Assignment 11: The N Leaf Rose
By: David Drew
EMAT 6680, J. Wilson
In this investigation we want to go through several explorations with what is known as the N Leaf Rose. What is the N Leaf Rose and how does a rose relate to math? It is just an equation that looks like a flower when you graph it for several different values. The equations we want to look at and compare and contrast are first and always in blue and †second and always in red with different values of k. The theta in our equation will range from 0 to 2pi.
We start with the most basic of our equations with a = b = 1 and k = 0.
We notice that our second equation doesnít even appear in the graph. Thatís because sin(0) = 0 so therefore there is nothing at all. In the other case we have a 1 so itís no surprise that our first graph should have radius of 1. Letís take it up a notch and see what happens if k = 1.
Now we start to see something interesting in the first graph and our second graph have caught up a little by just being visible. The image in graph one is obviously bigger than the second one, and its shape is that of a cardioid while the shape in the second graph is a lowly circle. Letís continue to increase the value of k by one.
Now the second graph has jumped passed the first one in the fact that we now have 4 leaves in our second image while the first graph just has two leaves. But notice that the second graph is still smaller than the first one. Letís take it up again by a value of 1.
Now we can see that it is basically the same shape just scaled by some factor. Letís do one more with a value of 4 and see what happens.
Now we can make a limited conjecture that when the value of k is even then the graph of †will have twice as many leaves, and be smaller than the graph of †with the values of a and b equal to one. And when both of the kís equal an odd number then they will have the same leaves but again †will be smaller.
Now that weíve looked at the values of a and b at 1 letís suppose a and b are equal to 2.
Once again we get a circle with radius equal to a in the first graph and nothing in the second graph because, again, sin(0) = 0. So letís do the same methods as before and increase k by 1.
Again we get a cardioid and a circle respectively in our first and second graph. But itís also no surprise that the objects are twice the size as before. Can we assume that the values of our earlier graphs hold true in these next examples, and can we guess that they will all be twice the size as before?
Do these look like they are twice as big as before? Donít let your eyes be fooled. I had to shrink graphs to fit the images inside the picture box. They are in fact twice as big as the ones previous. So can we expect that this will hold true for all our graphs? The answer is yes, and weíll do one more to show you.
Here are some cool looking designs with our equations. This one is our equation and the values of a and b are increasing from 2 to 10 in even intervals. Red is 2, yellow is 4, blue is 6, black is 8, and gray is 10.
In this one below we have our second equation †with b increasing from 1 to 6.
Perhaps this is where the term N Leaf Rose comes from?
One might wonder about the cosine function and itís involvement with all these values. Can we expect the cosine of each of these equations to act in the same fashion? Letís do a few quick observations to find out.
All we did on this graph is make the sine a cosine function. So now we have †where a is increasing from 1 to 6 in succession. Notice that we still have eight pedals in our flower, but they have rotated. It turns out that our cosine function is just a phase shift of our sine function and the shapes have stayed the same. So if you went back and did all this over again with the cosine function you would arrive at the same conclusions. In fact if youíre curious and you have some time click on these four links to do you own investigation with cosine and sine. Just click on the equations and the links will be there.
Write Up by David Drew