Chapter 2: Fractals! And Geometry in Nature

              Geometry. We all took it at the beginning of high school, and most of us have just passed it off and had little thoughts about it since. But besides basic algebra, it is the math most relevant and useful to everyday life. Whether you are measuring the dimensions for a new workbench in the garage, or fencing in the back yard, we all use a little bit of geometry every day. Although the biggest and mostly unknown use of geometry is in the world of nature around us through the exquisite design of fractals.

                So what exactly are fractals? Well the very mathematical definition is that Fractals are infinitely complex and repeating patterns that are self-similar across different scales and are no-where mathematically differentiable.^(1.) More simply, fractals are repeating patterns found that expand out in the same pattern that can be described by mathematical formulas. They are found everywhere in nature and biology; from trees, mountains, and rivers, to chemicals and basic atomic structure, and even in our very own DNA is a fractal pattern. ^(2.) So you actually have trillions of fractal patterns inside of you right now, and all around yourself.

 

 Two Fractal patterns found in nature

          So why does any of this matter? And why should you care? Well besides the fact that they are beautiful and immensely interesting, it is integral to understanding them to be able to study anything about our natural world. The leaves of any tree grow in a fractalated pattern. The way electricity flows through an object or substance, is in a fractalated pattern. The way a mountain is formed when two tectonic plates collide, is in a fractalated pattern. Are you beginning to see the big picture? Mathematics and geometry didn’t come out of thin air, and are integral to the world around us from the most micro parts of our world, to the absolute largest.

The mandelbrot equation and its resulting fractal graph

              We also use them in our non-natural world as well. Fractal art is a booming industry as many people deeply appreciate the beauty found in the patterns. Fractal patterns are also used extensively in the animation industry, as they are used to render characters in scenery for movies and video games. Since our body is composed of so many naturally occurring fractals, researchers at Harvard and Johns Hopkins have started using them in the medical field to help diagnose diseases such as cancer and emphysema. ^(3.)  Beyond that, they are used in fields such as electronics, geology, and astrology. So as you can see, math is all around us, you just have to know where to look.

 How fractal examination is used to detect lung cancer

 Fractal Art

 Animation rendering using fractals

Sources:

1.) "Fractal." Wikipedia. Wikimedia Foundation, 06 Nov. 2014. Web. 11 June 2014. <http://en.wikipedia.org/wiki/Fractal>.

2.) "Patterns of Visual Math - Fractals in Nature." Patterns of Visual Math - Fractals in Nature. N.p., n.d. Web. 11 June 2014. <http://www.miqel.com/fractals_math_patterns/visual-math-natural-fractals.html>.

3.) Haggit, Craig. "How Fractals Work." HowStuffWorks. HowStuffWorks.com, 26 Apr. 2011. Web. 11 June 2014. <http://science.howstuffworks.com/math-concepts/fractals4.htm>.

If you would like to learn more, please check out the great site Fractal Foundation:

http://fractalfoundation.org/

              As a child sitting in math class I always wondered how any of the stuff we learned would actually be used. Like why on earth do we need to study the quadratic formula? Well if you are like the vast majority of people, and work at the local hardware store or work a regular desk job, you never will. But I assure you, the quadratic formula and math in general are very important to how the world works. They are just used in ways that are hidden to most of us, so we rarely see it. This series of blogs will be focused on showing the ways math is used in the modern world, most of it in ways none of us ever new about. Despite being a math blog, I will try and keep all of the math basic and as simple as possible, so we can focus on the real content of how it is used.

Chapter 1: Taylor Series and Modern Computing

              If you have ever taken Calculus II you will remember learning about Taylor Series. If you haven’t don’t worry, they are rather simple and I will explain them in a second. They are one of the things Calculus students generally hate learning because they are hard to fully understand and most teachers don’t explain how they are applied in real life. As such many students simply learn what they need to in order to pass the test, and forgot all about them the second they know they have passed. Although Taylor series are very important in the modern world, and we use them every day, most of us just don’t ever realize it.

            For those of you who never learned about them (or forgot once you saw that B), Taylor Series are a representation of a function as an infinite of terms that are calculated from the values of the function's derivatives at a single point. More simply, Taylor series are a way of approximating a complex function as a simpler one. It isn’t a 100% accurate, but even a simple Taylor Series can approximate some very complex rational function to an astounding accuracy. To do this one simply repeated follows the Taylor formula and follows derivative laws until they have achieved the level of accuracy that is acceptable to them.

The Taylor Series formula:

Example of Taylor Approximation for sin(x):

            This approximation for sin(x) only goes to the fourth order and is very simple, but even so is very accurate. For example, when calculating an x value less than 1 and greater than -1, the error in approximation is less than 0.000003, which is very small and accurate enough for most applications. But I am not here to lecture you on calculus and this brings me to my real point. While sin(x) is a simple function, these simple Taylor Series are much easier to calculate once you start getting into complex functions. Especially for computers with limited CPU and RAM capabilities, they can use these Taylor Series to calculate values of complex functions within an acceptable accuracy level all while using much less memory and don’t need to store giant banks of trig function values. All most all computer calculator programs do this, from your basic four button calculator all the way up to some super computers. Because of the way computers “think” it is much easier for them to calculate in this way (especially for non-mathematical programming languages such as C+). For example, the function arctan(x) can be very complicated to calculate when dealing with higher numbers, so a computer would simply solve the Taylor Series instead, and they can solve it with only basic mathematical operators.

            So when you are working on homework and enter a function into your IPhone, it is actually solving something much different than what you entered and technically giving you a value that isn’t truly accurate. Now is this something that you will use every day? No. In fact, unless you’re a computer science major you may never use it since this is built into the basics of any mathematical programing. But that isn’t the point of learning, and now you know something that 99% of people don’t. Plus you now know how computers and the world work a little better; you can now finally say that calculus II wasn’t a total waste.

If you are interested in learning more about Taylor Series, Paul’s Online math does a great job going over them:

http://tutorial.math.lamar.edu/Classes/CalcII/TaylorSeries.aspx

More info on how they are used in the computer science field:

http://www.osix.net/modules/article/?id=796

Sources:

"Taylor Series." Wikipedia. Wikimedia Foundation, 06 June 2014. Web. 06 June 2014.   <http://en.wikipedia.org/wiki/Taylor_series>.

"Mathematical Foundations of Computer Science 2011." Google Books. N.p., n.d. Web. 06 June 2014. <http://books.google.com/books?id=C1K-Z6BIDZsC&pg=PA176&lpg=PA176&dq=taylor+series+computer+science&source=bl&ots=PnwdEU8FIi&sig=IbZsRoj4eNekcXDXHnVhbxmnZfg&hl=en&sa=X&ei=42ORU8L7K8vTsATO4YCADw&ved=0CH8Q6AEwCDgK#v=onepage&q=taylor%20series%20computer%20science&f=false>.