The Son of a Simpleton

They call him the man with many names. They call him Leonardo Pisano Bigollo, or Leonardo Pisano, or Leonardo Bonacci. They call him Leonardo Fibonacci and even, by some, “the most talented Western mathematician of the Middle Ages.” He did not however, actually discover the number sequence named after him (the Fibonacci numbers) as most would otherwise suspect. Most of Fibonacci’s fame should be attributed to his 1202 composition of “Liber Abaci” — a book that spread the Hindu-Arabic numeral system throughout Europe. 

Fibonacci’s early life appears to be a bit of a mystery. There are two separate theories which shed some light on why Fibonacci came to adopt so many names. The first theory presents Fibonacci’s father as a wealthy merchant named Guglielmo Bonacci. Some even go further claiming that Bonacci was the consul for Pisa. 

(Fibonacci was from Pisa, you know, the place with the off-kilter tower.) 

Alternatively, according to a history text by mathematician Tobias Dantzig, his father was "a lowly shipping clerk nicknamed Bonaccio, which, in the idiom of the period, meant 'simpleton’. This would explain the name given to his son... 'Fibonacci,' the 'son of a simpleton.’” Someone on Yahoo divulges that Leonardo's father, Guglielmo Bonacci, was a kind of customs officer in the North African town of Bugia now called Bougie where wax candles were exported to France. They explain that this is why candles are still called "bougies" in French. Despite the controversy concerning Bonacci’s profession, Fibonacci described his father in the introduction to “Liber Abaci” as “a public official… in the Bulgia customs house established for the Pisan merchants.” 

So why, if the sequence of numbers most commonly referred to as the “Fibonacci numbers” weren’t actually discovered by Fibonacci, what gives with the name? In “Liber Abaci,” the middle-age mathematician (that is, the mathematician from the middle ages, not to be confused with a middle-aged mathematician) posed, and solved a problem regarding the rate of reproduction in rabbits. The solution, generation by generation, turned out to be the same sequence of numbers we associate with the formula for the Golden Ratio: which occurs when a ratio is the same as the ratio of their sum to the larger of the two quantities. 

In the Fibonacci sequence of numbers, each number is the sum of the previous two numbers. Fibonacci began the sequence not with 0, 1, 1, 2, as modern mathematicians do but with 1,1, 2, etc. He carried the calculation up to the thirteenth place (fourteenth in modern counting), that is 233, though another manuscript carries it to the next place: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Fibonacci did not speak about the golden ratio as the limit of the ratio of consecutive numbers in this sequence.

Rabbits are far from the only animals affected by the Fibonacci number sequence. One of my favorite connections actually has to do with bees. The most profound example is by dividing the number of females in a colony by the number of males (females always outnumber males). The answer is typically something very close to 1.618. In addition, the family tree of honey bees also follows the familiar pattern. Males have one parent (a female), whereas females have two (a female and male). Thus, when it comes to the family tree, males have 2, 3, 5, and 8 grandparents, great-grandparents, gr-gr-grandparents, and gr-gr-gr-grandparents respectively. Following the same pattern, females have 2, 3, 5, 8, 13, and so on. Anyway, it seems as though Fibonacci has been a buzz word for a while now. 

We know Fibonacci is a man of many names, but he’s also a man with many things named after him.  For instance, there are many mathematical concepts named after Fibonacci, usually due to their connection to the Fibonacci numbers. Examples include the Brahmagupta–Fibonacci identity, the Fibonacci search technique, and the Pisano period. Beyond mathematics, namesakes of Fibonacci include the asteroid 6765 Fibonacci and the art rock band The Fibonaccis. 

It is my belief that every member of “The Fibonaccis” band would have burned at the stake if they presented this during Fibonacci’s time. 

While we’re on the topic of music, I’ll end by presenting a beautiful musical translation of the Fibonacci number sequence. Enjoy. 

The Mathematics of Composition

Moderns Film could be considered one of the most advanced forms of media we have today. When done well, films effectively combine the elements of sound, composition, story, talent and more to create one fluid cinematic piece. It takes years of hard work and sheer talent to master all of these elements. Until then, it’s a good idea to focus in on one area of film that interests you more then the others.

Chris Knight, writer of fstoppers.com, has created an ultimate guide that makes learning one such element much easier. Specifically, composition – a defining element of film, and pretty much the defining element of photography.

Wait, You mean that rule of thirds thingy?

Ah yes, but as Knight proves in his article, it goes far beyond that nine-box rule.

With pictures and examples guiding his definitions all the way though, Knight provides a clear guide to the mathematics behind beautiful composition.

After reading his article, and in an effort to beat Arturo to the punch, I’ve decided to review some of my own work to see if it stands up to the test.

Center Composition

The first and most basic type of composition Knight mentions is center composition. For this I used a picture I took while in the middle of nowhere. The road stretched on for miles and the lighting was fantastic. With flat land on both sides of the road, this was a perfect time to use symmetry to my advantage.

It's not mathematically perfect, but still an effective example.

Rule of Thirds

The classic. The 101 of photography. When unsure how to compose a photo it's best to stick with the good 'ol threes. I found this photo of a prairie dog which falls right on the sweet spot. 

Golden Triangles 

When playing with angles, golden triangles are a great way to check composition. Knight uses a picture of the eiffel tower as a great example. 

I found one picture from Santa Monica that again, is not mathematically perfect, but shows how this type of composition can have a compelling effect. 

Frame Within a Frame

Another type of composition he mentions is the frame within a frame effect. 

It's a great way to give an extra layer of depth to the photo while at the same time putting extra focus on the subject. I managed to achieve this once this summer as Erin walked through the main gate at Mount Rushmore. I was lucky enough to capture the moment by chance, but usually you need to plan out shots like this. 

In this picture her figure does not fall right on the rule of thirds line. In fact, she's quite a bit under it. But it slightly dwarfs her and makes Mount Rushmore seem even larger. The point is that in the end, art is art, and mathematics can't always determine its effectiveness. But it's a great starting point for discovering what might look cool. 

A Knight in Golden Armor

Get it? Chris KNIGHT? GOLDEN ratio? I digress.

I'm a big fan of this article, mainly because our man Chris Knight found an unpretentious, non-patronizing way to tell us not-so-savvy-with-a-camera people how to make small adjustments to better our poorly composed scenes. He understands what a film/television student really needs: big pictures with not a lot of words underneath.

This is a circle.

I've had a slew of high school teachers attempt to explain the golden ratio, and all I really got out of it was that I would never pursue a career in math and George Clooney was scientifically gorgeous (duh).

Symmetry is sexy.

But along came Mr. Knight with his diagrams and BAM! I am now a member of the club known as "Basic Understanding of Rectangles". Shot a little too crowded? Take ten paces to your left and try again. Subject look a little boring in the center of the frame? Shimmy over to your right and the interesting shot is yours.

The most adorable Rule of Thirds.

I was most affected by one of the smaller points of the article. Knight explained the basic technique of using a frame within the frame (a tunnel, a tree, a window, etc...). But, you can also use darkness as a frame. If you want to highlight a subject in the distance, keep your foreground dark. I always found lighting tricky, but this quick little tip really helped keep the basics of lighting in perspective.

Obviously these concepts are only the basics of composition, but it's a great start to the more complex aspects of setting up a shot. So put down your "Composition for Dummies" book, check out Chris Knight's "The Ultimate Guide to Composition", and explore the depths (of field... ha...) of framework. 

Composition & Bokeh

The part of these articles that I was drawn to was bokeh. I personally love the aesthetic of bokeh. Knight says “Obsession with bokeh is bad for your photography.” I somewhat agree with this statement. Too much can become repetitive and boring, however, I think if you use it in the right way and for the right photograph it can be amazing. Sometimes it’s better to see all the details and textures throughout the frame and other times those details don’t add anything. It really comes down to personal preference and judgement.

For example, in this photograph I think the bokeh works very well because it would be too busy with every detail in focus. You would get lost in all of the details and wouldn’t be drawn to the intended subject.

Chris Knight did a great job explaining some complicated concepts. Talking about composition rules such as the Baroque Diagonals, the Golden Ratio, the Rule of Thirds can become overwhelming. I like how he stated these are rules but are not rules at the same time. If you follow these rules it will not always lead to a provoking photograph. However, it is obviously important to read about and understand the rules for reference. I loved how well Knight explained what each rule was and how to apply most of them. After reading these articles I am more confident in what they are and how to integrate them. I’m looking forward to incorporating this new information into my future class, personal, and professional projects.