What Makes the Golden Ratio so Golden?

Now that I’ve concluded my Golden Ratio-themed comic serial, I wanted to get into a little what the Golden Ratio actually is. Just saying that it’s some number that’s approximately 1.618, doesn’t quite do it justice. What’s so special about that number? And what makes that a ratio?

Second point, first. A ratio is a comparison of two numbers, so it can cause confusion if only a single number is written, even if “to 1” is implied.

What makes it so special is what is being compared. Look at these two images, for example. They are basically the same thing, only the notation for the lengths are different.

Suppose we wanted to find a ratio of the length of the square to the length of the rectangle, and we wanted that ratio to equal the ratio of the length of the smaller rectangle to that of the bigger one, what value of x would accomplish that?

It terms out that we could set up the problem in either of these two ways (and many more, besides!). On the left, the square has a length of 1, the smaller rectangle has a width of x, so the bigger rectangle has a total length of x + 1. One the right, the square still has a length of 1, but the bigger rectangle has a total length of x, so the smaller rectangle has a length of x – 1.

Setting up the proportions and cross-multiplying we find:

In either case, we get the same quadratic equation: x² – x – 1 = 0.
That’s not something easily factorable, so we have to use Ye Olde Quadratic Formula, after which we discover that the roots are

If we take the positive value, we get x = 1.618033988749894848204586... But what about the negative value? If we subtract root 5 from 1 and divide by 2, we get x = -0.618033988749894848204586...

Look at the decimal portion. They are the same! Keep in mind, that we’re dealing with irrational numbers here, which aren’t supposed to conform to patterns, but this one is just brilliant. And, yes, there is a reason for it.

Take a look at that second proportion, above, on the right, with the removable 1 in the denominator.

This is saying that one less than the number is the reciprocal of that number! If you divide 1 by 1.618033988749894848204586..., you will get 0.618033988749894848204586..., the same as if you just subtracted one.

This works for the negative value as well: If you divide 1 by -0.618033988749894848204586..., you will get -1.618033988749894848204586..., the same as if you just subtracted one.

One last point, and a hat tip to William Ricker for mentioning it, how else can we write the reciprocal of x, 1/x? What exponent gives us the reciprocal of x? An exponent of -1. That means that this equation, this proportion, can be written as:

Absolutely brilliant. And quite Golden, if I say so myself.

A Little Bit of Fibonacci ...

Today's column isn't going to happen the way I wanted it to, so I'll try it for tomorrow. In the meantime, I'll leave you with this:

I just finished posted a series of comics with crazy numbers in the titles after the word "part", instead of the normal "Part 1/10, Part 2/10, ..." etc.

This is because those numbers were intended to be ratios, and not just any ratios, but Fibonacci number ratios. As you might have guessed from the first episode in that sequence, I would be working with Fibonacci numbers. You might also have guessed that I would have been done by the time I got to 21 -- That's okay! I was expecting to be done a lot sooner, too!

As I hope to get into a little more depth tomorrow, when you divide two consecutive Fibonacci numbers, you get a ratio that is close to the Golden Ratio, which is also known as "phi", and is approximately 1.6180339887. The larger the terms, the smaller the difference between the two ratios.

Here is a table showing you the first 15 terms:

But that's not the only thing interesting about the Golden Ratio. (Well, of course not! Books could -- and have -- been written!) There should be more about it in tomorrow's entry.

Golden

(Click on the cartoon to see the full image.)

(C)Copyright 2011, C. Burke. All rights reserved.


I wanted use 'Go Into' for 'Do Unto' but that gave the reciprocal value.

$1.98 Mathematics, Part 2

A couple years ago, I was walking about a 99-cent store and found a quad-ruled composition notebook and a box of colored pencils. Total: $1.98. I played around with them for a while, and then they were put in a draw and forgotten about. Until recently when I found the notebook. And then last week, I posted a sketch for those pages.

Here is another one:

This was actually the first sketch from the book, but is wasn't as colorful as the other one. And it seemed to be more boring. But is it really?

The sketch (and you can click on the image for a larger version) shows a Golden Spiral created by connecting the diagonals of adjoining squares. The length of the sides of each square are determined by using the next number in the Fibonacci sequence. Obviously, the squares increased in size so quickly that I couldn't finish the 34 x 34 square.

But there was something else I noticed. I had added extra diagonals to some of the rectangles that were created in addition to the squares. I highlighted one of them in red (on the scan -- it's in pencil on the original sketch). The red line appears to be the diagonal for many of the rectangles. Four of them, in fact.

How could that possibly be the case?
(If any of my students are reading this, STOP here and look at it. Investigate. See if you can figure it out. Come back when you have it or you've had enough. I'll wait.)

The four rectangles have the following sizes: 2 x 1, 5 x 3, 13 x 8 and 34 x 21. Zooming in shows that the red line really isn't a diagonal of the smallest rectangle, so let's discard that one for a moment. The others are close enough to be errors in sketching. Since the slope of a straight line has to be constant, if we calculate the slope at any two points, we should get the same number.

Slope can be calculated as rise over run or change in vertical over change in horizontal. (You remember that "delta y / delta x" thing I keep mentioning in class? Yeah, that.)

So we have slopes of 3/5, 8/13, and 21/34, which are definitely not equivalent fractions. (How do we know that?)

If we convert those fractions to decimals, look what we get:3/5 = 0.6
8/13 = 0.615384...
21/34 = 0.617647...
and if the paper had been bigger, we might have seen
55/89 = 0.6179775...
144/233 = 0.61802565...

So the slopes are nearly identical meaning that the diagonal of the big rectangle isn't really the diagonal of the others, but it's really, really close.

Extra points if anyone keeps going, or if they can tell me the significance of a particular number that starts 0.61803...