattice
.
Since
b
= 2
c
−
a
,
L
1
→
L
2
.
The
lattice
L
1
is
called
‘rectangular’
because
the
horizontal
and
vertical
lines
through
its
points
divide
the
plane
into
rectangles.
The
lattice
L
2
is
obtained
by
adding
to
L
1
the
midpoints
of
every
one
of
these
rectangles.
There
are
two
scale
parameters
in
the
description
of
L
–
the
lengths
of
the
vectors
a
and
b
.
The
usual
classification
of

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