Sunday, August 08, 2010

Our Plan for the Year: A Mathematical Rabbit Trail Part II

Some gentle readers might be thinking that those number patterns from yesterday are all fun and games. What do they have to do with the real world? How could they be of any use?

Prepare to be inspired if you aren't already.

We call those numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 . . .) the Fibonacci sequence, which gets the next number of the series by adding the previous two. The number after 89 will be 55 + 89 = 144. Hidden beneath the formulas and doldrums of mathematics are people interested in discovering patterns and making sense out of numbers. Fibonacci was born in Pisa at the end of the twelfth century, three years before construction started on the Leaning Tower. After completing one story, wars prevented further efforts, so Fibonacci never saw what we think of as Pisa's greatest landmark. His father worked at the customs house in Algeria, and the young boy was exposed to Moslem and Hindu cultures.

If your eyes glaze over at the thought of a guy who lived a thousand years ago, think again! Europe was still using the Roman numeral system, and it was Fibonacci who spared students from long division problems like MCMVIII divided by XII. (And, you thought you had it bad.) He wrote a clear explanation of how to apply the Hindu-Arabic decimal system and revolutionized how Europeans made arithemetic calculations. Why has nobody ever heard of him? He wrote his works in Latin, and how many mathematicians read a dead language?

A very interesting thing happens when you divide two Fibonacci numbers that are side by side, larger divided by the smaller. Do you see what happens as you divide larger and larger Fibonacci numbers? (Click the picture for a better view.)
As Fibonacci numbers grow larger and larger, the result (or ratio) becomes constant, which we call the Golden Ratio, or 1.61803398874989 . . . The ". . ." means that the Golden Ratio goes on forever and ever without repeating a pattern. It is so much like its cousin of pi (π) that we call it phi (φ).

The Golden Ratio is more than a silly number. A few months after we switched churches last year, a guy came up to me. He said, "I hear you are a math person. Well, I'm not. I'm making a table for my daughter." Pointing to a drawing on a napkin, he pointed and continued, "I need the ratio between this piece and that piece to be 1.6. If this piece is ___ inches long, how long would that piece need to be?"

This is a true story, I kind you not. I asked him if he was trying to use the golden ratio. He had no idea what I was talking about but said that woodworking magazines always recommended this ratio for making projects more beautiful. People find the golden ratio in the composition of da Vinci's "The Last Supper" and Seurat's "Circus Sideshow" (which scoffers think it is all flim-flam). Whether the Parthenon and other ancient structures were based on the golden ratio or the whole idea is the invention of German romantics, architects today incorporate phi into their designs. I know because an architect sat in my talk on mathematics at the ChildLightUSA conference this year and confirmed my claim.

If you have never seen Donald in Mathmagicland (worth renting on Netflix in my mathematical opinion), the following will sound familiar. Artists sometimes construct "golden rectangles" to figure out proportions in the composition of their works. First, imagine you have a 1 x 1 square.



Next to that, place another 1 x 1 square.


Below the two 1 x 1 squares, place a 2 x 2 square. We are going to form a series of rectangles that have ratios that get closer and closer to the golden ratio.


To the left of that, place a 3 x 3 square. The pattern is to continue adding squares of Fibonacci numbers going counterclockwise.


And so on.


And so on.


By now, you should get the idea.
Here comes the neat part. Using a compass (or free hand for the brave), you can draw arcs along the diagonal of the squares, going counterclockwise. You will end up with a beautiful spiral!

Martin Gardner, a proponent of mathematical recreations, said, "All mathematicians share . . .a sense of amazement over the infinite depth and the mysterious beauty and usefulness of mathematics."

3 comments:

Kathleen said...

Wow - that's really cool =) Thanks for sharing!

Stephanie said...

Are Fibonacci spirals found in nature? Say in a snail's shell?

The Glasers said...

Yes, some people do see his spirals in the nautilus shells, but the skeptics, and dare I say Fibonacci number atheists, scoff!