WOLFRAM|DEMONSTRATIONS PROJECT

Frieze Functions

​
function
1
2
3
4
5
6
7
m
1
1
2
3
4
5
n
1
1
2
3
4
5
m
2
1
2
3
4
5
n
2
1
2
3
4
5
a
0,0
a
m
1
,
n
1
a
m
2
,
n
2
This Demonstration makes use of the following table of recipes to create frieze functions, represented by series of the form
f(z)=
a
m,n
imz
e
-in
z
e
, with complex coefficients
a
m,n
. Let
ρ
2
represent a half-turn,
σ
y
and
σ
x
perpendicular mirror symmetries, and
γ
x
a glide reflection along the
x
axis.
groupname
symmetries
rules
p111
none
a
m,n
arbitrary
p211
ρ
2
a
m,n
=
a
-m,-n
p1m1
σ
y
a
m,n
=
a
n,m
p11m
σ
x
a
m,n
=
a
-n,-m
p11g
γ
x
a
m,n
=
m+n
(-1)
a
-n,-m
p2mm
ρ
2
,
σ
y
a
m,n
=
a
-m,-n
=
a
n,m
=
a
-n,-m
p2mg
γ
x
,
ρ
2
a
m,n
=
m+n
(-1)
a
-n,-m
=
a
-m,-n
=
m+n
(-1)
a
n,m
The functions used here are truncated to
f(z)=
a
0,0
+
a
m
1
,
n
1

i
m
1
z
e
-i
n
1
z
e
+...+
a
m
2
,
n
2

i
m
2
z
e
-i
n
2
z
e
+...
,
according to the rules. For example,
p211
gives
f
2
(z)=
a
0,0
+
a
m
1
,
n
1

i
m
1
z
e
-i
n
1
z
e
+
-i
m
1
z
e
i
n
1
z
e
+
a
m
2
,
n
2

i
m
2
z
e
-i
n
2
z
e
+
-i
m
2
z
e
i
n
2
z
e

and there would be four interior terms in the functions for
p2mm
and
p2mg
.
Only two copies of a unit cell are shown.