Counting rose petals
The simplicity and symmetry of Rhodonea or rose curves have fascinated mathematicians since they were first named by the Italian mathematician Guido Grandi in the 1700s. We were fascinated by an interesting pattern created by counting the number of petals of these curves.
Rose curves are not drawn using the coordinates corresponding to horizontal and vertical axes (called Cartesian coordinates). Instead, rose curves are drawn using polar coordinates. The first of the polar coordinates of a point is : the distance between the point and the pole (the pole is another name for the origin , and this distance is often referred to as the radius). The second of the polar coordinates is (read as "theta"): the angle formed by a ray (beginning at the pole extending to the point) and the polar axis (essentially the -axis). (You can read more about how to draw curves with polar coordinates in the article Polar power.)
The rose curve has the general equation of either or where is a non-zero real number and is an integer. You can see what happens with different values of in the example below. An interesting pattern emerges as you increase the value of from 1 to 20: see if you can spot it.
The rose curve plotted for a=1, and the values of n ranging from 1 to 20.
You might have observed that when is even, there are petals on the rose, and when is odd, there are only petals. This pattern emerges if is any integer, including if was negative. We will explore the maths behind this interesting pattern and we will see that this is always the case for the number of petals for a rose curve.
Let me count the ways
To begin explaining this pattern we must take a look at the equations that produce these curves. Suppose we examine . Using our knowledge of trigonometric functions like cosine, we know that must range from -1 to 1, and thus must range from to .
The graph of y=cos(4x) (where r=1, n=4) drawn in cartesian coordinates.
Thinking about what this means for the rose curve drawn in polar coordinates, the peaks of the petals are those points on the curves that are furthest from the pole; those points are where reaches or . Looking at the rose curve , we see 8 places where reaches 1 or -1, corresponding to the peaks of each of the petals.
The rose curve The graph of y=cos(4x) (where r=1, n=4) drawn using polar coordinates. The peaks of the petals correspond to the peaks and troughs of the cosine graph above.
To count the number of petals in a rose curve, we need to count the number of peaks. But which values of create a peak? Or in other words, which values of allow to reach or ? The function reaches or when is a multiple of . So the peaks of a rose curve occur when equals or , which is when is some multiple of .
Putting this mathematically, the peaks of the petals occur when , where is an integer. However, we are only considering values of between and . That's because the angle defined by in our polar coordinate system is the same as the angle defined by , and , so as soon as exceeds we are redrawing our rose.
The values of where
With some simple rearrangement we can find in terms of , and :
However, this seems to contradict our initial observation, that when is odd, there are only petals, not petals. To understand why this is not a contradiction, we must do some casework.
We can list out the polar coordinates for each peak for a rose curve with the formula . Taking all the values of from to , we get peaks when
We see that anytime that is odd, the first coordinate of these peaks is negative: . Since the first polar coordinate was defined as the distance between the corresponding point and the pole, this doesn't make sense: a distance can't be negative. However, there is a convention to deal with negative radius in the polar coordinate system: a point with a negative radius is exactly the same point as one with the same positive radius, but in the opposite direction from the pole. That is, is the same point as .
The point (3, π/3) is opposite the point (-3, π/3) from the pole. We can move one point to the other by adding π to the second polar coordinate. So we can rewrite (-3, π/3) as (3, 4π/3).
Thus for our list of the peaks of the rose curve, for those with a negative first coordinate , we can add to the second coordinate and switch the sign of the , and the points these new coordinates describe will still be equivalent:
Evens are exclusive
After rewriting our list of peaks of the rose curve, we have peaks at for even , and at for odd .
If is even, then is an odd number when is odd. So the rewritten coordinates for the odd do not correspond to the coordinates of any of the points for the even values of . Thus we end up with distinct points.
You can see eight distinct petals of the rose curve being drawn below:
Odd to overlap
If we try the case where is odd, we get something very different. When we list out the values that produce the peaks, and transform them such that all the radii are positive, we don't get distance points. Because is odd for the negative radii, and is odd as well, then the number is even. However this means that the rewritten coordinates for some odd value are just one of the coordinates we've already listed for some even value ! Therefore, each point is duplicated once and exactly one. Thus, instead of petals, we get petals.
You can see the overlapping petals for the rose curve being drawn below:
We've explored this pattern for rose curves described by , but the same line of reasoning explains the pattern for curves described by .
About this article
John Eckhart is a former High School Honors and AP mathematics teacher who has an interest in analytic geometry. He currently works as an education specialist at Marshall Space Flight Center.
Frank Lee is a high school junior at Sylvania Northview High School (Toledo, Ohio) who enjoys geometry. Challenged by his precalculus teacher to come up with a proof for this neat pattern found in rose curves, he found something equally neat