Symmetry Group S(
P
)
Symmetries in the Group
p111
Only translation
p1a1
Translation and glide-reflection
p1m1
Translation and horizontal reflection
pm11
Translation and vertical reflection
p112
Translation and rotation
pma2
Translation, vertical reflection, glide-
reflection, and rotation
pmm2
Translation, vertical reflection, and rotation
Since there are fifteen pattern types and seven symmetry groups, obviously there are
going to be multiple pattern types to a certain symmetry group.
The difference will come
in the induced groups.
Observe the figures below to see an example of each of the fifteen
strip patterns and an explanation to which symmetry group and the induced group goes
with each pattern type.
PS1
p111
c1
7

PS2
p1a1
c1
PS3
p1m1
c1
PS4
p1m1
d1
PS5
pm11
c1
PS6
pm11
d1
PS7
p112
c1
8

PS8
p112
d1
PS9
pma2
c1
PS10
pma2
c2
PS11
pma2
d1
PS12
pmm2 c1
9

PS13
pmm2 d1
PS14
pmm2 d1
PS15
pmm2 d2
Notice that in the case of PS13 and PS14 they both have an induced group of d1.
A motif
transitive subgroup is a subgroup of the symmetry group that maps any motif onto any
other copy of the motif in the pattern.
These two pattern types differ because they have
different motif-transitive proper subgroups.
I will not elaborate on this any further, but
you can refer to Grünbaum and Shephard for further details.
Pattern Types of Periodic Patterns
There are fifty-one different pattern types within the periodic patterns labeled PP1
through PP51.
A periodic pattern is a pattern that classifies further the seventeen
wallpaper groups.
The wallpaper groups are as follows taken from
Tilings and Patterns
(Grünbaum and Shephard, 1987).
Symmetry Groups
Description
p1
Only translation
pg
Two glide-reflections
pm
Two reflections
cm
One glide-reflection, one reflection
p2
Four rotations of period two
pgg
Two glide-reflections, two rotations of
period 2
10

pmg
Two glide-reflections, one reflection, two
rotations of period two
pmm
Four reflections, four rotations of period
two
cmm
Two glide-reflections, two reflections, four
rotations of period two
p3
Three rotations of period three
p31m
One glide-reflection, one reflection, two
rotations of period three
p3m1
One glide-reflection, one reflection, three
rotations of period three
p4
One rotation of period two, two rotations of
period four
p4g
Two glide-reflections, one reflection, one
rotation of period two, one rotation of
period four
p4m
One glide-reflection, three reflections, one
rotation of period two, one rotation of
period four
p6
One rotation of period two, one rotation of
period four, one rotation of period six
p6m
Two glide-reflections, two reflections, one
rotation of period two, one rotation of
period four, one rotation of period six
Furthermore, the wallpaper symmetry groups come from an induced group of either cn,
where
1
n
or dn, where
1
n
.
In a couple of cases, the symmetry group and induced
group of two pattern types are the same, so one must look further to the motif-transitive
subgroup to distinguish a difference in pattern types.
See an example of each of the fifty-
one different periodic pattern types along with their symmetry group and induced group
below.

#### You've reached the end of your free preview.

Want to read all 20 pages?

- Fall '08
- STAFF
- Number Theory, Symmetry group, symmetry groups, Grünbaum, pattern types