Donnie Darko (Tears for fears - Head over heels)

This is one of my favorite scenes ever and it starts with one of the most famous long takes in modern film. This scene from Donnie Darko is choreographed beautifully to another great song from Tears for Fears, Head Over Heels. I think it really captures what middle school in suburban America was like for me.

0, 1, 1, 2, 3, 5, 8, 13, 21... Fibonacci Life

Numbers are concrete, factual and logical, right? One thing I always loved about math class was how clearly defined truth is. It would always frustrate me when I'd see my grading on an essay in English class and whether I received an A or F, seemed almost arbitrary. There was no "right answer." But that doesn't apply to numbers and you can't argue with an equation. The ultimate truth is in numbers but some would argue that they lack in qualities of life. In life there isn't necessarily a right or wrong answer, and many things are not quantifiable. What number is equal to love? Further, the world we live in has imperfections or at least strays from what our idealistic mathematical models might predict. I think that numbers also have a magical quality and in many cases life and numbers share common qualities. It seems to me that numerical models represent the ideal, what reality and life tends towards. Pythagoras once said that numbers were the "cause of gods and demons," but I would say that numbers are the language of gods. Math is how we define the divine.


You have probably heard of the Fibonacci series, but if not it is a simple concept to learn. If we start with 0 and 1 then add the previous two numbers in the series to get the next, so 0 + 1 = 1 then 1 + 1 = 2 and 1 + 2 = 3 , this process leads to the Fibonacci series seen in the title. Its a simple idea, just add up the previous two numbers to get the next, but it appears everywhere in the natural world.

The series is sometimes referred to as the "bee ancestry code" because it represents the ideal genealogical history of a male bee. It is derived from three simple facts about bees:

1. An unfertilized egg (laid by a female) always hatches male.
2. A fertilized (by a male) egg always hatches female.
3. A male bee must have one parent (a female) while a female must have two parents.
   

"If one traces the ancestry of this male bee (1 bee), he has 1 female parent (1 bee). This female had 2 parents, a male and a female (2 bees). The female had two parents, a male and a female, and the male had one female (3 bees). Those two females each had two parents, and the male had one (5 bees). If one continues this sequence, it gives a perfectly accurate depiction of the Fibonacci sequence.

However, this statement is mostly theoretical. In reality, some ancestors of a particular bee will always be sisters or brothers, thus breaking the lineage of distinct parents." From here.

Another way Fibonacci shows up in life revolves around natural geometry. If you take the ratio of any two adjacent numbers in the series, they approach a common value as the numbers get larger in the series. For example, 3/2 = 1.5, 5/3 = 1.67, 8/5 = 1.6 and 13/8 = 1.63 after rounding. The value that this ratio tends to can be defined precisely, but lets just round to the nearest 1000th to get 1.618. Interestingly, if you take the ratio's the other way, so 2/3 = .67, 3/5 = .60, 5/8 = .625 and so on, you get the same number minus 1 or .618. This property makes this irrational number uniquely special and it is therefore called the golden ratio.

The golden ratio shows up in all sorts of places that you would never expect. Leonardo da Vinci discovered that beauty in art is often associated with the golden ratio and he strived to incorporate it into his works. He found that many of the proportions in the human body are close to this ratio and he depicted this in his Vertruvian Man and the Mona Lisa seen above and below. In these images the ratio of the length to the width of any rectangle and also the division of any line occurs at this ratio.

Its not just in art though, the golden ratio also leads to a spiral if you connect the corners of any two adjacent squares in the map above to form what is called the golden spiral. This is the spiral that the some sea shells form but its noteworthy that the nautilus does not. It is the spiral that you see in the seeds of a sunflower and other blooms. Hurricanes have this form and it is even found in the shape of our home, the Milky Way, and other galaxies.

Its even in the stock market. Many technical traders (those who use stock charts to determine whether to buy or sell) swear by the Fibonacci ratio and use it in their everyday strategies. The ratio often appears in charts on all time scales, 15 min, 1 week, 3 year, you name it. There are various levels of sophistication that are employed in using the golden ratio to trade, some of which are here. A simple approach is to connect the highest price and the lowest price in a pattern to find prices where the equity may pivot or change direction. I just drew up the Fibonacci ratios for the most recent pattern in the dow jones industrials weekly chart. Notice how the price tended to pivot at the 32.8% and 61.2% levels of the total (100%) move that recently completed.


The Fibonacci ratio appears in the lives of insects and the flowers they pollinate, the shells of the oceans and the hurricanes they feed, and the charts of stocks and the bodies of the Milky Way humans that trade them. This magical sequence of numbers goes beyond number theory and steps into life. Or maybe its life that is striving to become something divine. Any way you slice it the Fibonacci series is something substantially more than just a sequence of numbers.

note: Title image taken from: http://www.flickr.com/photos/myreflex/180524429/

Everybody Wants to Rule the World

My friend Indigo posted a great 80's music video on his blog this weekend which got me searching for my favorite song from the era of great movies and music. I grew up in the 80's and I'm thankful for it, pop music these days has become so formulaic and all about profit. This Tears for Fears music video is a little strange but it sure beats the filth you find on mtv today.

You can find pictures of some of my recent adventures here. I'll be back to my usual topic of posts soon with something on number theory.

Visual Multiplication

"The Thumbprint of God"

Arther C. Clarke explores the Mandelbrot set in this fantastic hour long documentary. Click on the link above for full screen or watch the video as sized below.



There are many free programs on the web for exploring the Mandelbrot set, here is one.


For the details of the calculation, this site has a nice simple explanation. That link is also where I got these beautiful images that seem... well, like they could be the fingerprint of a god.

I'll be on vacation for the next few weeks, so expect something inspired when I get back!

-pythagoruz

Simplicity in Chaos: The Logistic Map

When most hear the word chaos it conjures up thoughts of disarray, confusion and complexity. Webster defines it as "a state of things in which chance is supreme, especially the confused unorganized state of primordial mater before the creation of distinct forms and the inherent unpredictability in the behavior of a complex natural system (as the atmosphere, boiling water, or the beating heart." The beating heart!? But its also one of those everyday words that has been taken by science to name a theory in a more visual or mnemonic way, like string theory and for many years it was the leading edge of math and physics. Chaos Theory was the buzzword of the 1990's, but it has since been replaced by nanotech or solar energy. You must remember in Jurassic Park (also a film) how the lawyers had to call in the genius chaos theorist to give the dinosaur resurrection the go ahead. Well they didn't listen the chaos guy and well, things went to... chaos.

What is amazing to me, and a foundation of chaos theory, is that simple (albeit nonlinear) systems with simple laws can lead to extremely complex and "chaotic" results. In many cases these simple systems only lead to complex solutions under certain conditions, under other conditions they might yield a single simple answer. It seems to me that popularized fractals like the Mandelbrot Set(seen at right), the Koch Snowflake and the Sierpinski Triangle are mappings of the region between order and chaos. Not only are these maps extremely interesting from an academic standpoint, they have an inherent beauty that stretches to infinity. You can literally zoom in to these maps infinitely and find at least as much depth as the macroscopic view and in many cases much more. There is nothing else like them in the universe. Lets say you were looking at the galaxy from afar, you could zoom in further and further, but eventually you would hit a brick wall called the atom (or would you...). Evey thing we know has a limit, even physicists agree the universe is finite in size, although Einstein called it "infinity with bounds." Einstein was referring to the idea of the universe being finite but not having an "edge," a different idea, but the phrase still seems appropriate for the Logistic Map.

The logistic equation has been around since 1845 when it was developed as a method for modeling population growth, in fact you probably learned about it at some point if you know what a differential equation is. It basically assumes that the rate of population growth is proportional to the population but growth slows and stops as the population reaches the carrying capacity (maximum population). Maybe the equation itself will job your memory, from mathworld (also the top image):
While solving this equation might require some practice, it is fundamentally quite simple and based on some simple ideas. Actually, the logistic equation does not produce chaotic behavior. That is until the mid 1970's when Mitchell Feigenbaum (an interesting guy by the way) discovered that logistic equation could be modified to demonstrate strange behavior in some physical systems he was studying.

If you take the right side of the logistic equation and for a given value of r (rate constant) between 0 and 4, plug in an x value to give the next value of x. In other words, iterate the equation repeatedly using its previous value to give the next value. The initial value of x is another parameter that can be played with, but .4 works well. You just iterate this equation:
Yes, thats it.

The behavior is entirely dependent on the rate constant, defined for each iteration. For some values of r, like r is less than 1, the equation tends towards zero. For other values, for example r above 1 but less than 3, the value of x bifurcates and tends to two stable values, oscillating between them after a large number of iterations. As the rate constant increases further the system (x value ~ population) splits into 4, then 8 stable values of x. At some point the stability breaks down and they system no longer tends towards any given value, it has become chaotic. Plot all the values that the system yields after 50 or so iterations and you get the logistic map: The process gives orderly results for some ranges of r, chaotic results for others and in some cases their is a semi-stable result. If you zoom in to this picture you can see all the detail in the logistic map. In the midst of chaos there are regions of brief order where the map goes back to converging to a finite number of values. This is an perfect example of self-similarity in the sense that these smaller structures of order mirror the larger scale structure, looking almost identical in some cases. Just zoom in and the small scale order will look very familiar (at any zoom with a high res img).

What I find so amazing about this system is how simple the rules are. Take this function to modify a value of x and plug the result back into the equation. Now its not your normal approach to dealing with an equation, but it is simple. Some savvy electrical engineers have come up with an equivalent circuit to the logistic iteration, the output is shown below. I'm not sure how profound this really is, but it certainly suggests that some systems may be orderly under the right circumstances while the same system will be chaotic and completely unpredictable under other conditions... Hmm, that sounds alot like the stock market.
If you are interested in exploring the map check out the program at the bottom of this page, be sure to F1 for the list of commands once you get it running. Here is a program that should work on macs. Comments are appreciated.

I Heart Interactive Brokers

Yesterday I received this letter via Fed Ex with a check for the full 10th place prize. Lets just say I am very happy with IB right now and I will be buying at least 100 shares of IBKR for a long term investment with the money. I'd like to get my original ipo bid price if possible, which was 25$.