Threefold Symmetry from Rotated Plane Waves
Threefold Symmetry from Rotated Plane Waves
The plot of the function (z)=f(z)= represents a plane wave periodic in the direction of the imaginary axis. Threefold symmetry is created by taking the mean of the functions (z), (z)=f(z)=, and (z)=fz=, where =(-1±i are the two complex roots of the equation =1. So (z)=((z)+(z)+(z)) is invariant under rotation by ; in other words, it has threefold symmetry.
g
0
2πiIm(z)
e
g
0
g
1
ω
3
πi-Im(z)+
3
Re(z)e
g
2
2
ω
3
πi-Im(z)-
3
Re(z)e
ω
3
3
)23
z
f
3
1
3
g
1
g
2
g
3
2π/3
Similarly, (z) has -fold symmetry for , using the roots of unity, the solutions of =1.
f
n
n
n=4,5,6,7
th
n
n
z