History of Fractals

A fractal is an infinitely extending mathematical set that perpetuates the the same image over and over.  Contrary to what many people believe, a fractal does not always have a regular pattern.  Many fractals have a very irregular or even fractured appearance to them.  Fractals are not necessarily a first degree shape either.  Circles, squares, and other 2D shapes if increased, then the area is increased by a factor of 2 because that is the size of its dimensions; however, a fractal does not expand like 2D or even 3D objects.  Fractals grow by a value that is not always an integer.

Allegedly, fractals have been known about since about the 17th century by the mathematician and philosopher Gottfried Leibniz.  It was known by the late 19th century that fractals were recursive and that they were continuous but not always differentiable discovered by Georg Cantor, Felix Klein, and Henri Poincare.  In 1904 Helge von Koch started making figures that represented the ideas of fractals (which can be viewed at http://en.wikipedia.org/wiki/File:Von_Koch_curve.gif).  Later that century, around 1918, at about the same time, two French mathematicians Pierre Fatou and Gaston Julia, came up with how fractals map the complex numbers.  By 1975, the name "fractal" was finally coined for these figures.  The name is derived from the Latin fractus meaning "broken" or "fractured."  The name fractal was coined by Benoit Mandelbrot.  Although naming these objects fractals, debate is still had over these objects.  Many people do not know how to formally define a fractal and what its name should encompass, but Mandelbrot was known for saying that fractals are "beautiful, damn hard, and increasingly useful. That's fractals!"  Now the general consensus on what a fractal is, is infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions.

Fractals have been defined as many things, not just geometric objects.  Among other things, fractals can also describe processes of time.  Fractals have been studied in images, structures and sounds, and also found in nature, technology, art, and law.  My area of interest comes from fractals in nature.  Fractals in nature show extended,  but finite, scale ranges.  These fractals are being studied heavily and are helping in many ways.  One such way is leaves, based on the fractals of leaves it can be determined how much carbon is in a given tree.  Wikipedia.org gives the following extensive list of fractals in nature...

• clouds
• river networks
• fault lines
• mountain ranges
• craters
• lightning bolts
• coastlines
• Mountain Goat Horns
• Animal coloration patterns
• Broccoli and Cauliflower
• Heart rates
• Heartbeat
• Earthquakes
• Snowflakes
• Crystals
• Blood vessels and pulmonary vessels
• Ocean waves
• DNA
• Soil pores
• Psychological subjective perception

For those that are interested in fractals and in viewing more fractals, Wikipedia.org is a good source.  Some fractals that I found pretty cool were the Julia set (http://en.wikipedia.org/wiki/File:Julia_set_%28indigo%29.png), Karperien Strange Attractor 200 (http://en.wikipedia.org/wiki/File:Karperien_Strange_Attractor_200.gif), Uniform Triangle Mass Center grade 5 fractal (http://en.wikipedia.org/wiki/File:Uniform_Triangle_Mass_Center_grade_5_fractal.gif), Lichtenberg figure (http://en.wikipedia.org/wiki/File:Lichtenberg_figure_in_block_of_Plexiglas.jpg), and the Mandelbrot set (http://en.wikipedia.org/wiki/File:Animation_of_the_growth_of_the_Mandelbrot_set_as_you_iterate_towards_infinity.gif).

My information was either previously known information or found from http://en.wikipedia.org/wiki/Fractal.

What is math?

To me, math is the foundation of many other disciplines of study.  Math is the earliest type of science.  Math is a complex system of numbers, algorithms, and proofs.  Using math, many things can be done.  Math allows for its user to be able to study circumstances that they encounter.  The reason that I call math a "science" is because without math, no scientific studies would be able to be quantified, all studies would have to be qualitative and therefore much less precise.  Math is the building block onto which all other forms of number systems and theories can be and have been built.  For instance, the theory of gravity is only a theory because it can be quantified as a velocity of 9.8 meters per second.  Before the mathematical calculations, only the thought of gravity could be used, there was no precision as to how gravity affected all the things on the Earth.

There have been many discoveries in math.  I have ranked the top 5 in the order of importance that I find them for math below.

1. Algebra
2. Geometry
3. Pythagorean Theorem
4. Calculus
5. The Fibonacci Sequence

I have ranked algebra as number one because without the discovery of algebra then no other mathematical processes would be possible to do.  Algebra allows for variables in math to exist and for us to know what those variables represent and how to use them.  I have ranked Geometry as number two for two reasons.  The first reason is that it is a very visual representation of mathematics.  Geometry allows for the user to visualize what they are doing which is very helpful in other fields that incorporate math.  The other reason I have it ranked number two is because if I remember correctly, I had heard that through the discovery of Non-Euclidean Geometry mathematics became its own subject instead of just being used in conjunction with the sciences.  The Pythagorean Theorem and Calculus come in at three and four respectively because they are pivotal to the advancement of math.  While I do not know a whole lot about the discovery of the Pythagorean Theorem, I do know the importance of this discovery.  Calculus is also a very wide subject to be put at fourth.  The discovery of Calculus helped many other fields of study because they could/can use Calculus to solve for certain variables, an example, through derivatives and integrals, if given the formula for acceleration, velocity, or position, we can find the others.  I put the Fibonacci Sequence as fifth because I know that its discovery is very important, but I do not know much at all about the Fibonacci Sequence or its discovery.