Threefold Color-Turning Wallpaper Functions
Threefold Color-Turning Wallpaper Functions
This Demonstration illustrates threefold color-turning wallpaper functions using complex functions in Fourier series of the form , where (z)=(z)+(z)+(z)3 are color-turning waves, and = are called lattice waves.
f(x)=
∑
m,n∈Z
a
m,n
E
m,n
E
m,n
E
m,n
-1
ω
E
n,-(m+n)
-2
ω
E
-(m+n),n
E
m,n
2πi(mX+nY)
e
The hexagonal lattice has basis vectors , , and the lattice coordinates are , .
1
ω=(-1+i
3
)2X=x+y
3
Y=2y
3
The Fourier series are actually truncated to two terms: .
f(z)=1/2,(z)+1/4a,(z)
E
m
1
n
1
E
m
2
n
2