Periodic Functions with m-Fold Symmetry of Type k
Periodic Functions with m-Fold Symmetry of Type k
Suppose that and are relatively prime positive integers (i.e. ). Following [1, p. 14], define a curve with period to have -fold symmetry of type , of the form
m
k
gcd(k,m)=1
f(t)
2π
m
k
ft+=f(t)
2π
m
2πik/m
e
This Demonstration shows graphs of the function , where , , and are complex coefficients and the frequencies , , and are integers. In particular, for the frequencies , the curve has fivefold symmetry of type 1. Increase the frequencies by 1 to get . The new curve has fivefold symmetry of type 2. Increase the frequencies by 1 once again. The curve has fivefold symmetry of type 3.
f(t)=a+b+c
iAt
e
iBt
e
iCt
e
a
b
c
A
B
C
(A,B,C)=(1,6,-14)
(2,7,-13)
In the graphics, the green arrow shows the maximum modulus of the curve. Increasing the time from 0 to makes the green arrow skip two maxima and settle on a third. To get similar results for sevenfold symmetry, use the frequencies .
2π/5
(3,10,-11)