The n-leaf Rose
by Emily Kennedy

Note that the number of petals on the rose is equal to the number of times the graph passes through the origin in one cycle.

r = sin(3θ) 3 passes through the origin, 3 petals

r = sin(4θ) 8 passes through the origin, 8 petals

r = sin((3/2)θ) 6 passes through the origin, 6 petals

How many times does the graph of r = sin(nθ)
pass through the origin in one cycle?

The graph passes through the origin whenever r = 0,
so we need to find the number of values of θ such that sin(nθ) = 0.

To count the number of multiples of within the interval [0,c),
(and thus the number of petals)
we divide the cycle lengths c we found earlier by ,
and we find the following:

	# of petals
α and β both odd
either α or β even

 
  So the graph of has 3 petals,

the graph of also has 3 petals,

the graph of has 4 petals,

the graph of also has 4 petals,

the graph of has 6 petals,

etc.

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