Assignment 11

by

Johnie Forsythe

Computers allow us to make beautiful shapes by using equations. Lets explore some polar equations to make figure out how to construct a beautiful flower!

Lets investigate the equation:

In this equation, the variables a, b, and k are held constant or they are equal to one. Theta is between 0 and 2.

Let's continue to hold a and b constant, but this time....vary the values for k. We wil let k=2. This yields the following graph:

Observations:

When k=2, there are two leaves on the graph.

These leaves cross the x-axis at the origin and points (-2,0) and (2,0).

Continuing with our investigation...... what happens when we hold a and b constant and let k=3

There are three leaves in this graph.

Let's try somethihng different:Let k=1/2

Very interesting.....the graph looks like half of the other graph where k=1

It appears that k determines the number of leaves for each polar equation.

So, if our hypothesis is correct....then this equation should have 10 leaves.

And it DOES!!!

However, just to be on the safe side....let's graph the function when k=50.

There appears to be 100 very small leaves. Therfore, when a and b are equal, and k is an integer. the function forms a "n-leaf rose".

Now, we can investigate the polor equation without a.

Let's look at several equations where b is held contant at 1 and k is altered

Observations:

There is one leaf.

We can possibly imply that k determines the number of leaves for the function. Does it???

Apparantly, there are four leaves instead of the predicted two. Why is this so?

Let k=3:

What is going on?? Our predictions about the behavior of these polar equations are not matching up.

New hypothesis: The number of leaves is equal to 2k for even numbers and k for odd numbers. Let's try it!!

When k=6 we have an even number and 2(6)=12. This makes sense!!!
So let's set k equal to 9 and see what we derive.

Our new hypothesis seems to hold.

Let's explore the polar equation:

Here, k=1, a=0, and b=2. It seems that k may equal the number of leaves again.

WAIT....we see here that k does not equal the number of leaves.

The variable b determines the radi or length of the shape.

When k is even, the number of leaves is equal to 2k. When k is odd, the number of leaves is equal to k.

Well, as we can see.....polar equations can become artwork. If you are aware of what causes certain trends in mathematics, you can use mathematics to create artwork

So, how can we create a flower?

MATHEMATICALLY BEAUTIFUL!!

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