8/10/2019 2D Plane Group Flash Cards
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m
s group contains reflections
d glide reflections with parallel
es. There are no rotations in this
oup. The translations may be
lined at any angle to each
er, but the axes of the
ections bisect the angle formed
the translations, so the lattice is
ombic.
pg
This group contains glide
reflections. The direction of the
glide reflection is parallel to
one axis of translation and
perpendicular to the other axis
of translation. There are
neither rotations nor
reflections.
pm
This group contains
reflections. The axes of
reflection are parallel to one
axis of translation and
perpendicular to the other axis
of translation. The lattice is
rectangular. There are neither
rotations nor glide reflections.
p1
This is the simplest symme
group. It consists only of
translations. There are neit
reflections, glide-reflection
nor rotations. The two
translation axes may be
inclined at any angle to eac
other.
mm
is symmetry group contains
rpendicular axes of
flection, with 180 rotations
here the axes intersect.
pmg
This group contains reflections
and glide reflections which are
perpendicular to the reflection
axes. It has rotations of order
2 on the glide axes, halfway
between the reflection axes.
pgg
This group contains no
reflections, but it has glide-
reflections and 180 rotations.
There are perpendicular axes
for the glide reflections, and
the rotation centers do not lie
on the axes.
p2
This group differs only from p
that it contains 180 rotations
is, rotations of order 2. As i
symmetry groups there
translations, but there are ne
reflections nor glide reflect
The two translations axes ma
inclined at any angle to other.
3m1
is group is similar to the last in
at it contains reflections and
der-3 rotations. The axes of the
lections are again inclined at
to one another, but for this
oup all of the centers of rotation
lie on the reflection axes.
ere are some glide-reflections.
p31m
This group contains reflections
(whose axes are inclined at
60 to one another) and
rotations of order 3. Some of
the centers of rotation lie on
the reflection axes, and some
do not. There are some glide-reflections.
p3
This is the simplest group
that contains a 120-
rotation, that is, a rotation
of order 3. It has no
reflections or glide
reflections.
cmm
This group has perpendicu
reflection axes, as does gro
pmm, but it also has additio
rotations of order 2. The
centers of the additional
rotations do not lie on the
reflection axes.
6
This group contains 60
rotations, that is, rotations
of order 6. It also contains
rotations of orders 2 and
3, but no reflections or
glide-reflections.
p4m
This group also has both order 2 andorder 4 rotations. This
group has fouraxes of reflection. The axes ofreflection are
inclined to each other by45 so that four axes of reflection
passthrough each order 4 rotation center.Every rotation center lies
on somereflection axes. There are also twoglide reflections passing
through eachorder 2 rotation, with axes at 45 tothe reflection
axes. p4mis very
common and is easy to recognizebecause of its square
lattice.
p4g Like p4, this group contains
reflections and rotations of orders
2 and 4. There are two
perpendicular reflections passing
through each order 2 rotation.
However, the order 4 rotation
centers do not lie on any reflection
axis. There are four directions of
glide reflection.
p4
This group has a 90 rotatio
that is, a rotation of order 4
also has rotations of order
The centers of the order-2
rotations are midway betwe
the centers of the order-4
rotations. There are no
reflections.
