Fibonacci SeriesFrom: Jeff Jones Date: Mon, 7 Nov 94 14:08:53 EST Subject: Fibonacci series Hi, I was helping an Algebra student with a "bonus" problem recently. It asked something about drawing a spiral using the Fibonacci series. What is this series? Does it draw a spiral? Any help would be appreciated. Thanks, Jeff Jones Date: Mon, 7 Nov 1994 14:33:31 -0500 X-Sender: vmotto1@cc.swarthmore.edu Thanks for writing! Jeff, Looking back at an old calculus book of my own, I can give you the definition of the Fibonacci sequence, which is constructed using a function that is defined as follows: For each positive integer n, then / 1 , if n = 1 or n = 2 f(n) = / \ the sum of the two preceding values of f , if n > 2 \ For example, f(1) = 1 f(2) = 1 f(3) = f(1) + f(2) = 2 f(4) = f(2) + f(3) = 3 f(5) = f(3) + f(4) = 5 and so on. I am not that familiar with Fibonacci numbers myself, but if one of the other math doctors knows some more, they will write you soon. In the meantime, I hope this will give you an idea of what is going on with the sequence and how it formed. -Vanessa, M.D. From: Dr. Ethan Date: Mon, 7 Nov 1994 14:41:01 -0500 (EST) I am so excited--this is so great--I love the Fibonacci sequence. I am actually so pumped to be able to answer this question. It sure does draw a spiral and it is a really neat one as well. I am just freaking out in the joy I am receiving from answering this question. Unfortunately, describing how to get it to draw such a spiral might be a little tricky via email but we can give it a try. First think superimposed rectangles. Okay, let's start with 1,1. So draw a square of side length 1. Then choose one side, extend its length to 2 units, and draw the larger rectangle so it looks like a rectangle of side lengths 1 and 2 that is divided in half. Then we start to get a pattern. Expand along the direction of the shortest side to the length of the next Fibonacci number. This creates a cycle of ever expanding rectangles that produce a beautiful sprial. That I believe is the same spiral that conch shells make as they spiral out. Hope you can figure this out. If not, write again and I will try later. -Ethan, Dr. on call. From: Dr. Ken Date: Mon, 7 Nov 1994 16:50:10 -0500 (EST) Jeff! In looking at our responses to your question, I think the most helpful thing we could do is direct you to another source, one that I think gives a very interesting overview of Fibonacci Sequences and things that they are related to (as well as lots and lots of other interesting and beautiful mathematics): The book is called the Joy of Mathematics, and it's written by Theoni Pappas. I'm not sure who publishes it, but I know that its sequel (More Joy of Mathematics) is published by Wide World Publishing/Tetra, so that's a good place to check first. I'm almost positive that it's the same publisher. You won't be disappointed if you investigate the Fibonacci numbers further. There's a wealth of fascinating, accessible, and beautiful mathematics going on with them. Good luck! -Ken "Dr." Math From: Jeff Jones Date: Mon, 7 Nov 94 15:14:04 EST Hi Ken, Glad I could bring you such a pleasure today. The Algebra book from which I took the problem said to draw right triangles stacked up. For instance, draw a 45 degree right triangle with the hypotenuse sloping up and to the right, and the right angle on the bottom-right. Imagine that the 45 degree vertex at the left will be the origin of the spiral. Now, draw a line beginning at the far end of the hypotenuse, at a right angle to the hypotenuse, for a distance equal to the next Fibonacci number. It should look like below. Continue using the hypotenuse of one triangle as the base for the next triangle. I think you're right, the book said something about conch shells. Thanks for your help, Jeff 3 . . . . . . . . . . . . . . . . r . r. . . . . . . . . . 2 5 . . . . . . . . . . . . . . . r. . . . | . . . | . . . | . . . | 1 . . . | . . . | O____________r| 1 (the "r" is the right angle) From: Dr. Ken Date: Tue, 8 Nov 1994 14:18:17 -0500 (EST) Hello there, Jeff! I think that if you take a closer look, you'll find that the lengths of the sides are not 1,1,2,3,5,8,13,21,... (the Fibonacci numbers), but the square roots of those numbers. I mean, you may have already seen this, but I thought I'd point it out anyway. Enjoy! -Ken "Dr." Math |
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