The Synergistic Dynamics of Art, Mathematics, & Science
An Interdisciplinary Project at Stowe Middle School
[©Frederick David Abraham, project contributors, and sources]

Solutions to the Difference Equations for Archimedean Spirals

Remember that in developing the idea of the difference equation, we established that for the Archimedean spiral, we figured the distance, r, from the center, by adding a differenc to the r at the previous step. To to this we had to know the distance at the previous step. For example, in exercise 12 we figured that at step 4, we added or D(r) = .27777 to the r = .833 of step 3, and got r of step 4 as .83333 + .27778 = 1.11111. Now we can see that since D(r) stays constant at each step, that if we started with r at step 0, we have added D(r) four time by step 4. Adding some constant value (number) 4 times is the same as multiplying that number by four. Therefore, if we wish to know what r is at step 4 without having to figure it for all the steps preceding it, we only have to know the starting value at step 0, and add n*D(r), that is the number of steps times the amount added each time.


rn = r0 + n*D(r)
since in our examples, we start with r0 = 0, so this equation becomes:
rn = n*D(r)

In this example, for step 4,
r4 = 4 * .27778 = 1.1112, nearly the same as 1.1111, the difference being due to rounding error.

Exercise 13. Can you figure out r9; r10; r18; r27; r36?

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, an 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, an = ?

Equations of the type rn = r0 + 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.

Other Places To Go



of Spirals

Ex. 1-12

Defs Rel
to Spirals

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