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.

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: jwilson.coe.uga.edu

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: http://www.cet.ac.il/

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: flickr.com

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: wakeupworld.com

A Fibonacci Finger?

This kind of surprised me. If you look around online, you will find a number of claims that say that the ratio of the length of your forearm (elbow to wrist) to the length of your hand (same wrist to the tip of your middle finger), is about phi, or 1.618. I tried it and didn't come close. Maybe I have long fingers, I don't know. My ratio was 9:8, or about 1.125.

Photo credit: http://www.handanalysis.co.uk

But then I started measuring other parts of my hand and found this:

Looking at just my forefinger, I see that it is broken (no pun intended) into three segments. The tip, the middle section, and the longer main section, back to where my large knuckle is. From there, my hand continues down to the base of my thumb (not my wrist, but where my thumb kind of melds into my hand).

I brought out a ruler and here is what I found. Note that I am using rough estimates, but something easily enough seen on a standard ruler.

My fingertip (the last segment of my forefinger) is one (1) inch long. The second segment is also one (1) inch long. The third (base) segment of my forefinger (again, measured on top of my hand, back to the knuckle, is two (2) inches long. Finally, that last segment (not really my finger, but the rest of my hand) is three (3) inches long.

1, 1, 2, 3....Weird, huh? Of course, if you REALLY wanted to get into it, we have 2 hands, each with 5 digits, and technically just 8 fingers (2 thumbs). All Fibonacci numbers.

While you ponder this remarkable set of findings, watch these kittens meet a guinea pig, thanks to Animal Planet. After, I will give you some links to sites that explain some of the simple human anatomy relationships involved with Fibonacci.

Video courtesy of Animal Planet

Here are some links you might enjoy:

Human Hand & Foot

Golden Section

Cool video about the Golden Mean / Golden Ratio / Phi

Just a quick post here. This video is about seven minutes long and it does a really nice job of connecting the numbers in Fibonacci to the Golden Ratio, which we will soon explore in great detail. Thanks to Angie Zigkiri for posting it to YouTube. I think you'll like it!

Intrigued? It's pretty freaky, isn't it? Stick around for more! Next up will be the Golden Ratio and the human body.

Golden Ratios in Fibonacci

The magical and mysterious number: Phi
Photo credit: education-portal.com

By now we've seen some interesting and fairly unexpected places where values in the Fibonacci sequence can be found. I confess that I could spend hours dissecting even more intricate applications and examples of these numbers (don't worry, I probably still will). But for now, let's get back to the sequence itself. Here are the first 12 values in the sequence.

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144
Eventually, even the most ardent of mathematicians get a little bored looking at number sequences and they start playing around with patterns. Because the sequence always grows (after the first two numbers), we could try and find the ratio of any value to the one just before it. 

You may be familiar with the idea of a GEOMETRIC SEQUENCE, or one in which the ratio of consecutive elements is always the same. We call it a 'common ratio'. Below is an example of a geometric sequence in which the common ratio is 2:

1, 2, 4, 8, 16, 32, 64, 128, ...

So what happens when we look at ratios in Fibonacci? Well, at first, nothing really exciting. Here are the first few:

1/1 = 1

2/1 = 2

3/2 = 1.5

5/3 = 1.67

See? Nothing really exciting ...yet. But let's look at little more closely at these values. The first ratio (1) is fairly small but the second one gets larger (2). Then the third (1.5) is smaller, followed by the fourth one (1.67) which is larger. It's as if the pattern of ratios is bouncing around, up, then down, then up a little less, then down a little less....

Let's see what happens when we keep going from the previous ratio, which was 1.67

8/5 = 1.6 (smaller)

13/8 = 1.625 (larger)

(and here I'm going to start using approximations because the ratios will not come out to be nice, clean integers, fractions, or decimals).

21/13 = 1.615 (smaller)

34/21 = 1.619 (larger)

See? The pattern continues - up, then down, then up, then down..but something else is happening at the same time. Can you see it? Not only are the values bouncing up and down, but they seem to be getting closer together. That is, each time they jump up, they don't jump up as much as they did the previous time. And each time they jump down, that jump isn't as big as the previous jump down. Here are a few more:

55/34 = 1.6176 (smaller)

89/55 = 1.61818 (larger)

144/89 = 1.617977 (smaller)

The pattern of ups and down continues, but by now, we are really closing in on what seems to be a magic number. But what is that number?

Ratios of consecutive Fibonacci numbers
Photo credit: knoji.com

The ancient Greeks figured out what it was. It's 'phi'! If you keep finding ratios of consecutive Fibonacci numbers (and by all means, feel free to do so...), you will find that the ratios continue both bouncing around (slightly above and slightly below) and getting closer to this number, phi.

The value of the number is approximately 1.618. I say approximately, because phi is a number like pi, in that neither decimal ever ends or repeats. One interesting difference between phi and pi (besides the fact that they are different letters in the Greek alphabet) is that pi (approximately, 3.14159) is TRANSCENDENTAL. That means it is not the solution to any polynomial equation you could ever come up with. But strangely, phi is such a solution. So phi is not considered transcendental. That one might win you Final Jeopardy - might want to tuck it away for future reference. You're welcome.

What does one do with phi? Oh, we're just getting started answering that question. Stay tuned. For now, just know that it's out there, waiting for you. You can't ever actually find it, but you can get ever so close to it - kind of like true love, right?

Pine cones at a Wedding?

Photo Credit: Sand Hills Wedding Expo

You might wonder what a wedding has to do with the Fibonacci Sequence. Well, nothing really. Wait - that's not totally true. A part of me wonders if I should have waited two days for my own wedding so that it would have been on 8/13, but no, my wedding was wonderful (right honey?) and the date was the perfect one for us. So what's with the wedding photo?

Let me take you back. Last December, in the depths of what turned out to be a record-breaking Midwest winter, one which saw record amounts of snow and ice, saw Lake Michigan (Lake Michigan!!!) freeze over, I flew out to Illinois to take part in my niece's wedding. It was cold - really cold. In fact, in the photo below, taken in March, 2014, Jimmy Fallon, new host of The Tonight Show, emerges from Lake Michigan after completing the Polar Plunge. Look closely in the background and you can see the walls of ice that a BULLDOZER had to break up so there was a small channel for all of the people to run into the lake. By the way, who is that RIPPED guy on the left side of the picture? Yikes!

Jimmy Fallon emerges from Lake Michigan, fully clothed in a
suit, after completing the Polar Plunge to raise money for
Special Olympics (and prove to Mayor Rahm Emanuel that
he really is a tough guy).
Photo credit: redalertpolitics.com

Anyway, I was at the wedding, MC-ing the reception somewhat unexpectedly (thanks to my brother for bailing), and to my surprise, there were pine cones on the tables. Pine cones...at a wedding...in Illinois (are there even pine trees in Illinois?)...in December. In fact, here is a picture (Photo #1) of the exact pine cone that caught my eye. I brought it home to Washington state with me.

Photo #1
The actual pine cone from the wedding reception. Sent from
Oregon, found in Illinois, and brought home to
Washington for safekeeping.
Photo credit: Tom Robinson 

Now, there is a story, and eventually I will even connect this back to Fibonacci. But please indulge me. My niece, Anna, met her husband, Jeremy, while both were visiting my brother (her uncle) in Cannon Beach, Oregon, many years ago when they were just kids, barely teenagers. Their story is complicated and hard to follow, but suffice to say that the pine cones were a gift from the Oregon part of the family as wedding decorations to commemorate their first meeting many years ago. Ok, great, so where does Fibonacci come in?

As I ate my dinner and prepared myself to run the show (reception - again, last minute fill in for my brother), I looked closely at the pine cone and started counting. Actually, I first noticed that like the pineapple, this pine cone had spirals - that its pieces (anyone know what they are called? the little things that stick out of a pine cone?) were not arranged in a straight line, but in a spiral pattern. In fact, there were multiple spirals on it, just like on a pineapple.

And so I counted. First I started from the bottom (see Photo #2) and counted counter-clockwise spirals. There were....13.

Then I counted clockwise spirals. Yep - 8. Weird, isn't it? I mean...why? Why does it always seem to be THOSE numbers? I don't know. We may never know. 

Photo #2
I know it's a little hard to count the spirals, but feel free to give it a shot.
13 rotating counter-clockwise, 8 rotating clockwise.
Photo credit: Tom Robinson

Pineapples!

You don't have to travel to Hawaii to enjoy the sweet, decadent taste of a fresh, well.... ripe, pineapple. And what's even better is that you can experience a little bit of Fibonacci, while you enjoy a 'taste of the islands.'

Count the spirals coming out of the base of this pineapple (kind of tricky). There are exactly eight!
Photo credit: Tom Robinson

While I am not overly enamored with my picture taking skills, I did want to show you what I'm talking about. And these patterns you see here are found on just about any pineapple, whether you cut it off the stalk in Oahu or pick it up at the local grocery store. Look at the very bottom of the pineapple. This was where it was connected to the plant and was removed when harvested. In the image above, you can just make out the spiraling scales, drawn in black marker, but frustratingly hard to make out. If you count your way around the base of the pineapple, you will count exactly eight spirals. Eight is a Fibonacci number!

Truth be told, some pineapples' scales are harder to count than others, and they don't always make a nice, clean pattern. But check it out the next time you are at the store (or in Hawaii!). I bet you find the same number of spirals that I did.

Notice how these highlighted (in black marker) scales make a spiral path up the fruit in the opposite direction from the last group? Counting each of the scales from bottom to top, I found.... yep - 13 scales.
Photo credit: Tom Robinson

Now, back up from the pineapple a little bit and look at the scales spiraling the OTHER direction. You can see them in the image above circled in black marker. Counting from the bottom to the top, curving around the fruit in a spiral, I counted exactly thirteen scales - another Fibonacci number! You have to do this carefully. It turns out that not all pineapples have scales that grow in exactly a clean spiral. But keep looking - I am certain you will find what you are looking for.

I have included some better drawings and images below so you can see these patterns a little more clearly. Mahalo!

Notice the 'phyllotaxy' in this image. Yum! Actually,
you can see some of the
spirals a little more clearly here.
Photo credit: geochembio.com

Next time you find yourself with a pineapple at hand, use this handy guide to start counting the scales in the spirals. There are at least two distinct patterns indicated here.
Photo credit: thinkquest.org