An Interdisciplinary Project at Stowe Middle School

[©Frederick David Abraham, project contributors, and sources]

So:

r_{n} = r_{0} + n*D(r)

since in our examples, we start with r_{0} = 0, so this equation becomes:

r_{n} = n*D(r)

In this example, for step 4,

r_{4} = 4 * .27778 = 1.1112, nearly the same as 1.1111, the difference being due to rounding error.

Exercise 13. Can you figure out r_{9}; r_{10}; r_{18}; r_{27}; r_{36}?

Exercise 14. Where can you check your answers? That is, the answer to a previous question contains the answers. What question was that?

The following exercise might be a little harder:

Exercise 15. Can you figure the the angles, a_{n} the same way, using multiplication instead of addition by going through every step, just knowing the step number for which you want to compute the angle, a, and the length of the radius, r. Hint, remember that the D(a) = 10°. Compute the angles for each of the distances you computed in Exercise 13. Confirm your answers as in Exercise 14.

Exercise 16. Can you give an equation for the angle at step n, similar to that above for the distance at step n? That is, a_{n} = ?

Equations of the type r_{n} = r_{0} + n*D(r)

are called solutions to the difference equations. Difference equations depend on the difference between one step and the next, while the solutions are automatic ways of adding all the differences up to a particular step from some intitial step to the present step in one operation, instead of several successive operations.

We might mention, that these hopefully very easy relations between geometry and algebra are introducing you to some very complex subjects that are taught much later in your study of mathematics and science. That subject is called calculus. We won't get very fancy about these things, but we show you how easy it is by the methods shown here. We also hope that we show how important learning math, art, and science are and how much fun they can be. Many people think math is hard and are scared to learn, but in fact, it is very easy, and makes learning many other things much easier. They also are the key to understaning science, and demonstrate a lot about how art behaves like science and nature and why we enjoy them all.

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Created: 1/6/97 Updated: 1/7/97