Site Visits

Wednesday, June 18, 2014

Mona Phi-sa

By now you know that this number 'phi', which emerges as you look further down the list of Fibonacci numbers, explains some really unexpected relationships, from geometry to architecture, to the animal kingdom (Fibonacci Zoo, notwithstanding) and in the human body. One particular human, and technically, we aren't actually talking about a human, but a painting of a human, stands out as perhaps the most perfect of them all - Mona Lisa.

The Mona Lisa

This painting is arguably the most famous one ever made, certainly the most famous one Leonardo da Vinci (not OUR Leonardo!) painted. We aren't talking scupltures, so David is not in the conversation, by the way. The Mona Lisa hangs in the world-famous Louvre museum, and has captivated art buffs and casual fans for centuries. You can learn more about the painting at this Khan academy site.

So what makes this painting so famous? I've not had the chance to see it, but I understand it is actually quite small, and viewing it is a little stressful, with crowds gathering around it, cameras snapping right and left, and I am left to wonder, what's the big deal?

Well, I'm not going to make any claims about my art history credentials (which are none). Instead, I want to look at her face. Lisa, it is believed, was the wife of a man who asked da Vinci to paint her. That's it. But what makes her so interesting from a mathematical standpoint is that the dimensions of her face are nearly perfect. I guess maybe it's from one Italian Leonardo (Pisano) to another (da Vinci), but a close look at Mona Lisa reveals that her facial dimensions very closely match the Fibonacci spiral we saw in a previous post.
The Fibonacci Spiral in the painting of Mona Lisa

It turns out that some faces just look nicer than others, much like those rectangles we saw whose 'perfection' was hard to describe, but we knew it when we saw it. Facial recognition experts have long found the value of phi in faces that are widely accepted as being beautiful. Sample faces include Angelina Jolie, Jennifer Aniston, Beyonce, Johnny Depp, and Jessica Simpson, to name a few. At the Math of Beauty web site, you can see a little more into this phenomenon. And wouldn't you know it? Even Oprah got into the act on an episode that dealt with something called "The Beauty Equation.

So is it real? Do men and women with 'perfect' facial dimensions really look more attractive? I'll leave that to you to decide for yourself. As for me, my wife thinks so, and that's all that matters. Try searching for 'golden ratio' and 'beauty' and see what you find.

Monday, June 9, 2014

The Golden Rectangle

We're fast approaching some really fascinating applications of this number 'phi' and the Fibonacci sequence. But before we get to looking at those applications, famous faces, buildings, etc., I'd like to introduce you to something new. We've talked briefly about the Golden Ratio, or the Golden Mean (the number 'phi', which is approximately 1.618...). But there is a very simple geometric shape that defines this ratio and which serves as a starting point for just about all of the geometric instances of Fibonacci.

Photo credit:
The shape is a rectangle and this very special rectangle is one of those shown to the left. Can you figure out which one it is? Which rectangle looks 'just right'?

You see, some rectangles are long and skinny, while others are so even in their dimensions they almost look like squared (the green rectangle toward the bottom middle, for example).

But there are certain rectangles that just seem to look 'right'. I don't know what it is, but they are pleasant to look at and just catch your eye. One that immediately jumps out at me is near the top middle of this collection. It's orange, and it lies just below a long skinny orange rectangle.

Now why does this one rectangle stand out to me? What is it about its dimensions that make it seem so 'perfect'? Ah, this is where Fibonacci comes in.

Remember the Golden Ratio, the ratio of consecutive numbers in the sequence? For example, we looked at the ratios 8/5, 13/8, 21/13, 34/21, etc. Those values are not the same, and yet as we work our way through the sequence, the ratios all approach this special value we call 'phi', which is approximately 1.618. It's known as the Golden Ratio (among other names).

Here is a rectangle whose dimensions reflect the Golden Ratio, along with several that do not. That one in the middle is pretty sexy, isn't it?
Photo Credit:

Let's take a look quickly at how this 'golden' rectangle works. If you were to take a Golden Rectangle and cut from it a square whose length is the shorter side of the rectangle, it would look like this:
Photo Credit: Wikipedia

We now have a square, which measures b by b, and a rectangle. The thing is, we could then cut the new rectangle (A) into a square, again with side length equal to the shorter side of the rectangle, and that would leave another rectangle.

What's so interesting is that this pattern can continue forever. Each time you cut off a square, you end up with a new (smaller) rectangle. And that rectangle always has a length and width whose respective length ratio is phi.
Photo Credit:

Now this is where it gets really cool, and we'll probably stop here for today. You can start way down inside the repeating pattern of rectangles and squares and think of that innermost rectangle(the bright pink one) as TWO squares that each measure 1x1. Next, the brown square right below it is then 2x2, and the blue square that follows is 3x3. See a pattern here?

The green square measures 5x5, the red one is 8x8 and now it shouldn't surprise you to learn that the orange square is 13x13, followed by the yellow square which of course now measures 21x21. In fact, you could continue this pattern forever, and eventually, the series of squares will all form into a spiral...a Fibonacci Spiral. More on that in my next post. For now, I will leave you with a drawing of the numbered squares that form the Fibonacci Spiral.
Photo Credit: