Fibonacci Number Series

     After reading the post about Fibonacci's Number Series I was quite interested in the ramifications of his discovery and the possible applications of it. It is hard for me to comprehend how mathematical precision, certainty, and prediction could be found in something as wild and chaotic as the stock market or nature. I know it isn't necessarily Math History but I was interested in looking into all the different application of the Fibonacci Number Series for my independent study this week.
    Upon looking at the graph that was in your link in the Fibonacci lecture, I was amazed at the tendency of the stock to retrace itself with uncanny precision between those percentages which correlated to the percentages between some of the first few Fibonacci Numbers. Although the retracement wasn't picture perfect, it did seem to offer an educated guess at where the stock might go after reaching a certain percentage marker.
    After viewing the graph I decided to investigate other possibilities for the Fibonacci Number Series to be used. I was surprised at the application of the number series in nature. From the formation of galaxies to the formation of a Nautilus shell, which are both actually quite similar in formation and shape, just not size. As the Nautilus shell's spiral begins to fan out, its expansion, as the spiral begins to grow larger, can be measured and expressed in boxes of increasing size as seen below:
    
     http://ualr.edu/lasmoller/fibonacci.html?utm_source=http://ualr.edu/~lasmoller/fibonacci.html&utm_medium=700pxcustomerror404&utm_content=click&utm_campaign=custom404

    Another interesting application of the Number Series in nature is in flowers and the amount of petals that they contain. It was only after learning about the Fibonacci Series and its relevance in flowering plants that I realized that I haven't really seen alot of 4 petaled flowers, maybe a 4 leaf clover, but is that even considered a flower? Much of the Lilly family has 3 petals, Buttercups have 5 petals, some orchids don't exactly have distinguishable pedals, but mostly around 3 or so pedals on average. Even flowers with huge amounts of pedals such as daisies and other flowering plants from the Asteracae family have either 21, 55, or 89 petals, all Fibonacci numbers.
     Even in music, Fibonacci's Number Series seems to have relevance.  Many mathematicians agree that Mozart's Music shows strong indications of Fibonacci numbers even though there is no evidence to suggest that Mozart knew of Fibonacci or deliberately based music off of the series. In Mozart's Sonata No. 1 in C-major there are some striking Fibonacci numbers that stick out when counting "measures". The first section has 32 measures, and the last section has 68 measures, both Fibonacci numbers(http://www.lhup.edu/~dsimanek/pseudo/fibonacc.htm). This theme seems to be prevalent in modern day music and could just be due to the way the human ear recognizes and enjoys music. Even instruments bare a resemblance to the Fibonacci numbers, pianos especially. The scale consists of 13 keys, 8 of which are white and 5 of which are black, and split into groups of 3 and 2. It doesn't get more Fibonacci related than that.
    From spirals that make up our universe and Nautilus shells to stock reports and engineering marvels like the Parthenon. From flower's pedals to bunny rabbits subsequent generations, the Fibonacci Number Series is an interesting and somewhat mysterious mathematical tool that can make sense and order out of a seemingly chaotic world.

Greek Number System

Since Greek mathematicians, Philosophers, and thinkers seem to be one of the most prolific influences on western culture, and specifically the United States, I thought it would be fitting to begin by examining their number system and its roots.

      The Greek number system is largely similar to the Roman and Egyptian number systems. Identical to the Roman Number system because of the inheritance of many Greek ways by the Romans, such as religion, math and science. Similar to the Egyptian number system because of the large amount of trade done in that region during that time. In fact, although many believe that the Greek number system was developed by the Greeks between 475BC and 325BC, there has been some suggestion that in fact the System was adopted and amended by the Greeks after learning it in trade from the Egyptians (http://www.mlahanas.de/Greeks/Counting.htm).

      The Greek number system is a base 10 system, with actually two separate systems for naming and denoting of numbers. The first, using similar names as the second, but instead labeling numbers without symbols, but with the first letter of the name for that number. For example, the number 1 is called "Iota" and denoted by the letter "I". In the second system symbols are designated to represent numbers. For example, in the second system, the number 1 is called "Alpha" and is represented by an "A" or the symbol alpha seen in the first box on the left below.

     The latter system, called the "Ionian System", was used more often and was actually quite advanced and capable of some relatively complex calculations (at least for that time). Larger number were read by adding the value of the hundreds place with the value of the tens place and then with the value of the ones place, just as we do now days. For example, the number 523 is read as (500+20+3). Similarily, Φ+К+γ= ΦКγ. The first symbol in the previous additive equation is "Phi" and represents 5 in the hundreds spot. The second symbol in the sum is "Kappa" which represents 2 in the tens spot. Finally, the last symbol in the sum is "Gamma" which represents 3 in the ones spot. So, all together they represent "PhiKappaGamma" or 523.

     This Greek number system could also express fractions as well. Depending on the context, a sort of apostrophe type symbol following the number expressed a fraction. The apostrophe symbol signaled a 1 over whichever number or numbers was followed by the symbol. The "Ionian System" also allowed for more complex fractions by stating a number with a line over the top, this represented the numerator followed by the denominator which was again represented by an apostrophe symbol that followed the number

(http://www.math.tamu.edu/~dallen/history/gr_count/gr_count.html).