Frieze Functions
Frieze Functions
This Demonstration makes use of the following table of recipes to create frieze functions, represented by series of the form , with complex coefficients . Let represent a half-turn, and perpendicular mirror symmetries, and a glide reflection along the axis.
f(z)=
a
m,n
imz
e
-in
z
e
a
m,n
ρ
2
σ
y
σ
x
γ
x
x
groupname | symmetries | rules |
p111 | none | a m,n |
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
iz
m
1
e
-i
n
1
z
e
a
m
2
n
2
iz
m
2
e
-i
n
2
z
e
according to the rules. For example, gives
p211
f
2
a
0,0
a
m
1
n
1
iz
m
1
e
-i
n
1
z
e
-iz
m
1
e
i
n
1
z
e
a
m
2
n
2
iz
m
2
e
-i
n
2
z
e
-iz
m
2
e
i
n
2
z
e
and there would be four interior terms in the functions for and .
p2mm
p2mg
Only two copies of a unit cell are shown.