# Maddenation

## The Golden Ratio

I’m reading the new novel by Dan Brown, *The Da Vinci Code*, which is an adventure mystery about secret codes and secret societies and the powerful people who guard them and seek them. I won’t give away much of the plot, but at the beginning, the author talks about the Fibonacci series and *PHI*, the “golden ratio,” the “divine proportion.” Math aficionados will know that the Fibonacci series begins with the number 1 (purists would say 0 and 1) and continues by adding the previous two numbers in the series. Thus it starts: 1,1,2,3,5,8,13,21,…

One of the many interesting facts about his series is that the ratio of the last two numbers approaches the number PHI = 1.618… which is one of the most important numbers in the natural world. I first heard about the golden ratio in high school geometry class, where it was introduced as the “perfect” shape for a rectangle, the “golden rectangle,” in which the ratio of the short side to the long side was equal to the ratio of the long side to the sum of the two sides. In other words, x/y = y/(x+y). If you let x = 1, the equation becomes: y^{2} = 1 + y.

This solves to y = (1 + 5^{.5})/2 = 1.618+.

This ratio and the Fibonacci numbers appear widely in nature, and their properties have spawned the Fibonacci Society and the Fibonacci Quarterly. In Brown’s book, he points out a few of these “natural” facts, such as PHI is the ratio of female to male bees in a hive, or the ratio of successive chambers in a nautilus. He also talks about the relationship of PHI to the human body. If you divide your height by the height of your belly button (you more sophisticated folks might like to refer to it as the “navel”) you get PHI, or close to it. I tried it. My height is 75 inches and my belly button is 45.5 inches off the floor. The ratio, 1.65, is a little high, indicating that my legs are too short for me to be considered a true Adonis. What about you?

In fact, what about David’s science class? They could tabulate the appropriate measurements and plot height vs. navel height and see what the slope is. Does it really approximate *PHI*? What about successive lengths of finger joints? There’s a whole world out there.

## Comments

David • 09/28/03 • 10:14 PM:Dad,

Super coolness. I’ve been meaning to respond to this post for a while. I think I will work it into class somehow. Just two weeks ago as part of a graphing/measuring bellringer I had the kids find 5 circles and plot the circumference vs. diameter.

Also, tonight I’m going to start reading “Demons and Angels” also by Dan Brown. It’s the book before The Da Vinci Code. I was told to read it first. I’ll let you know how it goes!

David • 09/29/03 • 12:11 PM:I just roughly took my measurements (metric style of course) i’m ~ 189 cm/115 cm = 1.64 not too bad.

Dan • 10/25/03 • 11:38 PM:I’m 71.75 inches high. My belly button is 31.75 inches high. That makes my ratio 1.64 also.

Dan • 10/26/03 • 7:52 PM:And by 31.75 of course I really mean 43.75.

Patrick • 04/13/04 • 7:22 PM:I’m too tired to reconcile the apparent difference in math, but this article on the Golden Rectangle seems akin to the Golden Ratio, no? Certainly it fits with some stuff I’ve seen about Fibonacci numbers. Anyway, enjoy.

Dad • 04/14/04 • 11:40 PM:There is no difference in math, the sides of the golden rectangle are in exact proportion to the golden ratio (or Divine proportion if you like). The 3:2 ratio of 35mm film is significantly different from the golden ratio, although it’s close enough that it’s probably as pleasing to the eye. It also has the advantage of being simple whole numbers rather than an irrational number.

By the way, do you think it’s a stretch to say the Parthenon is based on the golden ratio when you have to include the peak of the roof in the in the height of the rectangle?

AJ • 04/15/04 • 9:12 PM:Architects are always stretching to say something meaningful about their building. So, yes it is a stretch, unless you’re an architect. Then it’s the greatest thing ever.

David • 04/19/04 • 7:36 PM:Check this site out. It’s kinda cool. And while you’re at it, just go to the regular home site www.johnkyrk.com - great flash graphics of neato stuff like DNA translation and meiosis. I use it in class all the time.

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